Content-Type: multipart/mixed; boundary="-------------0409221015955" This is a multi-part message in MIME format. ---------------0409221015955 Content-Type: text/plain; name="04-299.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-299.keywords" asymptotic expansion, Cauchy problem, formal power series, Gevrey estimates, Borel summability, linear partial differential equation ---------------0409221015955 Content-Type: application/postscript; name="revbalser4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="revbalser4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: revbalser4.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMSY10 CMR12 CMBX10 CMR10 CMMI10 CMBX12 CMMI8 %%+ CMR6 CMR8 MSBM10 CMSY8 CMEX10 CMSY6 CMR9 CMTI10 CMMI6 MSBM7 LASY10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o revbalser4.ps 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All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 73 /contintegraldisplay put dup 88 /summationdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 1477F5496111E055C7491002AFF272E46ECC46422F0380D093284870022523FB DA1716CC4F2E2CCAD5F173FCBE6EDDB874AD255CD5E5C0F86214393FCB5F5C20 9C3C2BB5886E36FC3CCC21483C3AC193485A46E9D22BD7201894E4D45ADD9BF1 CC5CF6A5010B5654AC0BE0DA903DB563B13840BA3015F72E51E3BC80156388BA F83C7D393392BCBC227771CDCB976E93302532471AAA096BDB7D3D00544FCEAE 86A3196E1683DB53C308F865EE7F6C22EC40D307AE114E328CC80446A3F982AF 9F6D58B72F017CA3A5E4241120081E58FA971E59192E4671C7792406813C6BB6 A9D5CF1CDD3F0B68897404FD1C8EBFE91EF253083495F0D92A7AAA065EDA79F8 7C9F31807E23979F8D77DA7F04B7995FCDF479274EA20690085B37441CBE529D FBC2DEFA2D2EEFEA08B828523F05516FF44856947BB35A3023E8D44E8C7843D0 DEBD78FEA96E4EBCCBE6CF0518DEC7AFF49CA4FEC889658CDDA8D815D37B985E 246422C972319DEAB98699ADF21AF8ABA7D738EBCE86D057D036DFFB2C173E0C BC673338506C9C9A74B4E62D0EDB093CF30BD650ED709367FEF5FB2B19DC9B35 AED8D861FA2E158432858FE2118E26D846DE0AD5CABC3836CEA858E2399C476F 7856E55CEEECDBEAE91490F7FF4F476763E3BFFFCFAD7142D0B5C9236CB6458C AA493DAED25DCD860649DA960E09AE71C2555E3912B5F172D197690E3F01F233 20F4F2D33028E2ACCB6A0C59DCE9CC50804B48F56C3F0F2E33115249EB9CA4C8 1A62D2F8CE2DEA453E3AB3E6EBFAB319EDF9987A5705A961E20BEE73E6FEC444 C4852729A8B16A8BB7FAF1AE130EB5127ED7BDCC6793A0A9A7884E4BC55D6897 6C658F4B179BA8EBDAA092590A92389280E95650577E7D3B61E8B305972F1960 35BDC106693FBD8768D2252F13A91303E5FFA223F02837E5C1680A261DF57931 425DFC60DAD14EDED35624461A6189D53F39114597AD475DA91FABA655BBBCC3 95B72C58931ADDB35C4CB7D40D8DB86BA0B51F1D1501D76A0DF93F6D845E43A3 16F209B1635563EFE9EC589656EDF513310426C0B419E26C8BD6A386A98EAC2F 4C7A3B01BCC7EC9D66827FB824CD52ABFA5533BFCE3349D17B6B7E0FB6D55EEF C02C1AE2BD439C03EDA5DD0C61AB99FC9BE9BF51326F6D272ECF595A5C6DB318 3EFB58A848C934BF553ACFAB4D358029EECD6E1720CDCD28BACFCB698BEFBDCB A692FE68CA153471E3B81432E43B9CE33514927CB9BC3FDE8AA8FBA4510B8B9D 32E9A96494FFC0801C016C19A5DCF4CE32343B83697028B030692EABFC9BE9BA 44369F890303BA92F1C99826DA55FE36B514689A34E3674C8571A0B3802DA65F 8CA0523DE8CFD942B57E0F9415BA4F9FCE75E07048156256745DD69B14167B0C 9D998EB938839C6A9144E6D3C4C01686E834C43497FA1A24FCA6A5FFAC749528 B4F7A3C5B6CE304667581B2CED32B2F8E4206E4FD33CEB36BF7FB5320D9AC731 37F506A336210D72F5873FC6156486569BE90B025DD93085D296BC86957784C7 8388D8F75A22860D5C97D75545C2A24803731EC4106A55B69839477E12980CCF 07FE04D7957D46448D501B55F2D7267DDFD43A0A1829C952821B16472BEBA6D6 DBC9F70331CC0B17EE00E53226A0CA1652DFCECF3056A62219DEF09F6CB38058 2E9A6101A9D9B61DAFDFB21B4DB052A1E9E399E27D337BE9E02B75470770B6F6 BDC265566365A3F0670C4F59C9C91B239C7B7E8017430151E60DF68B5F9DDDB3 06E224ED0AB7357ED7F7526D7CD7A6792130966A19CEA0748596B36C8E51B91C 6C258F503A44409E9AC7ECFAB6095E785D7D9F5B091E3B007A801435C455B5D0 21149CB7BB9DABAF51B302C92571CA9634F372C51A642E0186A4E8CBD238F1AE 1C57415857F43C6D7BEAA36F7A049B7DF630F182EA15D51C609CF41B42043E00 D30C990234CD8DE47906E5A95310F893DBFE5652CF678CAB1F8DCAE015CCB46E 60B03A80D8C496745F3512B20D0852F6491930A1D4F440048155088A1395290B CBB1C10427E70BD1571D57029A6F66A70C2B0B29B529B385949688F373A6184A 38B81441052137C0EAF5FCAA0616B2582F8EB07E599246D7E5F60D10ED152654 281A4DC6C31B259FEC9161E34CDE741F5DD32EDEA62CE8100216EF2F6428FA2C EFDF5322F5EEE0281B438A6A36620F40FC6B3AB299E1C97CF69039EAC5D212BC 2315876785CA00E0B06A485F1C4A7D1CF7688F3D339B73386F155040BAFA081B 54621BAB5C17C2C59EFBA1AF 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fl(2)p Fn(;\032)2043 2207 y Fd(and)34 b Ff(j)p Fo(t)p Ff(j)25 b Fo(<)g(R)q Fp(\()p Fo(\032)p Fp(\))p Fd(,)216 2374 y Fe(ii\))35 b Fd(Ther)-5 b(e)37 b(exists)f Fo(\016)e(>)c Fp(0)36 b Fd(such)g(that)h(for)f(al)5 b(l)36 b Fp(\()p Fo(z)1880 2388 y Fl(1)1920 2374 y Fo(;)15 b(z)2002 2388 y Fl(2)2042 2374 y Fp(\))31 b Ff(2)g(D)2270 2388 y Fl(1)p Fn(;\032)2387 2374 y Ff(\002)22 b(D)2550 2388 y Fl(2)p Fn(;\032)2646 2374 y Fd(,)36 b(the)g(series)g Fo(W)13 b Fp(\()p Fo(t;)i(z)3373 2388 y Fl(1)3412 2374 y Fo(;)g(z)3494 2388 y Fl(2)3534 2374 y Fp(\))36 b Fd(c)-5 b(an)37 b(b)-5 b(e)75 2486 y(analytic)g(al)5 b(ly)37 b(c)-5 b(ontinue)g(d)35 b(with)h(r)-5 b(esp)g(e)g(ct)36 b(to)f Fo(t)f Fd(in)g(a)h(se)-5 b(ctor)35 b Fo(S)2159 2501 y Fn(d;\016)2281 2486 y Fp(=)29 b Ff(f)p Fo(t)f Ff(2)h Fk(C)f Fp(:)g Ff(j)p Fo(d)23 b Ff(\000)e Fp(arg)q(\()p Fo(t)p Fp(\))p Ff(j)30 b Fo(<)e(\016)s Ff(g)p Fd(.)48 b(Mor)-5 b(e)g(over,)75 2599 y(ther)g(e)34 b(exist)e Fo(\016)553 2613 y Fl(1)618 2599 y Fo(<)25 b(\016)s Fd(,)34 b Fo(C)d(>)25 b Fp(0)p Fd(,)33 b(and)h Fo(K)e(>)25 b Fp(0)33 b Fd(such)f(that)1437 2816 y Fp(sup)1210 2904 y Fl(\()p Fn(z)1270 2913 y Fm(1)1305 2904 y Fn(;z)1358 2913 y Fm(2)1392 2904 y Fl(\))p Fj(2)1483 2887 y Fl(\026)1466 2904 y Fj(D)1521 2913 y Fm(1)p Fc(;\032)1606 2904 y Fj(\002)1678 2887 y Fl(\026)1661 2904 y Fj(D)1716 2913 y Fm(2)p Fc(;\032)1816 2816 y Ff(j)p Fo(W)13 b Fp(\()p Fo(t;)i(z)2090 2830 y Fl(1)2130 2816 y Fo(;)g(z)2212 2830 y Fl(2)2253 2816 y Fp(\))p Ff(j)25 b(\024)g Fo(C)7 b(e)2548 2778 y Fn(K)e Fj(j)p Fn(t)p Fj(j)2678 2755 y Fc(k)75 3098 y Fd(for)33 b(al)5 b(l)33 b Fo(t)25 b Ff(2)g Fo(S)550 3113 y Fn(d;\016)637 3122 y Fm(1)676 3098 y Fd(.)75 3317 y Fp(If)30 b(this)g(is)h(so,)f(the) h(Laplace)h(transform)d(of)i(order)f Fo(k)j Fp(of)e Fo(w)r Fp(\()p Fo(t;)15 b(z)2223 3331 y Fl(1)2264 3317 y Fo(;)g(z)2346 3331 y Fl(2)2386 3317 y Fp(\),)75 3590 y(\()p Fo(X)7 b Fp(\))792 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1251 3604 y Fl(1)1292 3590 y Fo(;)g(z)1374 3604 y Fl(2)1414 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4395 y Fo(X)40 b Fd(is)33 b Fo(k)s Ff(\000)p Fd(summable)g(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)e Fo(t)f Fd(in)h(dir)-5 b(e)g(ction)34 b Fo(d)p Fd(.)75 4561 y Fe(ii\))e Fd(F)-7 b(or)34 b(al)5 b(l)33 b Fo(n)25 b Ff(2)g Fk(N)738 4528 y Fn(r)775 4561 y Fd(,)33 b(the)g(series)1351 4769 y Fo(W)1437 4783 y Fn(n)1484 4769 y Fp(\()p Fo(t;)15 b(z)1634 4783 y Fl(2)1674 4769 y Fp(\))26 b(=)f Fo(@)1884 4732 y Fn(n)1879 4792 y(z)1912 4801 y Fm(1)1951 4769 y Fo(W)13 b Fp(\()p Fo(t;)i(z)2200 4783 y Fl(1)2240 4769 y Fo(;)g(z)2322 4783 y Fl(2)2362 4769 y Fp(\))2397 4788 y Fj(j)p Fn(z)2450 4797 y Fm(1)2484 4788 y Fl(=0)75 4977 y Fd(al)5 b(l)25 b(c)-5 b(onver)g(ge)25 b(for)g Ff(j)p Fo(t)p Ff(j)g(\024)g Fo(\034)10 b Fd(,)26 b Fo(z)1044 4991 y Fl(2)1109 4977 y Ff(2)f(D)1265 4991 y Fl(2)p Fn(;\032)1360 4977 y Fd(,)h(for)f(some)g Fo(\034)35 b(>)25 b Fp(0)g Fd(that)h(is)e(indep)-5 b(endent)26 b(of)e Fo(n)p Fd(.)39 b(Mor)-5 b(e)g(over,)27 b(ther)-5 b(e)25 b(exists)75 5090 y Fo(\016)115 5104 y Fl(1)191 5090 y Fo(>)37 b Fp(0)i Fd(so)g(that)i(for)e(al)5 b(l)39 b Fo(z)1027 5104 y Fl(2)1104 5090 y Ff(2)d(D)1271 5104 y Fl(2)p Fn(;\032)1366 5090 y Fd(,)k(al)5 b(l)40 b(the)f(functions)g Fo(W)2215 5104 y Fn(n)2262 5090 y Fp(\()p Fo(t;)15 b(z)2412 5104 y Fl(2)2452 5090 y Fp(\))40 b Fd(c)-5 b(an)39 b(b)-5 b(e)39 b(holomorphic)-5 b(al)5 b(ly)43 b(c)-5 b(ontinue)g(d)75 5203 y(with)38 b(r)-5 b(esp)g(e)g(ct)39 b(to)f Fo(t)f Fd(into)h(a)g(se)-5 b(ctor)38 b Fo(S)1363 5218 y Fn(d;\016)1450 5227 y Fm(1)1523 5203 y Fp(=)33 b Ff(f)p Fo(t)h Ff(2)g Fk(C)f Fp(:)h Ff(j)p Fo(d)25 b Ff(\000)e Fp(arg)r(\()p Fo(t)p Fp(\))p Ff(j)34 b Fo(<)g(\016)2618 5217 y Fl(1)2658 5203 y Ff(g)p Fd(.)56 b(Final)5 b(ly,)39 b(for)f(every)g Fo(\016)3569 5217 y Fl(2)3642 5203 y Fo(<)c(\016)3787 5217 y Fl(1)3827 5203 y Fd(,)75 5316 y(ther)-5 b(e)34 b(exist)e(c)-5 b(onstants)35 b Fo(C)q(;)15 b(K)33 b(>)25 b Fp(0)p Fd(,)32 b(indep)-5 b(endent)35 b(of)d Fo(n)p Fd(,)g(so)i(that)1462 5532 y Ff(j)p Fo(W)1573 5546 y Fn(n)1620 5532 y Fp(\()p Fo(t;)15 b(z)1770 5546 y Fl(2)1810 5532 y Fp(\))p Ff(j)26 b(\024)f Fo(C)2064 5495 y Fj(j)p Fn(n)p Fj(j)2149 5532 y Fo(n)p Fp(!)p Fo(e)2271 5495 y Fn(K)5 b Fj(j)p Fn(t)p Fj(j)2401 5471 y Fc(k)2443 5532 y Fo(;)75 5740 y Fd(for)33 b(al)5 b(l)33 b Fo(t)25 b Ff(2)g Fo(S)550 5755 y Fn(d;\016)637 5764 y Fm(3)676 5740 y Fd(,)32 b Fo(z)778 5754 y Fl(2)843 5740 y Ff(2)25 b(D)999 5754 y Fl(2)p Fn(;\032)1127 5740 y Fd(and)33 b Fo(n)25 b Ff(2)g Fk(N)1535 5707 y Fn(r)1573 5740 y Fd(.)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 75 100 a Fp(4)75 399 y Fe(iii\))32 b Fd(F)-7 b(or)34 b(al)5 b(l)33 b Fo(n)25 b Ff(2)g Fk(N)767 366 y Fn(r)804 399 y Fd(,)33 b(the)g(formal)h(series)1391 554 y Fp(^)1364 577 y Fo(X)1439 591 y Fn(n)1487 577 y Fp(\()p Fo(t;)15 b(z)1637 591 y Fl(2)1677 577 y Fp(\))26 b(=)f Fo(@)1887 540 y Fn(n)1882 600 y(z)1915 609 y Fm(1)1980 554 y Fp(^)1953 577 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)2186 591 y Fl(1)2226 577 y Fo(;)g(z)2308 591 y Fl(2)2348 577 y Fp(\))2383 596 y Fj(j)p Fn(z)2436 605 y Fm(1)2471 596 y Fl(=0)75 756 y Fd(ar)-5 b(e)38 b Fo(k)s Fd(-summable)g(in)g(the)f (dir)-5 b(e)g(ction)39 b Fo(d)p Fd(.)57 b(Mor)-5 b(e)g(over,)39 b(the)f(sums)g Fo(X)2425 770 y Fn(n)2472 756 y Fp(\()p Fo(t;)15 b(z)2622 770 y Fl(2)2663 756 y Fp(\))37 b Fd(ar)-5 b(e)39 b(holomorphic)h(in)e(a)f(r)-5 b(e)g(gion)75 869 y Fo(G)23 b Ff(\002)g(D)333 883 y Fl(2)p Fn(;\032)429 869 y Fd(,)37 b(indep)-5 b(endent)38 b(of)f Fo(n)p Fd(,)f(wher)-5 b(e)38 b Fo(G)f Fd(has)g(bisse)-5 b(cting)37 b(dir)-5 b(e)g(ction)38 b Fo(d)f Fd(and)g(op)-5 b(ening)37 b(gr)-5 b(e)g(ater)38 b(than)g Fo(\031)s(=k)s Fd(,)75 982 y(and)g(for)f(every)g (close)-5 b(d)38 b(subse)-5 b(ctor)1331 959 y Fp(\026)1315 982 y Fo(S)38 b Ff(\032)33 b Fo(G)p Fd(,)38 b(ther)-5 b(e)38 b(exist)f(c)-5 b(onstants)39 b Fo(C)q(;)15 b(K)41 b(>)33 b Fp(0)p Fd(,)38 b(indep)-5 b(endent)39 b(of)e Fo(n)p Fd(,)g(such)75 1095 y(that)1159 1208 y Fp(sup)1031 1295 y Fn(t)p Fj(2)1115 1279 y Fl(\026)1103 1295 y Fn(S)t(;z)1203 1304 y Fm(2)1237 1295 y Fj(2D)1339 1304 y Fm(2)p Fc(;\032)1439 1208 y Ff(j)p Fo(@)1517 1170 y Fn(l)1512 1230 y(t)1544 1208 y Fo(X)1619 1222 y Fn(n)1666 1208 y Fp(\()p Fo(t;)15 b(z)1816 1222 y Fl(2)1856 1208 y Fp(\))p Ff(j)26 b(\024)f Fo(C)7 b(K)2194 1170 y Fj(j)p Fn(n)p Fj(j)p Fl(+)p Fn(l)2356 1208 y Fo(n)p Fp(!)p Fo(l)r Fp(!\000\(1)21 b(+)f Fo(l)r(=k)s Fp(\))75 1436 y Fd(for)33 b(al)5 b(l)33 b Fo(n)25 b Ff(2)g Fk(N)582 1403 y Fn(r)652 1436 y Fd(and)34 b Fo(l)27 b Ff(2)e Fk(N)p Fd(.)75 1665 y Fe(Pro)s(of)31 b Fp(First,)f(w)m(e)g(sho)m (w)g(that)g Fe(i)p Fp(\))g(and)f Fe(ii)p Fp(\))h(are)g(equiv)-5 b(alen)m(t.)42 b(The)29 b(fact)i(that)f Fe(i)p Fp(\))g(implies)g Fe(ii)p Fp(\))g(follo)m(ws)h(directly)75 1778 y(from)d(the)i(standard)e (m)m(ulti-dimensional)i(Cauc)m(h)m(y)f(form)m(ula.)40 b(Indeed,)29 b(for)g(all)g Fo(n)c Ff(2)g Fk(N)3067 1745 y Fn(r)3105 1778 y Fp(,)k(the)g(series)g Fo(W)3642 1792 y Fn(n)3718 1778 y Fp(can)75 1890 y(b)s(e)h(written)g(in)g(term)h(of)f Fo(W)43 b Fp(as)31 b(follo)m(ws,)1004 2112 y Fo(W)1090 2126 y Fn(n)1137 2112 y Fp(\()p Fo(t;)15 b(z)1287 2126 y Fl(2)1327 2112 y Fp(\))26 b(=)1574 2051 y Fo(n)p Fp(!)p 1494 2091 241 4 v 1494 2174 a(\(2)p Fo(\031)s(i)p Fp(\))1695 2148 y Fn(r)1759 1988 y Fi(I)1810 2194 y Fn(C)1963 2051 y Fo(W)13 b Fp(\()p Fo(t;)i(z)2212 2065 y Fl(1)2252 2051 y Fo(;)g(z)2334 2065 y Fl(2)2374 2051 y Fp(\))p 1894 2091 584 4 v 1894 2185 a Fo(x)1946 2145 y Fn(n)1989 2154 y Fm(1)2023 2145 y Fl(+1)1946 2211 y(1)2133 2185 y Ff(\001)g(\001)g (\001)h Fo(x)2306 2146 y Fn(n)2349 2154 y Fc(r)2384 2146 y Fl(+1)2306 2197 y Fn(r)2488 2112 y Fo(dx)2587 2126 y Fl(1)2642 2112 y Ff(\001)f(\001)g(\001)h Fo(dx)2862 2126 y Fn(r)2900 2112 y Fo(;)75 2354 y Fp(with)35 b Fo(z)329 2368 y Fl(1)401 2354 y Fp(=)e(\()p Fo(x)592 2368 y Fl(1)632 2354 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)886 2368 y Fn(n)933 2354 y Fp(\),)37 b(where)d Fo(C)40 b Fp(=)32 b Ff(\002)1576 2321 y Fn(r)1576 2379 y(i)p Fl(=1)1694 2354 y Fo(C)1759 2368 y Fn(i)1788 2354 y Fp(,)k(and)e Fo(C)2095 2368 y Fn(i)2158 2354 y Fp(are)i(circles)g(of)f(radius)f(0)g Fo(<)f(\034)43 b(<)32 b Fp(min)3500 2368 y Fn(i)3543 2354 y Fo(\032)3590 2368 y Fn(i)3618 2354 y Fp(.)55 b(The)75 2467 y(fact)29 b(that)g Fe(ii)p Fp(\))g(implies)g Fe(i)p Fp(\))f(follo)m(ws)i(directly)f(from)f(the)g(standard)g(m)m (ulti-dimensional)h(T)-8 b(a)m(ylor)30 b(expansion)e(of)75 2580 y Fo(W)13 b Fp(\()p Fo(t;)i(z)324 2594 y Fl(1)364 2580 y Fo(;)g(z)446 2594 y Fl(2)486 2580 y Fp(\))31 b(with)f(resp)s (ect)g(to)h Fo(z)1223 2594 y Fl(1)1263 2580 y Fp(.)41 b(Indeed,)30 b(the)g(series)h Fo(W)43 b Fp(can)30 b(b)s(e)g(written)h (in)f(term)g(of)h(the)f Fo(W)3457 2594 y Fn(n)3504 2580 y Fp(,)h Fo(n)25 b Ff(2)g Fk(N)3792 2547 y Fn(r)3829 2580 y Fp(,)75 2692 y(as)31 b(follo)m(ws)75 2896 y(\()p Fo(T)13 b(W)g Fp(\))874 b Fo(W)13 b Fp(\()p Fo(t;)i(z)1433 2910 y Fl(1)1472 2896 y Fo(;)g(z)1554 2910 y Fl(2)1595 2896 y Fp(\))25 b(=)1772 2810 y Fi(X)1751 3007 y Fn(n)p Fj(2)p Fb(N)1889 2988 y Fc(r)1949 2835 y Fo(W)2035 2849 y Fn(n)2082 2835 y Fp(\()p 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3366 y(^)1539 3389 y Fo(X)8 b Fp(,)30 b(with)g(resp)s(ect)g(of)h Fo(z)2341 3403 y Fl(1)2381 3389 y Fp(,)f(whic)m(h)g(yields)75 3615 y(\()p Fo(T)13 b(X)7 b Fp(\))904 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1429 3629 y Fl(1)1470 3615 y Fo(;)g(z)1552 3629 y Fl(2)1592 3615 y Fp(\))25 b(=)1769 3529 y Fi(X)1748 3726 y Fn(n)p Fj(2)p Fb(N)1886 3707 y Fc(r)1946 3554 y Fo(X)2021 3568 y Fn(n)2068 3554 y Fp(\()p Fo(t;)15 b(z)2218 3568 y Fl(2)2259 3554 y Fp(\))p 1946 3594 348 4 v 2080 3678 a Fo(n)p Fp(!)2304 3615 y Fo(x)2356 3575 y Fn(n)2399 3584 y Fm(1)2356 3641 y Fl(1)2453 3615 y Ff(\001)g(\001)g(\001)h Fo(x)2626 3578 y Fn(n)2669 3586 y Fc(r)2626 3638 y Fn(r)2707 3615 y Fo(;)75 3882 y Fp(and)30 b(the)g(expansion)h(\()p Fo(T)13 b(W)g Fp(\),)30 b(whic)m(h)g(giv)m(es)i(the)e(relation)1104 4126 y Fo(X)1179 4140 y Fn(n)1227 4126 y Fp(\()p Fo(t;)15 b(z)1377 4140 y Fl(2)1417 4126 y Fp(\))26 b(=)f Fo(t)1607 4089 y Fj(\000)p Fn(k)1719 4002 y Fi(Z)1810 4029 y Fj(1)p Fl(\()p Fn(\015)t Fl(\))1770 4209 y(0)1995 4126 y Fo(W)2081 4140 y Fn(n)2128 4126 y Fp(\()p Fo(\034)5 b(;)15 b(z)2290 4140 y Fl(2)2330 4126 y Fp(\))p Fo(e)2407 4089 y Fj(\000)p Fl(\()p Fn(\034)8 b(=t)p Fl(\))2615 4065 y Fc(k)2660 4126 y Fo(d\034)2757 4089 y Fn(k)2800 4126 y Fo(;)75 4346 y Fp(for)31 b(all)i Fo(n)26 b Ff(2)h Fk(N)578 4313 y Fn(r)616 4346 y Fp(.)43 b(F)-8 b(rom)32 b(the)g(latter)h(relations,)g (one)f(can)g(sho)m(w)f(that)h(the)g(estimates)h(on)e Fo(W)3259 4360 y Fn(n)3337 4346 y Fp(implies)h(those)75 4459 y(for)g Fo(X)291 4473 y Fn(n)338 4459 y Fp(.)46 b(On)32 b(the)g(other)g(hand,)g(supp)s(ose)f(that)i Fe(iii)p Fp(\))f(holds.)46 b(The)32 b(estimates)h(on)f Fo(X)2984 4473 y Fn(n)3064 4459 y Fp(allo)m(w)h(to)g(de\014ned)e(the)75 4572 y(series)25 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)545 4586 y Fl(1)586 4572 y Fo(;)g(z)668 4586 y Fl(2)708 4572 y Fp(\))25 b(b)m(y)f(the)h(form)m(ula)g(\()p Fo(T)13 b(X)7 b Fp(\).)39 b(W)-8 b(e)26 b(de\014ne)e Fo(W)13 b Fp(\()p Fo(t;)i(z)2306 4586 y Fl(1)2346 4572 y Fo(;)g(z)2428 4586 y Fl(2)2468 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b(this)g(section)i(w)m(e)f(study)f(formal)h(series)1666 734 y(^)1640 757 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1873 771 y Fl(1)1913 757 y Fo(;)g(z)1995 771 y Fl(2)2035 757 y Fp(\))39 b(of)g(the)g(form)f(\()2670 734 y(^)2644 757 y Fo(X)7 b Fp(\))39 b(whic)m(h)g(are)g(solutions)g(of)f(the)75 870 y(linear)31 b(partial)g(di\013eren)m(tial)h(equation)75 1170 y(\()p Fo(P)13 b(D)s(E)5 b Fp(\))547 b Fo(@)966 1132 y Fn(S)1009 1141 y Fm(1)961 1192 y Fn(z)994 1201 y Fm(1)1048 1170 y Fo(@)1101 1132 y Fn(S)1144 1141 y Fm(2)1096 1192 y Fn(z)1129 1201 y Fm(2)1209 1147 y Fp(^)1183 1170 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1416 1184 y Fl(1)1456 1170 y Fo(;)g(z)1538 1184 y Fl(2)1578 1170 y Fp(\))26 b(=)f Ff(\000)p Fp(\()1878 1051 y Fn(Q)1841 1083 y Fi(X)1845 1278 y Fn(q)r Fl(=0)1987 1170 y Fo(P)2045 1184 y Fn(q)2083 1170 y Fp(\()p Fo(@)2166 1184 y Fn(z)2199 1193 y Fm(1)2239 1170 y Fo(;)15 b(@)2327 1184 y Fn(z)2360 1193 y Fm(2)2399 1170 y Fp(\))p Fo(@)2487 1125 y Fn(q)2482 1194 y(t)2526 1170 y Fp(\))2587 1147 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Fo(;)15 b(k)807 2044 y Fl(2)847 2030 y Fp(\))26 b Ff(2)f Fk(N)1060 1997 y Fl(2)1125 2030 y Fp(:)h(0)g Ff(\024)f Fo(k)1390 2044 y Fl(1)1450 2030 y Fp(+)20 b Fo(k)1588 2044 y Fl(2)1653 2030 y Ff(\024)25 b Fo(K)1826 2044 y Fn(q)1864 2030 y Ff(g)p Fp(,)32 b(whose)e(co)s(e\016cien)m(ts)i(are)f(holomorphic)g (functions)f(on)75 2143 y Fo(D)150 2161 y Fl(2)p Fn(R)p Fl(\()p Fn(\032)p Fl(\))354 2143 y Ff(\002)20 b Fo(D)520 2157 y Fn(\032)556 2166 y Fm(1)616 2143 y Ff(\002)g Fo(D)782 2157 y Fn(\032)818 2166 y Fm(2)887 2143 y Fp(written)31 b(in)f(the)g(form)1107 2382 y Fo(b)1146 2397 y Fn(q)r(;k)1237 2406 y Fm(1)1271 2397 y Fn(;k)1328 2406 y Fm(2)1366 2382 y Fp(\()p Fo(t;)15 b(z)1516 2396 y Fl(1)1557 2382 y Fo(;)g(z)1639 2396 y Fl(2)1679 2382 y Fp(\))25 b(=)1880 2296 y Fi(X)1835 2493 y Fn(l)1856 2502 y Fm(1)1891 2493 y Fn(;l)1932 2502 y Fm(2)1966 2493 y Fj(\025)p Fl(0)2072 2382 y Fo(b)2111 2397 y Fn(q)r(;k)2202 2406 y Fm(1)2235 2397 y Fn(;k)2292 2406 y Fm(2)2326 2397 y Fn(;l)2367 2406 y Fm(1)2402 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y Fn(;l)1858 3038 y Fm(1)1892 3029 y Fn(;l)1933 3038 y Fm(2)1971 3014 y Fp(\()p Fo(t)p Fp(\))p Ff(j)p 1423 3056 677 4 v 1716 3139 a Fo(l)1743 3153 y Fl(0)1782 3139 y Fp(!)2135 3077 y(=)25 b Ff(j)p Fo(b)p Ff(j)2320 3092 y Fn(q)r(;k)2411 3101 y Fm(1)2446 3092 y Fn(;k)2503 3101 y Fm(2)2537 3092 y Fn(;l)2578 3101 y Fm(1)2612 3092 y Fn(;l)2653 3101 y Fm(2)2687 3092 y Fn(;l)2728 3101 y Fm(0)2766 3077 y Fo(;)75 3373 y Fp(with)30 b(the)h(prop)s(ert)m(y)e(that)i(the)g(series) 934 3595 y Ff(j)p Fo(b)p Ff(j)1023 3610 y Fn(q)r(;k)1114 3619 y Fm(1)1148 3610 y Fn(;k)1205 3619 y Fm(2)1243 3595 y Fp(\()p Fo(t)1311 3558 y Fn(K)1379 3595 y Fo(;)15 b(z)1461 3609 y Fl(1)1502 3595 y Fo(;)g(z)1584 3609 y Fl(2)1624 3595 y Fp(\))25 b(=)1863 3509 y Fi(X)1780 3707 y Fn(l)1801 3716 y Fm(0)1836 3707 y Fn(;l)1877 3716 y Fm(1)1911 3707 y Fn(;l)1952 3716 y Fm(2)1986 3707 y Fj(\025)p Fl(0)2092 3595 y Ff(j)p Fo(b)p Ff(j)2181 3610 y Fn(q)r(;k)2272 3619 y Fm(1)2306 3610 y Fn(;k)2363 3619 y Fm(2)2397 3610 y Fn(;l)2438 3619 y Fm(1)2472 3610 y Fn(;l)2513 3619 y Fm(2)2548 3610 y Fn(;l)2589 3619 y Fm(0)2627 3595 y Fo(t)2660 3558 y Fn(K)5 b(l)2746 3567 y Fm(0)2784 3595 y Fo(z)2830 3555 y Fn(l)2851 3564 y Fm(1)2826 3621 y Fl(1)2890 3595 y Fo(z)2936 3555 y Fn(l)2957 3564 y Fm(2)2932 3621 y Fl(2)75 3913 y Fp(are,)31 b(for)f(all)h(in)m(tegers)h Fo(K)g(>)25 b Fp(0,)31 b(con)m(v)m(ergen)m(t)i(in)d(a)h(neigh)m(b)s (orho)s(o)s(d)e(of)h(the)h(origin)f(in)h Fk(C)3058 3880 y Fl(3)3097 3913 y Fp(.)216 4086 y(In)44 b(the)h(next)f(lemma)i(w)m(e)f (describ)s(e)f(formal)g(series)h(solutions)g(of)g(\()p Fo(P)13 b(D)s(E)5 b Fp(\))45 b(in)g(term)f(of)h(di\013eren)m(tial)75 4198 y(relations.)75 4402 y Fe(Lemma)35 b(1)45 b Fd(R)-5 b(ewrite)34 b(the)e(formal)i(series)e Fp(\()1635 4379 y(^)1608 4402 y Fo(X)8 b Fp(\))33 b Fd(in)f(a)g(formal)i(series)e(with) i(r)-5 b(esp)g(e)g(ct)33 b(to)g(the)g(variables)g Fo(z)3665 4416 y Fl(1)3705 4402 y Fo(;)15 b(z)3787 4416 y Fl(2)3827 4402 y Fd(,)1246 4659 y Fp(^)1220 4682 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1452 4696 y Fl(1)1493 4682 y Fo(;)g(z)1575 4696 y Fl(2)1615 4682 y Fp(\))26 b(=)1838 4595 y Fi(X)1771 4791 y Fn(n)1814 4800 y Fm(1)1849 4791 y Fn(;n)1912 4800 y Fm(2)1946 4791 y Fj(\025)p Fl(0)2087 4597 y Fp(^)2061 4620 y Fo(X)2136 4634 y Fn(n)2179 4643 y Fm(1)2214 4634 y Fn(;n)2277 4643 y Fm(2)2315 4620 y Fp(\()p Fo(t)p Fp(\))p 2061 4661 358 4 v 2121 4744 a Fo(n)2176 4758 y Fl(1)2215 4744 y Fp(!)p Fo(n)2295 4758 y Fl(2)2334 4744 y Fp(!)2429 4682 y Fo(z)2475 4641 y Fn(n)2518 4650 y Fm(1)2471 4708 y Fl(1)2556 4682 y Fo(z)2602 4641 y Fn(n)2645 4650 y Fm(2)2598 4708 y Fl(2)2684 4682 y Fo(:)75 5008 y Fd(Then,)369 4985 y Fp(^)343 5008 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)576 5022 y Fl(1)616 5008 y Fo(;)g(z)698 5022 y Fl(2)738 5008 y Fp(\))36 b Fd(is)f(a)h(formal)g(series)g(solution)g(of)g Fp(\()p Fo(P)13 b(D)s(E)5 b Fp(\))36 b Fd(if)f(and)h(only)g(if)f(the)h (fol)5 b(lowing)36 b(di\013er)-5 b(ential)75 5121 y(r)g(elation)34 b Fp(\()p Fo(D)s(R)q Fp(\))f Fd(is)g(satis\014e)-5 b(d,)320 5341 y Fp(^)294 5364 y Fo(X)369 5378 y Fn(m)431 5387 y Fm(1)467 5378 y Fl(+)p Fn(S)565 5387 y Fm(1)599 5378 y Fn(;m)681 5387 y Fm(2)716 5378 y Fl(+)p Fn(S)814 5387 y Fm(2)852 5364 y Fp(\()p Fo(t)p Fp(\))p 294 5405 662 4 v 410 5488 a(\()p Fo(m)525 5502 y Fl(1)564 5488 y Fp(\)!\()p Fo(m)739 5502 y Fl(2)780 5488 y Fp(\)!)991 5426 y(=)25 b Ff(\000)1211 5308 y Fn(Q)1173 5340 y Fi(X)1176 5534 y Fn(q)r Fl(=0)1424 5340 y Fi(X)1319 5541 y Fl(\()p Fn(k)1383 5550 y Fm(1)1418 5541 y Fn(;k)1475 5550 y Fm(2)1509 5541 y Fl(\))p Fj(2S)1626 5549 y Fc(q)1780 5340 y Fi(X)1676 5537 y Fn(l)1697 5546 y Fm(1)1731 5537 y Fl(+)p Fn(n)1829 5546 y Fm(1)1864 5537 y Fl(=)p Fn(m)1981 5546 y Fm(1)2135 5340 y Fi(X)2031 5537 y Fn(l)2052 5546 y Fm(2)2086 5537 y Fl(+)p Fn(n)2184 5546 y Fm(2)2219 5537 y Fl(=)p Fn(m)2336 5546 y Fm(2)2386 5426 y Fo(b)2425 5441 y Fn(q)r(;k)2516 5450 y Fm(1)2550 5441 y Fn(;k)2607 5450 y Fm(2)2641 5441 y Fn(;l)2682 5450 y Fm(1)2716 5441 y Fn(;l)2757 5450 y Fm(2)2795 5426 y Fp(\()p Fo(t)p Fp(\))2908 5363 y Fo(@)2961 5319 y Fn(q)2956 5387 y(t)3026 5340 y Fp(^)3000 5363 y Fo(X)3075 5378 y Fn(n)3118 5387 y Fm(1)3153 5378 y Fl(+)p Fn(k)3245 5387 y Fm(1)3279 5378 y Fn(;n)3342 5387 y Fm(2)3376 5378 y Fl(+)p Fn(k)3468 5387 y Fm(2)3506 5363 y Fp(\()p Fo(t)p Fp(\))p 2908 5405 702 4 v 3069 5488 a(\()p Fo(n)3159 5502 y Fl(1)3199 5488 y Fp(\)!\()p Fo(n)3349 5502 y Fl(2)3389 5488 y Fp(\)!)3620 5426 y Fo(;)75 5740 y Fd(for)33 b(al)5 b(l)33 b Fo(m)430 5754 y Fl(1)470 5740 y Fo(;)15 b(m)590 5754 y Fl(2)654 5740 y Ff(\025)25 b Fp(0)p Fd(.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 75 100 a Fp(6)75 399 y Fe(Pro)s(of)31 b Fp(W)-8 b(e)32 b(\014rst)d(ha)m(v)m(e)j(that)874 646 y Fo(@)927 609 y Fn(k)964 618 y Fm(1)922 669 y Fn(z)955 678 y Fm(1)1002 646 y Fo(@)1055 609 y Fn(k)1092 618 y Fm(2)1050 669 y Fn(z)1083 678 y Fm(2)1131 646 y Fo(@)1184 602 y Fn(q)1179 670 y(t)1249 623 y Fp(^)1223 646 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1455 660 y Fl(1)1495 646 y Fo(;)g(z)1577 660 y Fl(2)1617 646 y Fp(\))26 b(=)1841 560 y Fi(X)1774 756 y Fn(n)1817 765 y Fm(1)1851 756 y Fn(;n)1914 765 y Fm(2)1948 756 y Fj(\025)p Fl(0)2064 583 y Fo(@)2117 539 y Fn(q)2112 607 y(t)2181 560 y Fp(^)2155 583 y Fo(X)2230 598 y Fn(n)2273 607 y Fm(1)2308 598 y Fl(+)p Fn(k)2400 607 y Fm(1)2434 598 y Fn(;n)2497 607 y Fm(2)2531 598 y Fl(+)p Fn(k)2623 607 y Fm(2)2661 583 y Fp(\()p Fo(t)p Fp(\))p 2064 625 702 4 v 2224 709 a(\()p Fo(n)2314 723 y Fl(1)2354 709 y Fp(\)!\()p Fo(n)2504 723 y Fl(2)2544 709 y Fp(\)!)2775 646 y Fo(z)2821 606 y Fn(n)2864 615 y Fm(1)2817 672 y Fl(1)2903 646 y Fo(z)2949 606 y Fn(n)2992 615 y Fm(2)2945 672 y Fl(2)3031 646 y Fo(:)75 939 y Fp(So)k(that)158 1167 y Fo(P)216 1181 y Fn(q)254 1167 y Fp(\()p Fo(@)337 1181 y Fn(z)370 1190 y Fm(1)410 1167 y Fo(;)15 b(@)498 1181 y Fn(z)531 1190 y Fm(2)570 1167 y Fp(\))p Fo(@)658 1123 y Fn(q)653 1191 y(t)723 1144 y Fp(^)697 1167 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)929 1181 y Fl(1)969 1167 y Fo(;)g(z)1051 1181 y Fl(2)1091 1167 y Fp(\))26 b(=)744 1334 y Fi(X)658 1530 y Fn(m)720 1539 y Fm(1)755 1530 y Fn(;m)837 1539 y Fm(2)871 1530 y Fj(\025)p Fl(0)977 1238 y Fi(0)977 1402 y(@)1161 1334 y(X)1056 1536 y Fl(\()p Fn(k)1120 1545 y Fm(1)1155 1536 y Fn(;k)1212 1545 y Fm(2)1246 1536 y Fl(\))p Fj(2S)1363 1544 y Fc(q)1517 1334 y Fi(X)1413 1532 y Fn(l)1434 1541 y Fm(1)1468 1532 y Fl(+)p Fn(n)1566 1541 y Fm(1)1600 1532 y Fl(=)p Fn(m)1717 1541 y Fm(1)1872 1334 y Fi(X)1768 1532 y Fn(l)1789 1541 y Fm(2)1823 1532 y Fl(+)p Fn(n)1921 1541 y Fm(2)1955 1532 y Fl(=)p Fn(m)2072 1541 y Fm(2)2123 1421 y Fo(b)2162 1436 y Fn(q)r(;k)2253 1445 y Fm(1)2286 1436 y Fn(;k)2343 1445 y Fm(2)2377 1436 y Fn(;l)2418 1445 y Fm(1)2453 1436 y Fn(;l)2494 1445 y Fm(2)2532 1421 y Fp(\()p Fo(t)p Fp(\))2645 1357 y Fo(@)2698 1313 y Fn(q)2693 1382 y(t)2763 1334 y Fp(^)2737 1357 y Fo(X)2812 1372 y Fn(n)2855 1381 y Fm(1)2890 1372 y Fl(+)p Fn(k)2982 1381 y Fm(1)3016 1372 y Fn(;n)3079 1381 y Fm(2)3113 1372 y Fl(+)p Fn(k)3205 1381 y Fm(2)3243 1357 y Fp(\()p Fo(t)p Fp(\))p 2645 1400 V 2806 1483 a(\()p Fo(n)2896 1497 y Fl(1)2935 1483 y Fp(\)!\()p Fo(n)3085 1497 y Fl(2)3125 1483 y Fp(\)!)3357 1238 y Fi(1)3357 1402 y(A)3451 1421 y Fo(z)3497 1380 y Fn(m)3559 1389 y Fm(1)3493 1447 y Fl(1)3599 1421 y Fo(z)3645 1380 y Fn(m)3707 1389 y Fm(2)3641 1447 y Fl(2)3746 1421 y Fo(:)75 1724 y Fp(The)k(lemma)h(then)f(easily)h(follo)m(ws.)2452 b Fa(2)75 1936 y Fe(Example:)40 b Fp(F)-8 b(or)31 b(the)g(heat)g (equation)1176 2139 y Fo(@)1229 2101 y Fl(2)1224 2161 y Fn(z)1257 2170 y Fm(1)1322 2116 y Fp(^)1296 2139 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1529 2153 y Fl(1)1569 2139 y Fo(;)g(z)1651 2153 y Fl(2)1691 2139 y Fp(\))26 b(=)f(\()p Ff(\000)p Fo(@)2007 2101 y Fl(2)2002 2161 y Fn(z)2035 2170 y Fm(2)2094 2139 y Fp(+)20 b Fo(@)2233 2153 y Fn(t)2262 2139 y Fp(\))2324 2116 y(^)2297 2139 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)2530 2153 y Fl(1)2571 2139 y Fo(;)g(z)2653 2153 y Fl(2)2693 2139 y Fp(\))p Fo(;)75 2342 y Fp(the)31 b(di\013eren)m(tial)g(relation)h(writes,)1103 2522 y(^)1077 2545 y Fo(X)1152 2559 y Fn(m)1214 2568 y Fm(1)1249 2559 y Fl(+2)p Fn(;m)1421 2568 y Fm(2)1460 2545 y Fp(\()p Fo(t)p Fp(\))26 b(=)f Ff(\000)1781 2522 y Fp(^)1756 2545 y Fo(X)1831 2559 y Fn(m)1893 2568 y Fm(1)1928 2559 y Fn(;m)2010 2568 y Fm(2)2044 2559 y Fl(+2)2139 2545 y Fp(\()p Fo(t)p Fp(\))20 b(+)g Fo(@)2401 2559 y Fn(t)2457 2522 y Fp(^)2431 2545 y Fo(X)2506 2559 y Fn(m)2568 2568 y Fm(1)2603 2559 y Fn(;m)2685 2568 y Fm(2)2724 2545 y Fp(\()p Fo(t)p Fp(\))p Fo(:)216 2748 y Fp(In)40 b(the)g(next)h(prop)s (osition,)i(w)m(e)d(giv)m(e)i(su\016cien)m(t)f(conditions)g(for)f(the)h (existence)g(and)f(uniqueness)f(of)75 2861 y(formal)31 b(series)f(\()666 2838 y(^)639 2861 y Fo(X)8 b Fp(\))31 b(that)g(are)f(solutions)h(of)g(\()p Fo(P)13 b(D)s(E)5 b Fp(\).)75 3047 y Fe(Prop)s(osition)36 b(2)46 b Fd(We)34 b(assume)h(that)h(the)f(tuple)g Fp(\()p Fo(S)1882 3061 y Fl(1)1922 3047 y Fo(;)15 b(S)2018 3061 y Fl(2)2057 3047 y Fp(\))35 b Fd(do)-5 b(es)36 b(not)f(b)-5 b(elong)35 b(to)h(the)f(c)-5 b(onvex)35 b(hul)5 b(l)34 b(in)h Fk(R)3687 3014 y Fl(2)3687 3070 y(+)3780 3047 y Fd(of)75 3160 y(the)e(set)g(of)g (tuples)g Fp(\()p Fo(k)817 3174 y Fl(1)857 3160 y Fo(;)15 b(k)944 3174 y Fl(2)984 3160 y Fp(\))33 b Fd(for)g(which)h Fo(b)1492 3175 y Fn(q)r(;k)1583 3184 y Fm(1)1617 3175 y Fn(;k)1674 3184 y Fm(2)1744 3160 y Fd(is)f(not)g(the)g(zer)-5 b(o)34 b(function,)e(for)i(al)5 b(l)33 b Fp(0)25 b Ff(\024)g Fo(q)j Ff(\024)d Fo(Q)33 b Fd(and)g(that)1558 3363 y Fo(S)1614 3377 y Fl(1)1674 3363 y Fp(+)20 b Fo(S)1821 3377 y Fl(2)1885 3363 y Ff(\025)58 b Fp(max)1981 3424 y Fl(0)p Fj(\024)p Fn(q)r Fj(\024)p Fn(Q)2231 3363 y Fo(K)2308 3377 y Fn(q)2346 3363 y Fo(:)75 3630 y Fd(Mor)-5 b(e)g(over,)54 b(assume)49 b(that)h(se)-5 b(quenc)g(es)48 b Fp(\()1542 3607 y(^)1516 3630 y Fo(X)1591 3644 y Fn(\026)1633 3653 y Fm(1)1669 3644 y Fn(;\026)1731 3653 y Fm(2)1770 3630 y Fp(\()p Fo(t)p Fp(\)\))1908 3644 y Fn(\026)1950 3653 y Fm(2)1985 3644 y Fj(\025)p Fl(0)2080 3630 y Fd(,)k(for)d(al)5 b(l)49 b Fp(0)55 b Ff(\024)f Fo(\026)2747 3644 y Fl(1)2840 3630 y Ff(\024)g Fo(S)3021 3644 y Fl(1)3092 3630 y Ff(\000)31 b Fp(1)49 b Fd(and)g(se)-5 b(quenc)g(es)75 3743 y Fp(\()136 3720 y(^)110 3743 y Fo(X)185 3757 y Fn(\026)227 3766 y Fm(1)263 3757 y Fn(;\026)325 3766 y Fm(2)364 3743 y Fp(\()p Fo(t)p Fp(\)\))502 3757 y Fn(\026)544 3766 y Fm(1)579 3757 y Fj(\025)p Fl(0)674 3743 y Fd(,)32 b(for)h(al)5 b(l)33 b Fp(0)26 b Ff(\024)f Fo(\026)1231 3757 y Fl(2)1296 3743 y Ff(\024)g Fo(S)1448 3757 y Fl(2)1507 3743 y Ff(\000)20 b Fp(1)33 b Fd(ar)-5 b(e)33 b(given.)216 3856 y(Then,)48 b(ther)-5 b(e)46 b(exists)f(a)h(unique)e(se)-5 b(quenc)g(e)45 b Fp(\()1836 3833 y(^)1810 3856 y Fo(X)1885 3870 y Fn(\026)1927 3879 y Fm(1)1962 3870 y Fn(;\026)2024 3879 y Fm(2)2063 3856 y Fp(\()p Fo(t)p Fp(\)\))2201 3870 y Fn(\026)2243 3879 y Fm(1)2279 3870 y Fn(;\026)2341 3879 y Fm(2)2425 3856 y Fd(of)g(formal)h(series)f(which)h(satis\014es)g(the)75 3969 y(di\013er)-5 b(ential)40 b(r)-5 b(elation)41 b Fp(\()p Fo(D)s(R)q Fp(\))p Fd(.)61 b(As)39 b(a)g(r)-5 b(esult,)42 b(for)d(given)g(se)-5 b(quenc)g(es)39 b Fp(\()2569 3946 y(^)2543 3969 y Fo(X)2618 3983 y Fn(\026)2660 3992 y Fm(1)2695 3983 y Fn(;\026)2757 3992 y Fm(2)2796 3969 y Fp(\()p Fo(t)p Fp(\)\))2934 3983 y Fn(\026)2976 3992 y Fm(2)3012 3983 y Fj(\025)p Fl(0)3106 3969 y Fd(,)i(for)e(al)5 b(l)40 b Fp(0)d Ff(\024)g Fo(\026)3708 3983 y Fl(1)3784 3969 y Ff(\024)75 4082 y Fo(S)131 4096 y Fl(1)182 4082 y Ff(\000)12 b Fp(1)29 b Fd(and)h(se)-5 b(quenc)g(es)28 b Fp(\()976 4059 y(^)949 4082 y Fo(X)1024 4096 y Fn(\026)1066 4105 y Fm(1)1102 4096 y Fn(;\026)1164 4105 y Fm(2)1203 4082 y Fp(\()p Fo(t)p Fp(\)\))1341 4096 y Fn(\026)1383 4105 y Fm(1)1419 4096 y Fj(\025)p Fl(0)1513 4082 y Fd(,)i(for)f(al)5 b(l)29 b Fp(0)d Ff(\024)f Fo(\026)2060 4096 y Fl(2)2124 4082 y Ff(\024)g Fo(S)2276 4096 y Fl(2)2327 4082 y Ff(\000)12 b Fp(1)p Fd(,)30 b(ther)-5 b(e)30 b(exists)f(a)g(unique)f(formal)i (series)75 4195 y Fp(\()136 4172 y(^)110 4195 y Fo(X)8 b Fp(\))p Fd(,)33 b(solution)g(of)g Fp(\()p Fo(P)13 b(D)s(E)5 b Fp(\))p Fd(.)75 4431 y Fe(Pro)s(of)37 b Fp(By)g(assumption,)h(\()p Fo(S)1119 4445 y Fl(1)1159 4431 y Fo(;)15 b(S)1255 4445 y Fl(2)1294 4431 y Fp(\))37 b(do)s(es)g(not)g(b)s(elong)f(to)i(the)f (con)m(v)m(ex)h(h)m(ull)e(in)h Fk(R)2991 4398 y Fl(2)2991 4453 y(+)3086 4431 y Fp(of)g(the)g(set)g(of)g(tuples)75 4544 y(\()p Fo(k)157 4558 y Fl(1)197 4544 y Fo(;)15 b(k)284 4558 y Fl(2)324 4544 y Fp(\))34 b(for)f(whic)m(h)g Fo(b)837 4559 y Fn(q)r(;k)928 4568 y Fm(1)962 4559 y Fn(;k)1019 4568 y Fm(2)1090 4544 y Fp(is)g(not)h(the)f(zero)h(function,)g(for)f (all)h(0)d Ff(\024)e Fo(q)k Ff(\024)d Fo(Q)p Fp(.)49 b(F)-8 b(rom)34 b(the)f(classical)j(Hahn-)75 4657 y(Banac)m(h)31 b(separation)h(theorem,)f(w)m(e)g(deduce)f(that)h(there)f(exist)h Fo(\032)2347 4671 y Fl(1)2387 4657 y Fo(;)15 b(\032)2474 4671 y Fl(2)2539 4657 y Ff(2)25 b Fk(R)30 b Fp(suc)m(h)g(that)75 4860 y(\()p Fo(C)7 b(o)p Fp(\))703 b Fo(\032)p Fp(\()p Fo(S)1102 4874 y Fl(1)1142 4860 y Fo(;)15 b(S)1238 4874 y Fl(2)1277 4860 y Fp(\))26 b(=)f Fo(\032)1481 4874 y Fl(1)1520 4860 y Fo(S)1576 4874 y Fl(1)1636 4860 y Fp(+)20 b Fo(\032)1774 4874 y Fl(2)1813 4860 y Fo(S)1869 4874 y Fl(2)1934 4860 y Fo(>)25 b(\032)2077 4874 y Fl(1)2116 4860 y Fo(k)2163 4874 y Fl(1)2223 4860 y Fp(+)20 b Fo(\032)2361 4874 y Fl(2)2400 4860 y Fo(k)2447 4874 y Fl(2)2512 4860 y Fp(=)25 b Fo(\032)p Fp(\()p Fo(k)2737 4874 y Fl(1)2777 4860 y Fo(;)15 b(k)2864 4874 y Fl(2)2905 4860 y Fp(\))p Fo(;)75 5063 y Fp(for)30 b(all)h(\()p Fo(k)422 5077 y Fl(1)463 5063 y Fo(;)15 b(k)550 5077 y Fl(2)590 5063 y Fp(\))30 b(for)g(whic)m(h)h Fo(b)1094 5078 y Fn(q)r(;k)1185 5087 y Fm(1)1218 5078 y Fn(;k)1275 5087 y Fm(2)1344 5063 y Fp(is)f(not)h(the)f(zero)i(function,)e(for)g(all)h(0)26 b Ff(\024)f Fo(q)j Ff(\024)d Fo(Q)p Fp(.)216 5176 y(On)34 b(the)g(other)h(hand,)f(from)g(the)g(h)m(yp)s(othesis)g Fo(S)1918 5190 y Fl(1)1980 5176 y Fp(+)23 b Fo(S)2130 5190 y Fl(2)2201 5176 y Ff(\025)32 b Fp(max)2473 5190 y Fl(0)p Fj(\024)p Fn(q)r Fj(\024)p Fn(Q)2726 5176 y Fo(K)2803 5190 y Fn(q)2842 5176 y Fp(,)j(w)m(e)g(can)g(replace)g Fo(\032)p Fp(\()p Fo(k)3652 5190 y Fl(1)3692 5176 y Fo(;)15 b(k)3779 5190 y Fl(2)3819 5176 y Fp(\))75 5289 y(b)m(y)32 b Fo(\032)p Fp(\()p Fo(k)332 5303 y Fl(1)372 5289 y Fo(;)15 b(k)459 5303 y Fl(2)500 5289 y Fp(\))22 b(+)f Fo(s)p Fp(\()p Fo(k)774 5303 y Fl(1)835 5289 y Fp(+)g Fo(k)974 5303 y Fl(2)1014 5289 y Fp(\))33 b(in)f(the)h(inequalities)h(\()p Fo(C)7 b(o)p Fp(\),)33 b(for)f(all)h Fo(s)c(>)f Fp(0.)47 b(In)32 b(suc)m(h)g(a)h(w)m(a)m(y)-8 b(,)35 b(w)m(e)e(ma)m(y)g(assume) 75 5402 y(in)d(the)h(follo)m(wing)g(that)g Fo(\032)965 5416 y Fl(1)1005 5402 y Fo(;)15 b(\032)1092 5416 y Fl(2)1157 5402 y Fo(>)25 b Fp(0)31 b(in)f(the)g(inequalities)i(\()p Fo(C)7 b(o)p Fp(\).)216 5514 y(T)-8 b(ak)m(e)36 b(\()p Fo(\026)530 5528 y Fl(1)569 5514 y Fo(;)15 b(\026)664 5528 y Fl(2)704 5514 y Fp(\))p Fo(;)g Fp(\()p Fo(\027)859 5528 y Fl(1)899 5514 y Fo(;)g(\027)984 5528 y Fl(2)1024 5514 y Fp(\))32 b Ff(2)f Fk(N)1249 5481 y Fl(2)1288 5514 y Fp(,)36 b(w)m(e)f(sa)m(y)f(b)m(y)h(de\014nition,)g(that)f(\()p Fo(\026)2502 5528 y Fl(1)2542 5514 y Fo(;)15 b(\026)2637 5528 y Fl(2)2677 5514 y Fp(\))34 b(is)g(lo)m(w)m(er)i(than)e(\()p Fo(\027)3379 5528 y Fl(1)3419 5514 y Fo(;)15 b(\027)3504 5528 y Fl(2)3543 5514 y Fp(\))35 b(and)e(is)75 5627 y(denoted)e(b)m(y)g (\()p Fo(\026)636 5641 y Fl(1)676 5627 y Fo(;)15 b(\026)771 5641 y Fl(2)811 5627 y Fp(\))27 b Fo(<)944 5641 y Fn(\032)1010 5627 y Fp(\()p Fo(\027)1090 5641 y Fl(1)1130 5627 y Fo(;)15 b(\027)1215 5641 y Fl(2)1255 5627 y Fp(\))31 b(if)h Fo(\032)p Fp(\()p Fo(\026)1543 5641 y Fl(1)1582 5627 y Fo(;)15 b(\026)1677 5641 y Fl(2)1717 5627 y Fp(\))27 b Fo(<)f(\032)p Fp(\()p Fo(\027)2003 5641 y Fl(1)2043 5627 y Fo(;)15 b(\027)2128 5641 y Fl(2)2168 5627 y Fp(\).)43 b(The)31 b(relation)h Fo(<)2864 5641 y Fn(\032)2936 5627 y Fp(is)f(a)h(partial)g (ordering)f(on)75 5740 y Fk(N)141 5707 y Fl(2)180 5740 y Fp(.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fp(7)216 399 y(F)-8 b(rom)31 b(inequalities)h(\()p Fo(C)7 b(o)p Fp(\),)31 b(w)m(e)f(deduce)g(that)940 605 y(\()p Fo(k)1022 619 y Fl(1)1082 605 y Fp(+)20 b Fo(m)1253 619 y Fl(1)1313 605 y Ff(\000)g Fo(l)1431 619 y Fl(1)1470 605 y Fo(;)15 b(k)1557 619 y Fl(2)1617 605 y Fp(+)20 b Fo(m)1788 619 y Fl(2)1848 605 y Ff(\000)g Fo(l)1966 619 y Fl(2)2005 605 y Fp(\))26 b Fo(<)2137 619 y Fn(\032)2202 605 y Fp(\()p Fo(S)2293 619 y Fl(1)2353 605 y Fp(+)20 b Fo(m)2524 619 y Fl(1)2563 605 y Fo(;)15 b(S)2659 619 y Fl(2)2719 605 y Fp(+)20 b Fo(m)2890 619 y Fl(2)2929 605 y Fp(\))p Fo(;)75 811 y Fp(for)g(all)h Fo(m)400 825 y Fl(1)439 811 y Fo(;)15 b(m)559 825 y Fl(2)599 811 y Fo(;)g(l)666 825 y Fl(1)706 811 y Fo(;)g(l)773 825 y Fl(2)838 811 y Ff(\025)25 b Fp(0)c(and)f(all)h(\()p Fo(k)1365 825 y Fl(1)1405 811 y Fo(;)15 b(k)1492 825 y Fl(2)1532 811 y Fp(\))21 b(for)f(whic)m(h)g Fo(b)2006 826 y Fn(q)r(;k)2097 835 y Fm(1)2131 826 y Fn(;k)2188 835 y Fm(2)2246 811 y Fp(is)g(not)h(the)f(zero)h(function,)h (for)f(all)g(0)k Ff(\024)g Fo(q)j Ff(\024)d Fo(Q)p Fp(.)216 934 y(In)j(this)h(w)m(a)m(y)-8 b(,)30 b(the)f(term)1101 911 y(^)1075 934 y Fo(X)1150 948 y Fn(m)1212 957 y Fm(1)1247 948 y Fl(+)p Fn(S)1345 957 y Fm(1)1379 948 y Fn(;m)1461 957 y Fm(2)1496 948 y Fl(+)p Fn(S)1594 957 y Fm(2)1661 934 y Fp(in)g(the)f(di\013eren)m(tial)i(relation)g(\()p Fo(D)s(R)q Fp(\),)g(whic)m(h)e(can)h(b)s(e)f(called)i(in)75 1055 y(this)j(case)g(di\013eren)m(tial)h(recurrence,)f(can)g(b)s(e)f (uniquely)g(expressed)g(in)h(terms)f(of)2946 1032 y(^)2920 1055 y Fo(X)2995 1069 y Fn(\026)3037 1078 y Fm(1)3072 1069 y Fn(;\026)3134 1078 y Fm(2)3206 1055 y Fp(with)g(\()p Fo(\026)3505 1069 y Fl(1)3545 1055 y Fo(;)15 b(\026)3640 1069 y Fl(2)3679 1055 y Fp(\))30 b Fo(<)3815 1069 y Fn(\032)75 1168 y Fp(\()p Fo(m)190 1182 y Fl(1)250 1168 y Fp(+)20 b Fo(S)397 1182 y Fl(1)436 1168 y Fo(;)15 b(m)556 1182 y Fl(2)616 1168 y Fp(+)20 b Fo(S)763 1182 y Fl(2)802 1168 y Fp(\))31 b(for)f(all)h Fo(m)1213 1182 y Fl(1)1252 1168 y Fo(;)15 b(m)1372 1182 y Fl(2)1437 1168 y Ff(\025)25 b Fp(0.)41 b(The)30 b(result)g(of)h(the)f(prop)s(osition)g(follo)m(ws.) 642 b Fa(2)216 1384 y Fp(The)43 b(next)h(result)f(giv)m(es)i (su\016cien)m(t)f(conditions)g(for)f(the)g(existence)i(of)f Fo(\024)p Fp(-summable)f(formal)h(series)75 1496 y(solutions)31 b(\()516 1473 y(^)490 1496 y Fo(X)8 b Fp(\))30 b(of)h(\()p Fo(P)13 b(D)s(E)5 b Fp(\),)31 b(for)f(w)m(ell)i(c)m(hosen)f Fo(\024)p Fp(.)75 1687 y Fe(Theorem)k(1)46 b Fd(Consider)33 b(the)g(line)-5 b(ar)34 b(p)-5 b(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation,)75 1979 y Fp(\()p Fo(P)13 b(D)s(E)5 b Fp(\))547 b Fo(@)966 1942 y Fn(S)1009 1951 y Fm(1)961 2002 y Fn(z)994 2011 y Fm(1)1048 1979 y Fo(@)1101 1942 y Fn(S)1144 1951 y Fm(2)1096 2002 y Fn(z)1129 2011 y Fm(2)1209 1956 y Fp(^)1183 1979 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1416 1993 y Fl(1)1456 1979 y Fo(;)g(z)1538 1993 y Fl(2)1578 1979 y Fp(\))26 b(=)f Ff(\000)p Fp(\()1878 1861 y Fn(Q)1841 1893 y Fi(X)1845 2087 y Fn(q)r Fl(=0)1987 1979 y Fo(P)2045 1993 y Fn(q)2083 1979 y Fp(\()p Fo(@)2166 1993 y Fn(z)2199 2002 y Fm(1)2239 1979 y Fo(;)15 b(@)2327 1993 y Fn(z)2360 2002 y Fm(2)2399 1979 y Fp(\))p Fo(@)2487 1935 y Fn(q)2482 2003 y(t)2526 1979 y Fp(\))2587 1956 y(^)2561 1979 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)2793 1993 y Fl(1)2834 1979 y Fo(;)g(z)2916 1993 y Fl(2)2956 1979 y Fp(\))p Fo(:)75 2274 y Fd(Assume)32 b(that)h(the)g(c)-5 b(onditions)34 b(of)e(the)g(pr)-5 b(op)g(osition)36 b(2)d(ar)-5 b(e)33 b(ful\014l)5 b(le)-5 b(d.)42 b(Assume)32 b(mor)-5 b(e)g(over)34 b(that)f(ther)-5 b(e)33 b(exists)75 2387 y(an)g(inte)-5 b(ger)33 b Fo(K)f(>)25 b Fp(1)33 b Fd(such)g(that)1411 2502 y Fo(S)1467 2516 y Fl(1)1527 2502 y Fp(+)20 b Fo(S)1674 2516 y Fl(2)1738 2502 y Ff(\025)58 b Fp(max)1834 2562 y Fl(0)p Fj(\024)p Fn(q)r Fj(\024)p Fn(Q)2068 2502 y Fp(\()p Fo(q)s(K)28 b Fp(+)19 b Fo(K)2419 2516 y Fn(q)2458 2502 y Fp(\))p Fo(;)75 2724 y Fd(and)28 b Fp(\(0)p Fo(;)15 b(S)422 2738 y Fl(1)463 2724 y Fo(;)g(S)559 2738 y Fl(2)598 2724 y Fp(\))28 b Fd(do)-5 b(es)28 b(not)g(b)-5 b(elong)28 b(to)g(the)g(c)-5 b(onvex)27 b(hul)5 b(l)28 b(in)f Fk(R)2162 2691 y Fl(3)2162 2747 y(+)2248 2724 y Fd(of)g(the)h(set)g(of)f(tuples)h Fp(\()p Fo(q)s(K)q(;)15 b(k)3232 2738 y Fl(1)3273 2724 y Fo(;)g(k)3360 2738 y Fl(2)3400 2724 y Fp(\))p Fd(,)29 b(for)e(which)75 2837 y Fo(b)114 2852 y Fn(q)r(;k)205 2861 y Fm(1)239 2852 y Fn(;k)296 2861 y Fm(2)366 2837 y Fd(is)33 b(not)g(the)g(zer)-5 b(o)34 b(function.)216 2960 y(We)c(assume)i(that)g(the)f(series)1310 2937 y Fp(^)1284 2960 y Fo(X)1359 2974 y Fn(\026)1401 2983 y Fm(1)1436 2974 y Fn(;\026)1498 2983 y Fm(2)1537 2960 y Fp(\()p Fo(t)p Fp(\))g Fd(for)h(al)5 b(l)31 b Fo(\026)1998 2974 y Fl(2)2062 2960 y Ff(\025)25 b Fp(0)p Fd(,)32 b Fp(0)25 b Ff(\024)g Fo(\026)2484 2974 y Fl(1)2549 2960 y Ff(\024)g Fo(S)2701 2974 y Fl(1)2756 2960 y Ff(\000)16 b Fp(1)31 b Fd(and)h(the)f(series)3524 2937 y Fp(^)3498 2960 y Fo(X)3573 2974 y Fn(\026)3615 2983 y Fm(1)3650 2974 y Fn(;\026)3712 2983 y Fm(2)3751 2960 y Fp(\()p Fo(t)p Fp(\))75 3073 y Fd(for)i(al)5 b(l)33 b Fo(\026)405 3087 y Fl(1)469 3073 y Ff(\025)25 b Fp(0)p Fd(,)33 b Fp(0)25 b Ff(\024)g Fo(\026)892 3087 y Fl(2)957 3073 y Ff(\024)g Fo(S)1109 3087 y Fl(2)1168 3073 y Ff(\000)19 b Fp(1)p Fd(,)32 b(ar)-5 b(e)34 b Fp(1)p Fo(=)p Fp(\()p Fo(K)27 b Ff(\000)19 b Fp(1\))p Fd(-summable)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)f Fo(t)f Fd(in)g(a)h(dir)-5 b(e)g(ction)34 b Fo(d)25 b Ff(2)g Fk(R)p Fd(.)216 3187 y(Mor)-5 b(e)g(over,)40 b(we)e(assume)g(that)h(their)f(sums)g Fo(X)1836 3201 y Fn(\026)1878 3210 y Fm(1)1913 3201 y Fn(;\026)1975 3210 y Fm(2)2014 3187 y Fp(\()p Fo(t)p Fp(\))g Fd(ar)-5 b(e)38 b(holomorphic)j(in)d(a)f(se)-5 b(ctorial)40 b(r)-5 b(e)g(gion)38 b Fo(G)d Ff(\032)75 3300 y Fo(D)150 3318 y Fn(R)p Fl(\()p Fn(\032)p Fl(\))299 3300 y Fd(,)c(indep)-5 b(endent)32 b(of)f Fo(\026)1020 3314 y Fl(1)1059 3300 y Fo(;)15 b(\026)1154 3314 y Fl(2)1194 3300 y Fd(,)30 b(which)i(has)g(bise)-5 b(cting)30 b(dir)-5 b(e)g(ction)32 b Fo(d)f Fd(and)h(op)-5 b(ening)31 b(gr)-5 b(e)g(ater)32 b(than)g Fp(\()p Fo(K)23 b Ff(\000)16 b Fp(1\))p Fo(\031)75 3412 y Fd(and)34 b(satisfy)f(the)g(fol)5 b(lowing)34 b(estimates.)75 3526 y(F)-7 b(or)34 b(al)5 b(l)33 b(close)-5 b(d)34 b(se)-5 b(ctor)919 3503 y Fp(\026)903 3526 y Fo(S)31 b Ff(\032)25 b Fo(G)p Fd(,)32 b(ther)-5 b(e)34 b(exist)f Fo(C)q(;)1787 3503 y Fp(~)1762 3526 y Fo(K)f(>)25 b Fp(0)33 b Fd(such)g(that)1037 3732 y Fp(sup)1046 3820 y Fn(t)p Fj(2)1130 3803 y Fl(\026)1118 3820 y Fn(S)1189 3732 y Ff(j)p Fo(@)1267 3692 y Fn(n)1310 3701 y Fm(0)1262 3756 y Fn(t)1350 3732 y Fo(X)1425 3746 y Fn(n)1468 3755 y Fm(1)1502 3746 y Fn(;n)1565 3755 y Fm(2)1603 3732 y Fp(\()p Fo(t)p Fp(\))p Ff(j)26 b(\024)f Fo(C)1949 3709 y Fp(~)1925 3732 y Fo(K)2009 3695 y Fn(n)2052 3704 y Fm(0)2086 3695 y Fl(+)p Fn(n)2184 3704 y Fm(1)2218 3695 y Fl(+)p Fn(n)2316 3704 y Fm(2)2355 3732 y Fp(\()p Fo(K)7 b(n)2529 3746 y Fl(0)2568 3732 y Fp(\)!)p Fo(n)2683 3746 y Fl(1)2722 3732 y Fp(!)p Fo(n)2802 3746 y Fl(2)2842 3732 y Fp(!)p Fo(;)75 3998 y Fd(for)33 b(al)5 b(l)33 b Fp(0)26 b Ff(\024)f Fo(n)572 4012 y Fl(1)636 3998 y Ff(\024)g Fo(S)788 4012 y Fl(1)848 3998 y Ff(\000)19 b Fp(1)33 b Fd(and)h(al)5 b(l)33 b Fo(n)1378 4012 y Fl(0)1417 3998 y Fo(;)15 b(n)1512 4012 y Fl(2)1577 3998 y Ff(2)25 b Fk(N)32 b Fd(and)h(for)h(al)5 b(l)33 b Fp(0)25 b Ff(\024)g Fo(n)2434 4012 y Fl(2)2499 3998 y Ff(\024)g Fo(S)2651 4012 y Fl(2)2710 3998 y Ff(\000)20 b Fp(1)33 b Fd(and)g(al)5 b(l)33 b Fo(n)3240 4012 y Fl(0)3280 3998 y Fo(;)15 b(n)3375 4012 y Fl(1)3439 3998 y Ff(2)25 b Fk(N)p Fd(.)216 4111 y(Then)33 b(the)g(formal)h(series)1175 4088 y Fp(^)1149 4111 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1382 4125 y Fl(1)1422 4111 y Fo(;)g(z)1504 4125 y Fl(2)1544 4111 y Fp(\))p Fd(,)33 b(solution)g(of)g Fp(\()p Fo(P)13 b(D)s(E)5 b Fp(\))p Fd(,)33 b(is)g Fp(1)p Fo(=)p Fp(\()p Fo(K)27 b Ff(\000)20 b Fp(1\))p Fd(-summable)34 b(with)f(r)-5 b(esp)g(e)g(ct)75 4224 y(to)33 b Fo(t)g Fd(in)f(the)h(dir)-5 b(e)g(ction)34 b Fo(d)26 b Ff(2)f Fk(R)p Fd(.)75 4465 y Fe(Pro)s(of)37 b Fp(The)f(goal)i(of)e(the)h(pro)s(of)e(is)i(to)g(sho) m(w)f(that)h(the)f(conditions)h Fe(iii\))f Fp(in)g(Prop)s(osition)g(1)h (are)g(ful\014lled.)75 4578 y(First)d(of)g(all,)i(under)d(the)h(giv)m (en)h(assumptions,)f(it)g(follo)m(ws)h(from)f(the)g(relation)h(\()p Fo(D)s(R)q Fp(\))f(and)f(Prop)s(osition)h(2)75 4691 y(\(using)39 b(the)g(general)h(results)f(on)g(deriv)-5 b(ation)39 b(of)g Fo(k)s Fp(-summable)g(series,)j(see)e([2]\),)i(that)e(the)f (formal)g(series)101 4781 y(^)75 4804 y Fo(X)150 4818 y Fn(\026)192 4827 y Fm(1)227 4818 y Fn(;\026)289 4827 y Fm(2)328 4804 y Fp(\()p Fo(t)p Fp(\))e(are)f(1)p Fo(=)p Fp(\()p Fo(K)c Ff(\000)23 b Fp(1\)-summable)37 b(in)e(the)h(direction)h Fo(d)e Ff(2)f Fk(R)p Fp(,)j(for)f(all)h Fo(\026)2795 4818 y Fl(1)2834 4804 y Fo(;)15 b(\026)2929 4818 y Fl(2)3003 4804 y Ff(\025)34 b Fp(0.)58 b(Moreo)m(v)m(er,)39 b(their)75 4917 y(sums)29 b Fo(X)378 4931 y Fn(\026)420 4940 y Fm(1)456 4931 y Fn(;\026)518 4940 y Fm(2)556 4917 y Fp(\()p Fo(t)p Fp(\))i(are)g(holomorphic)g(in)f(the)g(sectorial)i(region)f Fo(G)p Fp(.)216 5031 y(The)j(rest)g(of)g(the)g(pro)s(of)g(will)g (consist)h(in)f(pro)m(ving)g(that)h(the)f(estimates)i(in)d Fe(iii\))h Fp(of)g(Prop)s(osition)h(1)f(are)75 5143 y(satis\014ed)c (for)h(the)f(formal)h(series)1276 5120 y(^)1249 5143 y Fo(X)1324 5157 y Fn(\026)1366 5166 y Fm(1)1402 5157 y Fn(;\026)1464 5166 y Fm(2)1503 5143 y Fp(\()p Fo(t)p Fp(\).)216 5257 y(W)-8 b(e)32 b(consider)e(no)m(w)g(the)h(sequence)g (of)f(p)s(ositiv)m(e)i(real)f(n)m(um)m(b)s(ers)1374 5463 y Fo(w)1439 5477 y Fn(n)1482 5486 y Fm(0)1516 5477 y Fn(;n)1579 5486 y Fm(1)1613 5477 y Fn(;n)1676 5486 y Fm(2)1740 5463 y Fp(=)25 b(sup)1845 5551 y Fn(t)p Fj(2)1929 5534 y Fl(\026)1917 5551 y Fn(S)1988 5463 y Ff(j)p Fo(@)2066 5423 y Fn(n)2109 5432 y Fm(0)2061 5487 y Fn(t)2148 5463 y Fo(X)2223 5477 y Fn(n)2266 5486 y Fm(1)2301 5477 y Fn(;n)2364 5486 y Fm(2)2402 5463 y Fp(\()p Fo(t)p Fp(\))p Ff(j)p Fo(;)75 5740 y Fp(for)30 b Fo(n)269 5754 y Fl(0)308 5740 y Fo(;)15 b(n)403 5754 y Fl(1)443 5740 y Fo(;)g(n)538 5754 y Fl(2)602 5740 y Ff(\025)25 b Fp(0,)31 b(where)1077 5717 y(\026)1062 5740 y Fo(S)k Fp(is)c(a)g(closed)g(sector)g(in)f Fo(G)p Fp(.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 75 100 a Fp(8)216 399 y(F)-8 b(rom)31 b(the)g(Leibniz)f(form)m(ula,)h(w)m(e)g(ha)m(v)m(e)g(that)93 665 y Fo(@)146 624 y Fn(n)189 633 y Fm(0)141 689 y Fn(t)228 665 y Fp(\()p Fo(b)302 680 y Fn(q)r(;k)393 689 y Fm(1)427 680 y Fn(;k)484 689 y Fm(2)518 680 y Fn(;l)559 689 y Fm(1)593 680 y Fn(;l)634 689 y Fm(2)673 665 y Fp(\()p Fo(t)p Fp(\))786 602 y Fo(@)839 557 y Fn(q)834 626 y(t)877 602 y Fo(X)952 617 y Fn(n)995 626 y Fm(1)1030 617 y Fl(+)p Fn(k)1122 626 y Fm(1)1156 617 y Fn(;n)1219 626 y Fm(2)1253 617 y Fl(+)p Fn(k)1345 626 y Fm(2)1384 602 y Fp(\()p Fo(t)p Fp(\))p 786 644 702 4 v 1017 727 a Fo(n)1072 741 y Fl(1)1111 727 y Fp(!)p Fo(n)1191 741 y Fl(2)1231 727 y Fp(!)1497 665 y(\))26 b(=)1681 550 y Fn(n)1724 559 y Fm(0)1654 578 y Fi(X)1654 776 y Fn(h)p Fl(=0)1966 603 y Fo(n)2021 617 y Fl(0)2060 603 y Fp(!)p 1810 644 432 4 v 1810 727 a(\()p Fo(n)1900 741 y Fl(0)1960 727 y Ff(\000)20 b Fo(h)p Fp(\)!)p Fo(h)p Fp(!)2251 665 y Fo(@)2304 624 y Fn(n)2347 633 y Fm(0)2382 624 y Fj(\000)p Fn(h)2299 689 y(t)2482 665 y Fo(b)2521 680 y Fn(q)r(;k)2612 689 y Fm(1)2646 680 y Fn(;k)2703 689 y Fm(2)2736 680 y Fn(;l)2777 689 y Fm(1)2812 680 y Fn(;l)2853 689 y Fm(2)2891 665 y Fp(\()p Fo(t)p Fp(\))3004 602 y Fo(@)3057 557 y Fn(h)p Fl(+)p Fn(q)3052 626 y(t)3192 602 y Fo(X)3267 617 y Fn(n)3310 626 y Fm(1)3344 617 y Fl(+)p Fn(k)3436 626 y Fm(1)3471 617 y Fn(;n)3534 626 y Fm(2)3568 617 y Fl(+)p Fn(k)3660 626 y Fm(2)3698 602 y Fp(\()p Fo(t)p Fp(\))p 3004 644 797 4 v 3284 727 a Fo(n)3339 741 y Fl(1)3378 727 y Fp(!)p Fo(n)3458 741 y Fl(2)3497 727 y Fp(!)3811 665 y Fo(:)75 943 y Fp(On)30 b(the)g(other)h(hand,)e(b)m(y)h(classical)j(estimates,)f (w)m(e)f(ha)m(v)m(e)h(that)1685 1124 y Fo(n)1740 1138 y Fl(0)1779 1124 y Fp(!)p 1685 1165 120 4 v 1706 1248 a Fo(h)p Fp(!)1840 1186 y Ff(\024)1945 1124 y Fp(\()p Fo(K)7 b(n)2119 1138 y Fl(0)2159 1124 y Fp(\)!)p 1945 1165 274 4 v 1966 1248 a(\()p Fo(K)g(h)p Fp(\)!)2229 1186 y Fo(;)75 1434 y Fp(for)30 b(all)h Fo(n)395 1448 y Fl(0)460 1434 y Ff(\025)25 b Fp(0)30 b(and)g(all)h(0)26 b Ff(\024)f Fo(h)h Ff(\024)f Fo(n)1330 1448 y Fl(0)1369 1434 y Fp(.)40 b(F)-8 b(rom)31 b(the)g(latter)h(estimates)g(and)d(\()p Fo(C)7 b(I)g Fp(\),)31 b(w)m(e)g(deduce)f(that)173 1713 y(sup)182 1801 y Fn(t)p Fj(2)266 1784 y Fl(\026)254 1801 y Fn(S)325 1581 y Fi(\014)325 1636 y(\014)325 1690 y(\014)325 1745 y(\014)355 1713 y Fo(@)408 1672 y Fn(n)451 1681 y Fm(0)403 1737 y Fn(t)490 1713 y Fp(\()p Fo(b)564 1728 y Fn(q)r(;k)655 1737 y Fm(1)690 1728 y Fn(;k)747 1737 y Fm(2)781 1728 y Fn(;l)822 1737 y Fm(1)856 1728 y Fn(;l)897 1737 y Fm(2)935 1713 y Fp(\()p Fo(t)p Fp(\))1048 1650 y Fo(@)1101 1606 y Fn(q)1096 1674 y(t)1140 1650 y Fo(X)1215 1665 y Fn(n)1258 1674 y Fm(1)1293 1665 y Fl(+)p Fn(k)1385 1674 y Fm(1)1419 1665 y Fn(;n)1482 1674 y Fm(2)1516 1665 y Fl(+)p Fn(k)1608 1674 y Fm(2)1646 1650 y Fp(\()p Fo(t)p Fp(\))p 1048 1692 702 4 v 1280 1775 a Fo(n)1335 1789 y Fl(1)1374 1775 y Fp(!)p Fo(n)1454 1789 y Fl(2)1493 1775 y Fp(!)1760 1713 y(\))1795 1581 y Fi(\014)1795 1636 y(\014)1795 1690 y(\014)1795 1745 y(\014)1430 2046 y Ff(\024)1553 1932 y Fn(n)1596 1941 y Fm(0)1526 1960 y Fi(X)1527 2158 y Fn(h)p Fl(=0)1683 1985 y Fp(\()p Fo(K)7 b(n)1857 1999 y Fl(0)1896 1985 y Fp(\)!)p 1683 2025 274 4 v 1704 2109 a(\()p Fo(K)g(h)p Fp(\)!)1966 2046 y Ff(j)p Fo(b)p Ff(j)2055 2061 y Fn(q)r(;k)2146 2070 y Fm(1)2181 2061 y Fn(;k)2238 2070 y Fm(2)2272 2061 y Fn(;l)2313 2070 y Fm(1)2347 2061 y Fn(;l)2388 2070 y Fm(2)2422 2061 y Fn(;n)2485 2070 y Fm(0)2519 2061 y Fj(\000)p Fn(h)2629 1952 y Fp(sup)2766 1975 y Fn(t)p Fj(2)2850 1959 y Fl(\026)2838 1975 y Fn(S)2904 1847 y Fi(\014)2904 1902 y(\014)2904 1956 y(\014)2934 1952 y Fo(@)2987 1908 y Fn(h)p Fl(+)p Fn(q)2982 1976 y(t)3121 1952 y Fo(X)3196 1967 y Fn(n)3239 1976 y Fm(1)3274 1967 y 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Fi(X)1156 2849 y Fl(\()p Fn(k)1220 2858 y Fm(1)1254 2849 y Fn(;k)1311 2858 y Fm(2)1345 2849 y Fl(\))p Fj(2S)1462 2857 y Fc(q)1616 2647 y Fi(X)1512 2845 y Fn(l)1533 2854 y Fm(1)1568 2845 y Fl(+)p Fn(n)1666 2854 y Fm(1)1700 2845 y Fl(=)p Fn(m)1817 2854 y Fm(1)1971 2647 y Fi(X)1867 2845 y Fn(l)1888 2854 y Fm(2)1923 2845 y Fl(+)p Fn(n)2021 2854 y Fm(2)2055 2845 y Fl(=)p Fn(m)2172 2854 y Fm(2)2249 2619 y Fn(n)2292 2628 y Fm(0)2222 2647 y Fi(X)2222 2845 y Fn(h)p Fl(=0)2379 2670 y Ff(j)p Fo(b)p Ff(j)2468 2685 y Fn(q)r(;k)2559 2694 y Fm(1)2593 2685 y Fn(;k)2650 2694 y Fm(2)2684 2685 y Fn(;l)2725 2694 y Fm(1)2759 2685 y Fn(;l)2800 2694 y Fm(2)2834 2685 y Fn(;n)2897 2694 y Fm(0)2931 2685 y Fj(\000)p Fn(h)p 2379 2713 653 4 v 2589 2796 a Fp(\()p Fo(K)g(h)p Fp(\)!)3051 2670 y Fo(w)3116 2685 y Fn(h)p Fl(+)p Fn(q)r(;n)3309 2694 y Fm(1)3343 2685 y Fl(+)p Fn(k)3435 2694 y Fm(1)3469 2685 y Fn(;n)3532 2694 y Fm(2)3566 2685 y Fl(+)p Fn(k)3658 2694 y Fm(2)p 3051 2713 646 4 v 3184 2796 a Fp(\()p Fo(n)3274 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Ff(2)g(D)925 4718 y Fn(\032)996 4704 y Fp(if)1367 4817 y Fo(E)5 b Fp(\()p Fo(t;)15 b Fp(0)p Fo(;)g(z)1674 4831 y Fl(2)1716 4817 y Fp(\))p Fo(;)g(:)g(:)g(:)i(;)e(@)2006 4780 y Fn(S)2049 4789 y Fm(1)2084 4780 y Fj(\000)p Fl(1)2001 4840 y Fn(z)2034 4849 y Fm(1)2178 4817 y Fo(E)5 b Fp(\()p Fo(t;)15 b Fp(0)p Fo(;)g(z)2485 4831 y Fl(2)2527 4817 y Fp(\))75 4982 y(are)31 b(prescrib)s(ed)e(holomorphic)h(functions)g(on)g Ff(fj)p Fo(t)p Ff(j)c Fo(<)f(R)q Fp(\()p Fo(\032)p Fp(\))p Ff(g)c(\002)f Fo(D)2372 4996 y Fn(\032)2408 5005 y Fm(2)2478 4982 y Fp(and)1367 5182 y Fo(E)5 b Fp(\()p Fo(t;)15 b(z)1589 5196 y Fl(1)1630 5182 y Fo(;)g Fp(0\))p Fo(;)g(:)g(:)g(:)j(;)d(@)2006 5144 y Fn(S)2049 5153 y Fm(2)2084 5144 y Fj(\000)p Fl(1)2001 5204 y Fn(z)2034 5213 y Fm(2)2178 5182 y Fo(E)5 b Fp(\()p Fo(t;)15 b(z)2400 5196 y Fl(1)2441 5182 y Fo(;)g Fp(0\))75 5381 y(are)23 b(prescrib)s(ed)f(holomorphic)h(functions)f(on)h Ff(fj)p Fo(t)p Ff(j)j Fo(<)f(R)q Fp(\()p Fo(\032)p Fp(\))p Ff(g)5 b(\002)g Fo(D)2304 5395 y 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Fo(;)g Fp(0\),)75 5740 y(0)34 b Ff(\024)g Fo(j)39 b Ff(\024)34 b Fo(S)496 5754 y Fl(2)559 5740 y Ff(\000)23 b Fp(1)36 b(to)g(b)s(e)f(of)h(the)f(form)g Fo(@)1522 5696 y Fn(j)1517 5752 y(z)1550 5761 y Fm(2)1589 5740 y Fo(E)5 b Fp(\()p Fo(t;)15 b(z)1811 5754 y Fl(1)1852 5740 y Fo(;)g Fp(0\))35 b(=)e Fo(W)2197 5754 y Fn(j)2234 5740 y Fp(\()p Fo(t)2302 5707 y Fn(K)2370 5740 y Fo(;)15 b(z)2452 5754 y Fl(1)2492 5740 y Fp(\),)38 b(where)d Fo(W)2944 5754 y Fn(j)3015 5740 y Fp(are)h(holomorphic)g(in)f(a)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fp(9)75 399 y(neigh)m(b)s(orho)s(o)s (d)24 b(of)h(the)g(origin.)40 b(F)-8 b(rom)25 b(the)h(latter)g (assumptions)e(and)h(due)f(the)i(fact)g(that)f(the)g(co)s(e\016cien)m (ts)i(of)75 511 y(the)f(equation)h(\()p Fo(C)7 b(K)g Fp(\))25 b(are)h(also)h(of)f(the)g(form)f Fo(\013)p Fp(\()p Fo(t)1757 478 y Fn(K)1827 511 y Fo(;)15 b(z)1909 525 y Fl(1)1949 511 y Fo(;)g(z)2031 525 y Fl(2)2071 511 y Fp(\),)27 b(w)m(e)f(deduce)g(that)g(the)g(solution)g 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Fm(2)1141 1642 y Fo(@)1194 1598 y Fn(q)r(K)1189 1666 y(t)1297 1642 y Fo(V)20 b Fp(\()p Fo(t)1438 1604 y Fn(K)1507 1642 y Fo(;)15 b(z)1589 1656 y Fl(1)1629 1642 y Fo(;)g(z)1711 1656 y Fl(2)1751 1642 y Fp(\))475 1811 y(=)671 1725 y Fi(X)571 1916 y Fn(m)633 1925 y Fm(0)668 1916 y Fn(;m)750 1925 y Fm(1)785 1916 y Fn(;m)867 1925 y Fm(2)901 1811 y Fp(\()1041 1725 y Fi(X)936 1923 y Fn(l)957 1932 y Fm(1)992 1923 y Fl(+)p Fn(n)1090 1932 y Fm(1)1125 1923 y Fl(=)p Fn(m)1242 1932 y Fm(1)1396 1725 y Fi(X)1292 1923 y Fn(l)1313 1932 y Fm(2)1347 1923 y Fl(+)p Fn(n)1445 1932 y Fm(2)1480 1923 y Fl(=)p Fn(m)1597 1932 y Fm(2)1751 1725 y Fi(X)1647 1923 y Fn(l)1668 1932 y Fm(0)1702 1923 y Fl(+)p Fn(n)1800 1932 y Fm(0)1835 1923 y Fl(=)p Fn(m)1952 1932 y Fm(0)2002 1811 y Ff(j)p Fo(b)p Ff(j)2091 1826 y Fn(q)r(;k)2182 1835 y Fm(1)2216 1826 y Fn(;k)2273 1835 y Fm(2)2307 1826 y Fn(;l)2348 1835 y Fm(1)2382 1826 y Fn(;l)2423 1835 y Fm(2)2458 1826 y Fn(;l)2499 1835 y Fm(0)2547 1748 y Fo(v)2591 1763 y Fn(n)2634 1772 y Fm(0)2668 1763 y Fl(+)p Fn(q)r(;n)2820 1772 y Fm(1)2854 1763 y Fl(+)p Fn(k)2946 1772 y Fm(1)2980 1763 y Fn(;n)3043 1772 y Fm(2)3077 1763 y Fl(+)p Fn(k)3169 1772 y Fm(2)p 2547 1790 661 4 v 2621 1874 a Fp(\()p Fo(K)7 b(n)2795 1888 y Fl(0)2834 1874 y Fp(\)!)p Fo(n)2949 1888 y Fl(1)2989 1874 y Fp(!)p Fo(n)3069 1888 y Fl(2)3108 1874 y Fp(!)3218 1811 y(\))p Fo(t)3286 1774 y Fn(K)e(m)3413 1783 y Fm(0)3451 1811 y Fo(z)3497 1771 y Fn(m)3559 1780 y Fm(1)3493 1837 y Fl(1)3599 1811 y Fo(z)3645 1771 y Fn(m)3707 1780 y Fm(2)3641 1837 y Fl(2)3746 1811 y Fo(:)75 2111 y Fp(F)-8 b(rom)24 b(this,)i(w)m(e)e(deduce)f(a)i(m)m(ultiv)-5 b(ariate)25 b(linear)g(recurrence)e(\()p Fo(M)10 b(R)q Fp(\))24 b(satis\014ed)g(b)m(y)g(the)g(co)s(e\016cien)m(ts)i Fo(v)3554 2125 y Fn(n)3597 2134 y Fm(0)3631 2125 y Fn(;n)3694 2134 y Fm(1)3728 2125 y Fn(;n)3791 2134 y Fm(2)3829 2111 y Fp(,)254 2349 y Fo(v)298 2363 y Fn(n)341 2372 y Fm(0)376 2363 y Fn(;m)458 2372 y Fm(1)492 2363 y Fl(+)p Fn(S)590 2372 y Fm(1)625 2363 y Fn(;m)707 2372 y Fm(2)741 2363 y Fl(+)p Fn(S)839 2372 y Fm(2)p 254 2391 624 4 v 285 2474 a Fp(\()p Fo(K)7 b(n)459 2488 y Fl(0)498 2474 y Fp(\)!)p Fo(m)638 2488 y Fl(1)678 2474 y Fp(!)p Fo(m)783 2488 y Fl(2)822 2474 y Fp(!)913 2412 y(=)1047 2293 y Fn(Q)1009 2325 y Fi(X)1013 2520 y Fn(q)r Fl(=0)1261 2325 y Fi(X)1156 2527 y Fl(\()p Fn(k)1220 2536 y Fm(1)1254 2527 y Fn(;k)1311 2536 y Fm(2)1345 2527 y Fl(\))p Fj(2S)1462 2535 y Fc(q)1616 2325 y Fi(X)1512 2523 y Fn(l)1533 2532 y Fm(1)1568 2523 y Fl(+)p Fn(n)1666 2532 y Fm(1)1700 2523 y Fl(=)p Fn(m)1817 2532 y Fm(1)1971 2325 y Fi(X)1867 2523 y Fn(l)1888 2532 y Fm(2)1923 2523 y Fl(+)p Fn(n)2021 2532 y Fm(2)2055 2523 y Fl(=)p Fn(m)2172 2532 y Fm(2)2249 2297 y Fn(n)2292 2306 y Fm(0)2222 2325 y Fi(X)2222 2523 y Fn(h)p Fl(=0)2379 2348 y Ff(j)p Fo(b)p Ff(j)2468 2363 y Fn(q)r(;k)2559 2372 y Fm(1)2593 2363 y Fn(;k)2650 2372 y Fm(2)2684 2363 y Fn(;l)2725 2372 y Fm(1)2759 2363 y Fn(;l)2800 2372 y Fm(2)2834 2363 y Fn(;n)2897 2372 y Fm(0)2931 2363 y Fj(\000)p Fn(h)p 2379 2391 653 4 v 2589 2474 a Fp(\()p Fo(K)g(h)p Fp(\)!)3051 2348 y Fo(v)3095 2363 y Fn(h)p Fl(+)p Fn(q)r(;n)3288 2372 y Fm(1)3322 2363 y Fl(+)p Fn(k)3414 2372 y Fm(1)3448 2363 y Fn(;n)3511 2372 y Fm(2)3545 2363 y Fl(+)p Fn(k)3637 2372 y Fm(2)p 3051 2391 625 4 v 3173 2474 a Fp(\()p Fo(n)3263 2488 y Fl(1)3302 2474 y Fp(\)!\()p Fo(n)3452 2488 y Fl(2)3492 2474 y Fp(\)!)75 2718 y(for)35 b(all)h Fo(n)405 2732 y Fl(0)444 2718 y Fo(;)15 b(m)564 2732 y Fl(1)604 2718 y Fo(;)g(m)724 2732 y Fl(2)797 2718 y Ff(\025)34 b Fp(0.)56 b(F)-8 b(rom)36 b(the)f(holomorph)m(y)g(of)h Fo(V)20 b Fp(,)37 b(w)m(e)f(deduce)f(the)g(follo)m(wing)i(estimates)g(for)e (the)75 2831 y(co)s(e\016cien)m(ts)d Fo(v)579 2845 y Fn(n)622 2854 y Fm(0)657 2845 y Fn(;n)720 2854 y Fm(1)754 2845 y Fn(;n)817 2854 y Fm(2)855 2831 y Fp(.)40 b(There)30 b(exist)h Fo(C)q(;)1529 2808 y Fp(~)1504 2831 y Fo(K)i(>)25 b Fp(0,)31 b(suc)m(h)f(that)75 3047 y(\()p Fo(E)5 b(s)p Fp(\))1083 b Ff(j)p Fo(v)1412 3061 y Fn(n)1455 3070 y Fm(0)1490 3061 y Fn(;n)1553 3070 y Fm(1)1587 3061 y Fn(;n)1650 3070 y Fm(2)1688 3047 y Ff(j)25 b(\024)g Fo(C)1930 3024 y Fp(~)1906 3047 y Fo(K)1990 3009 y Fj(j)p Fn(n)p Fj(j)p Fl(+)p Fn(n)2171 3018 y Fm(0)2208 3047 y Fp(\()p Fo(K)7 b(n)2382 3061 y Fl(0)2421 3047 y Fp(\)!)p Fo(n)p Fp(!)p Fo(;)75 3253 y Fp(where)30 b Fo(n)25 b Fp(=)g(\()p Fo(n)604 3267 y Fl(1)643 3253 y Fo(;)15 b(n)738 3267 y Fl(2)777 3253 y Fp(\),)31 b(for)f(all)i Fo(n)25 b Ff(2)f Fk(N)1365 3220 y Fl(2)1435 3253 y Fp(and)30 b Fo(n)1667 3267 y Fl(0)1731 3253 y Ff(2)25 b Fk(N)p Fp(.)216 3417 y(F)-8 b(rom)31 b(Theorem)f(H,)g(w)m(e)h(see)f(that)h(the)f(sequence)h(\()p Fo(v)2051 3431 y Fn(n)2094 3440 y Fm(0)2129 3431 y Fn(;n)2192 3440 y Fm(1)2226 3431 y Fn(;n)2289 3440 y Fm(2)2327 3417 y Fp(\))2362 3431 y Fn(n)2405 3440 y Fm(0)2440 3431 y Fn(;n)2503 3440 y Fm(1)2537 3431 y Fn(;n)2600 3440 y Fm(2)2668 3417 y Fp(is)f(uniquely)g(determined)f(b)m(y)h(the)75 3530 y(sequences)j(\()p Fo(v)569 3544 y Fn(n)612 3553 y Fm(0)647 3544 y Fn(;n)710 3553 y Fm(1)744 3544 y Fn(;n)807 3553 y Fm(2)845 3530 y Fp(\))880 3544 y Fn(n)923 3553 y Fm(0)957 3544 y Fn(;n)1020 3553 y Fm(2)1059 3530 y Fp(,)g(0)c Ff(\024)f Fo(n)1345 3544 y Fl(1)1413 3530 y Ff(\024)h Fo(S)1569 3544 y Fl(1)1629 3530 y Ff(\000)22 b Fp(1)33 b(and)e(the)i(sequences)g(\()p Fo(v)2631 3544 y Fn(n)2674 3553 y Fm(0)2709 3544 y Fn(;n)2772 3553 y Fm(1)2806 3544 y Fn(;n)2869 3553 y Fm(2)2907 3530 y Fp(\))2942 3544 y Fn(n)2985 3553 y Fm(0)3020 3544 y Fn(;n)3083 3553 y Fm(1)3121 3530 y Fp(,)g(0)c Ff(\024)g Fo(n)3408 3544 y Fl(2)3475 3530 y Ff(\024)g Fo(S)3631 3544 y Fl(2)3692 3530 y Ff(\000)21 b Fp(1.)75 3643 y(W)-8 b(e)32 b(c)m(ho)s(ose)f(no)m (w)f(the)h(latter)h(sequences)e(as)h(follo)m(ws.)1549 3849 y Fo(v)1593 3863 y Fn(n)1636 3872 y Fm(0)1670 3863 y Fn(;n)1733 3872 y Fm(1)1767 3863 y Fn(;n)1830 3872 y Fm(2)1894 3849 y Fp(:=)25 b Fo(w)2080 3863 y Fn(n)2123 3872 y Fm(0)2157 3863 y Fn(;n)2220 3872 y Fm(1)2254 3863 y Fn(;n)2317 3872 y Fm(2)2356 3849 y Fo(;)75 4055 y Fp(for)30 b(all)h(0)26 b Ff(\024)f Fo(n)562 4069 y Fl(1)626 4055 y Ff(\024)g Fo(S)778 4069 y Fl(1)838 4055 y Ff(\000)20 b Fp(1,)31 b Fo(n)1085 4069 y Fl(0)1124 4055 y Fo(;)15 b(n)1219 4069 y Fl(2)1283 4055 y Ff(\025)25 b Fp(0)31 b(and)f(for)g(all)h(0)26 b Ff(\024)f Fo(n)2119 4069 y Fl(2)2183 4055 y Ff(\024)g Fo(S)2335 4069 y Fl(2)2394 4055 y Ff(\000)20 b Fp(1,)31 b Fo(n)2641 4069 y Fl(0)2680 4055 y Fo(;)15 b(n)2775 4069 y Fl(1)2840 4055 y Ff(\025)25 b Fp(0.)216 4169 y(F)-8 b(rom)31 b(the)g(estimates)h(in)e(the)g (assumptions)g(of)g(the)h(theorem)g(1,)g(w)m(e)g(deduce)f(that)h(the)f (series)1367 4375 y Fo(E)5 b Fp(\()p Fo(t;)15 b Fp(0)p Fo(;)g(z)1674 4389 y Fl(2)1716 4375 y Fp(\))p Fo(;)g(:)g(:)g(:)i(;)e(@) 2006 4337 y Fn(S)2049 4346 y Fm(1)2084 4337 y Fj(\000)p Fl(1)2001 4397 y Fn(z)2034 4406 y Fm(1)2178 4375 y Fo(E)5 b Fp(\()p Fo(t;)15 b Fp(0)p Fo(;)g(z)2485 4389 y Fl(2)2527 4375 y Fp(\))75 4581 y(are)31 b(con)m(v)m(ergen)m(t)i(on)d Ff(fj)p Fo(t)p Ff(j)c Fo(<)f(R)q Fp(\()p Fo(\032)p Fp(\))p Ff(g)c(\002)f Fo(D)1477 4595 y Fn(\032)1513 4604 y Fm(2)1582 4581 y Fp(and)1367 4787 y Fo(E)5 b Fp(\()p Fo(t;)15 b(z)1589 4801 y Fl(1)1630 4787 y Fo(;)g Fp(0\))p Fo(;)g(:)g(:)g(:)j(;)d(@)2006 4749 y Fn(S)2049 4758 y Fm(2)2084 4749 y Fj(\000)p Fl(1)2001 4809 y Fn(z)2034 4818 y Fm(2)2178 4787 y Fo(E)5 b Fp(\()p Fo(t;)15 b(z)2400 4801 y Fl(1)2441 4787 y Fo(;)g Fp(0\))75 4993 y(are)37 b(con)m(v)m(ergen)m(t)h(on)f Ff(fj)p Fo(t)p Ff(j)e Fo(<)g(R)q Fp(\()p Fo(\032)p Fp(\))p Ff(g)25 b(\002)f Fo(D)1522 5007 y Fn(\032)1558 5016 y Fm(1)1597 4993 y Fp(.)59 b(Moreo)m(v)m(er,)40 b(due)35 b(to)i(the)g(relations)g(\()p Fo(M)10 b(R)q(I)d Fp(\))37 b(and)f(\()p Fo(M)10 b(R)q Fp(\),)38 b(w)m(e)75 5106 y(deduce)30 b(that)1561 5221 y Fo(w)1626 5235 y Fn(n)1669 5244 y Fm(0)1704 5235 y Fn(;n)1767 5244 y Fm(1)1801 5235 y Fn(;n)1864 5244 y Fm(2)1927 5221 y Ff(\024)25 b Fo(v)2067 5235 y Fn(n)2110 5244 y Fm(0)2145 5235 y Fn(;n)2208 5244 y Fm(1)2242 5235 y Fn(;n)2305 5244 y Fm(2)2343 5221 y Fo(;)75 5390 y Fp(for)30 b(all)h Fo(n)395 5404 y Fl(0)434 5390 y Fo(;)15 b(n)529 5404 y Fl(1)569 5390 y Fo(;)g(n)664 5404 y Fl(2)703 5390 y Fp(.)216 5503 y(F)-8 b(rom)26 b(the)f(estimates)i(\()p Fo(E)5 b(s)p Fp(\),)26 b(w)m(e)g(\014nally)f(deduce)f(that)i(there)f (exist)h Fo(C)q(;)2688 5480 y Fp(~)2663 5503 y Fo(K)33 b(>)25 b Fp(0,)i(suc)m(h)d(that)i(the)f(sequence)75 5616 y Fo(w)140 5630 y Fn(n)183 5639 y Fm(0)217 5630 y Fn(;n)280 5639 y Fm(1)315 5630 y Fn(;n)378 5639 y Fm(2)446 5616 y Fp(satis\014es)1358 5740 y Fo(w)1423 5754 y Fn(n)1466 5763 y Fm(0)1500 5754 y Fn(;n)1563 5763 y Fm(1)1597 5754 y Fn(;n)1660 5763 y Fm(2)1724 5740 y Ff(\024)g Fo(C)1915 5717 y Fp(~)1892 5740 y Fo(K)1976 5703 y Fj(j)p Fn(n)p Fj(j)p Fl(+)p Fn(n)2157 5712 y Fm(0)2193 5740 y Fp(\()p Fo(K)7 b(n)2367 5754 y Fl(0)2406 5740 y Fp(\)!)p Fo(n)p Fp(!)p Fo(;)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 75 100 a Fp(10)75 399 y(where)30 b Fo(n)25 b Fp(=)g(\()p Fo(n)604 413 y Fl(1)643 399 y Fo(;)15 b(n)738 413 y Fl(2)777 399 y Fp(\),)31 b(for)f(all)i Fo(n)25 b Ff(2)f Fk(N)1365 366 y Fl(2)1435 399 y Fp(and)30 b Fo(n)1667 413 y Fl(0)1731 399 y Ff(2)25 b Fk(N)p Fp(.)216 515 y Fd(Final)5 b(ly,)32 b(we)f(c)-5 b(onclude)31 b(that)h(the)g(c)-5 b(ondition)32 b Fe(iii\))e Fd(in)h(Pr)-5 b(op)g(osition)33 b(1)f(is)e(satis\014e)-5 b(d)33 b(for)e(the)g(formal)h(series)101 605 y Fp(^)75 628 y Fo(X)150 642 y Fn(n)193 651 y Fm(1)228 642 y Fn(;n)291 651 y Fm(2)329 628 y Fp(\()p Fo(t)p Fp(\),)d(so)f(that) h(the)f(formal)g(series)1494 605 y(^)1468 628 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1700 642 y Fl(1)1741 628 y Fo(;)g(z)1823 642 y Fl(2)1863 628 y Fp(\),)29 b(solution)f(of)g(\()p Fo(P)13 b(D)s(E)5 b Fp(\),)29 b(is)f(1)p Fo(=)p Fp(\()p Fo(K)23 b Ff(\000)15 b Fp(1\)-summable)29 b(with)75 741 y(resp)s(ect)h(to)h Fo(t)g Fp(in)f(the)g(direction)h Fo(d)26 b Ff(2)f Fk(R)p Fp(.)2304 b Fa(2)75 916 y Fe(Example:)40 b Fp(In)30 b(the)g(case)i(of)e(the)h(heat)g(equation)1176 1133 y Fo(@)1229 1095 y Fl(2)1224 1155 y Fn(z)1257 1164 y Fm(1)1322 1110 y Fp(^)1296 1133 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)1529 1147 y Fl(1)1569 1133 y Fo(;)g(z)1651 1147 y Fl(2)1691 1133 y Fp(\))26 b(=)f(\()p Ff(\000)p Fo(@)2007 1095 y Fl(2)2002 1155 y Fn(z)2035 1164 y Fm(2)2094 1133 y Fp(+)20 b Fo(@)2233 1147 y Fn(t)2262 1133 y Fp(\))2324 1110 y(^)2297 1133 y Fo(X)8 b Fp(\()p Fo(t;)15 b(z)2530 1147 y Fl(1)2571 1133 y Fo(;)g(z)2653 1147 y Fl(2)2693 1133 y Fp(\))p Fo(;)75 1349 y Fp(one)31 b(can)f(construct)h(formal)g (series)f(solutions)1246 1586 y(^)1220 1609 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)1452 1623 y Fl(1)1493 1609 y Fo(;)g(z)1575 1623 y Fl(2)1615 1609 y Fp(\))26 b(=)1838 1523 y Fi(X)1771 1719 y Fn(n)1814 1728 y Fm(1)1849 1719 y Fn(;n)1912 1728 y Fm(2)1946 1719 y Fj(\025)p Fl(0)2087 1524 y Fp(^)2061 1547 y Fo(X)2136 1561 y Fn(n)2179 1570 y Fm(1)2214 1561 y Fn(;n)2277 1570 y Fm(2)2315 1547 y Fp(\()p Fo(t)p Fp(\))p 2061 1588 358 4 v 2121 1672 a Fo(n)2176 1686 y Fl(1)2215 1672 y Fp(!)p Fo(n)2295 1686 y Fl(2)2334 1672 y Fp(!)2429 1609 y Fo(z)2475 1569 y Fn(n)2518 1578 y Fm(1)2471 1635 y Fl(1)2556 1609 y Fo(z)2602 1569 y Fn(n)2645 1578 y Fm(2)2598 1635 y Fl(2)2684 1609 y Fo(;)75 1916 y Fp(whic)m(h)31 b(are)h(1-summable)g(with)f(resp)s(ect)g(to)h Fo(t)g Fp(in)f(a)h(direction)g Fo(d)27 b Ff(2)g Fk(R)p Fp(,)k(under)f(the)i (follo)m(wing)h(assumptions:)75 2029 y(The)40 b(series)551 2006 y(^)525 2029 y Fo(X)600 2043 y Fl(0)p Fn(;\026)697 2052 y Fm(2)737 2029 y Fp(\()p Fo(t)p Fp(\))g(and)1093 2006 y(^)1067 2029 y Fo(X)1142 2043 y Fl(1)p Fn(;\026)1239 2052 y Fm(2)1278 2029 y Fp(\()p Fo(t)p Fp(\))h(are)f(1-summable)h(with) f(resp)s(ect)g(to)h Fo(t)f Fp(in)g(a)g(direction)h Fo(d)h Ff(2)f Fk(R)p Fp(,)i(for)75 2142 y(all)38 b Fo(\026)263 2156 y Fl(2)340 2142 y Ff(\025)f Fp(0.)62 b(Moreo)m(v)m(er,)42 b(their)c(sums)e Fo(X)1549 2156 y Fl(0)p Fn(;\026)1646 2165 y Fm(2)1685 2142 y Fp(\()p Fo(t)p Fp(\))i(and)f Fo(X)2085 2156 y Fl(1)p Fn(;\026)2182 2165 y Fm(2)2222 2142 y Fp(\()p Fo(t)p Fp(\))h(are)g(holomorphic)f(in)g(a)h(sectorial)i (region)75 2255 y Fo(G)27 b Ff(\032)f Fo(D)345 2273 y Fn(R)p Fl(\()p Fn(\032)p Fl(\))494 2255 y Fp(,)31 b(indep)s(enden)m(t)f (of)h Fo(\026)1225 2269 y Fl(2)1264 2255 y Fp(,)g(whic)m(h)g(has)g (bisecting)h(direction)f Fo(d)g Fp(and)g(op)s(ening)f(greater)i(than)f Fo(\031)j Fp(and)75 2382 y(satisfy)d(the)f(follo)m(wing)i(estimates.)43 b(F)-8 b(or)31 b(all)g(closed)g(sector)2171 2359 y(\026)2156 2382 y Fo(S)f Ff(\032)25 b Fo(G)p Fp(,)31 b(there)f(exist)i Fo(C)q(;)3044 2359 y Fp(~)3019 2382 y Fo(K)g(>)25 b Fp(0)31 b(suc)m(h)f(that)1059 2598 y(sup)1068 2686 y Fn(t)p Fj(2)1152 2669 y Fl(\026)1140 2686 y Fn(S)1212 2598 y Ff(j)p Fo(@)1290 2558 y Fn(n)1333 2567 y Fm(0)1285 2622 y Fn(t)1372 2598 y Fo(X)1447 2612 y Fn(n)1490 2621 y Fm(1)1524 2612 y Fn(;n)1587 2621 y Fm(2)1626 2598 y Fp(\()p Fo(t)p Fp(\))p Ff(j)c(\024)f Fo(C)1971 2575 y Fp(~)1948 2598 y Fo(K)2032 2561 y Fn(n)2075 2570 y Fm(0)2108 2561 y Fl(+)p Fn(n)2206 2570 y Fm(1)2240 2561 y Fl(+)p Fn(n)2338 2570 y Fm(2)2377 2598 y Fp(\()p Fo(n)2467 2612 y 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