Content-Type: multipart/mixed; boundary="-------------0409221000153" This is a multi-part message in MIME format. ---------------0409221000153 Content-Type: text/plain; name="04-297.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-297.keywords" Partial differential equations, power series, summability, hypergeometric functions. ---------------0409221000153 Content-Type: application/postscript; name="para2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="para2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: para2.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMSY10 CMR12 CMBX10 CMR10 CMBX12 CMMI10 CMMI8 %%+ CMR8 CMR6 CMMI6 CMEX10 CMSY8 MSBM10 CMSY6 CMR9 CMTI10 LASY10 %%+ TeX-cmex8 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o para2.ps para2.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 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All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 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 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec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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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % 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. 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%!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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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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: 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 dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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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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 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 92 /intersectiondisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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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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. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 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{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % 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 (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 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{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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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. 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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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Fp(+)20 b Fo(w)j Fp(+)d Fo(j)5 b Fp(\))868 5206 y Fn(m)930 5215 y Fk(1)985 5244 y Fl(\001)15 b(\001)g(\001)h Fp(\()p Fo(c)1180 5259 y Fn(k)1217 5268 y Fk(0)1277 5244 y Fp(+)k Fo(w)i Fp(+)e Fo(j)5 b Fp(\))1623 5206 y Fn(m)1685 5218 y Fj(k)1718 5233 y Fk(0)1762 5244 y Fo(p)1808 5258 y Fn(j)t Fm(+1)1935 5244 y Fp(\()p Fo(w)r Fp(\))26 b(=)f Fo(p)2240 5258 y Fn(j)2277 5244 y Fp(\()p Fo(w)r Fp(\)\()p Fo(b)2488 5258 y Fm(1)2549 5244 y Fp(+)20 b Fo(w)i Fp(+)e Fo(j)5 b Fp(\))15 b Fl(\001)g(\001)g(\001)j Fp(\()p Fo(b)3107 5259 y Fn(k)3170 5244 y Fp(+)i Fo(w)j Fp(+)c Fo(j)5 b Fp(\))p Fo(;)75 5432 y Fp(with)30 b Fo(p)328 5446 y Fm(0)367 5432 y Fp(\()p Fo(w)r Fp(\))d(=)e Fo(\025)680 5446 y Fm(0)719 5432 y Fp(\()p Fo(w)r Fp(\),)32 b(so)f(that)g(for)f Fo(j)h Fl(\025)25 b Fp(1,)1122 5669 y Fo(p)1168 5683 y Fn(j)1204 5669 y Fp(\()p Fo(w)r Fp(\))i(=)1574 5607 y(\()p Fo(b)1648 5621 y Fm(1)1708 5607 y Fp(+)19 b Fo(w)r Fp(\))1900 5621 y Fn(j)1953 5607 y Fl(\001)c(\001)g(\001)h Fp(\()p Fo(b)2148 5622 y Fn(k)2211 5607 y Fp(+)k Fo(w)r Fp(\))2404 5621 y Fn(j)p 1474 5648 V 1474 5746 a Fp(\()p Fo(c)1548 5760 y Fm(1)1608 5746 y Fp(+)g Fo(w)r Fp(\))1801 5705 y Fn(m)1863 5714 y Fk(1)1801 5773 y Fn(j)1918 5746 y Fl(\001)15 b(\001)g(\001)h Fp(\()p Fo(c)2113 5761 y Fn(k)2150 5770 y Fk(0)2210 5746 y Fp(+)k Fo(w)r Fp(\))2403 5688 y Fn(m)2465 5700 y Fj(k)2498 5715 y Fk(0)2403 5773 y Fn(j)2552 5669 y Fo(\025)2605 5683 y Fm(0)2644 5669 y Fp(\()p Fo(w)r Fp(\))p Fo(:)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 75 100 a Fp(4)75 399 y(The)30 b(lemma)h(follo)m (ws)g(from)f(the)h(latter)g(form)m(ula.)1956 b Fb(2)216 555 y Fp(W)-8 b(e)38 b(consider)f(no)m(w)g(the)g(follo)m(wing)i (auxiliary)f(function)e(asso)s(ciated)j(to)f(the)f(formal)g(series)g (\()6 b(^)-51 b Fo(u)q Fp(\),)39 b(con-)75 668 y(structed)30 b(in)g(Lemma)h(1,)1488 845 y Fo(v)s Fp(\()p Fo(t;)15 b(z)t Fp(\))27 b(=)1877 732 y Fh(1)1847 759 y Fi(X)1851 954 y Fn(j)t Fm(=0)2067 783 y Fo(u)2119 797 y Fn(j)2156 783 y Fp(\()p Fo(z)t Fp(\))p 2003 824 335 4 v 2003 908 a(\()p Fo(j)5 b Fp(!\)\()p Fo(j)g Fp(!\))2277 881 y Fn(A)2347 845 y Fo(t)2380 808 y Fn(j)2416 845 y Fo(:)75 1098 y Fp(In)33 b(order)g(to)h(in)m(v)m(estigate)j(analytic)e(con)m(tin)m (uation)g(prop)s(erties)e(of)h Fo(v)s Fp(\()p Fo(t;)15 b(z)t Fp(\),)36 b(w)m(e)e(sp)s(ecify)f(the)g(path)h Fo(\015)3578 1112 y Fn(\017)3610 1098 y Fp(,)h(from)75 1211 y(no)m(w)30 b(on)h(and)e(to)j(the)e(end)g(of)g(the)h(pap)s(er.)216 1324 y(Consider)g(complex)h(n)m(um)m(b)s(ers)1319 1300 y(~)1322 1324 y Fo(b)1361 1338 y Fn(q)1426 1324 y Fp(=)27 b Fo(b)1563 1338 y Fn(q)1622 1324 y Fl(\000)20 b Fo(r)1754 1338 y Fn(q)1792 1324 y Fp(,)32 b(1)c Fl(\024)e Fo(q)k Fl(\024)d Fo(k)s Fp(,)32 b(where)f Fo(r)2600 1338 y Fn(q)2665 1324 y Fl(2)c Fg(N)2819 1338 y Fm(0)2889 1324 y Fp(and)33 b(~)-47 b Fo(c)3106 1338 y Fn(q)3172 1324 y Fp(=)26 b Fo(c)3308 1338 y Fn(q)3347 1324 y Fp(,)32 b(1)27 b Fl(\024)g Fo(q)j Fl(\024)d Fo(k)3790 1338 y Fm(0)3829 1324 y Fp(,)75 1436 y(satisfying)k(the)g(follo)m(wing)g(conditions.)105 1593 y Fd(i\))65 b Fp(The)30 b(in)m(tersection)811 1793 y Fo(R)880 1808 y Fn(b)940 1793 y Fp(=)25 b Fl(\\)1097 1754 y Fn(A)p Fm(+1)1097 1819 y Fn(q)r Fm(=1)1243 1793 y Fp(\()p Fl(\000)p Fp(Re)q(\()1489 1769 y(~)1492 1793 y Fo(b)1531 1807 y Fn(q)1569 1793 y Fp(\))p Fo(;)15 b Fp(1)22 b Fl(\000)e Fp(Re\()1941 1769 y(~)1944 1793 y Fo(b)1983 1807 y Fn(q)2021 1793 y Fp(\)\))2106 1706 y Fi(\\)2223 1793 y Fl(\\)2284 1755 y Fn(k)2284 1815 y(q)r Fm(=)p Fn(A)p Fm(+2)2520 1793 y Fp(\()p Fl(\000)p Fp(Re\()2765 1769 y(~)2768 1793 y Fo(b)2807 1807 y Fn(q)2846 1793 y Fp(\))p Fo(;)15 b Fp(+)p Fl(1)p Fp(\))75 1978 y(is)30 b(not)h(empt)m(y)-8 b(.)75 2091 y Fd(ii\))65 b Fp(F)-8 b(or)31 b(all)g(1)26 b Fl(\024)f Fo(q)j Fl(\024)d Fo(k)907 2105 y Fm(0)947 2091 y Fp(,)30 b(for)g(all)i Fo(\034)j Fl(2)25 b Fp([)p Fo(A)c Fp(+)e(2)i(+)f Fo(\033)1842 2105 y Fn(q)r Fh(\000)p Fm(1)1970 2091 y Fo(;)15 b(A)21 b Fp(+)f(1)g(+)g Fo(\033)2398 2105 y Fn(q)2436 2091 y Fp(],)1634 2277 y(Re\()r(~)-47 b Fo(c)1815 2291 y Fn(q)1854 2277 y Fp(\))25 b Fo(>)g Fp(Re)q(\()2150 2253 y(~)2153 2277 y Fo(b)2192 2291 y Fn(\034)2235 2277 y Fp(\))p Fo(:)75 2463 y Fp(The)41 b(real)h(n)m(um)m(b)s(er)e Fo(\017)h Fp(is)h(no)m(w)f(assumed)g(to)h(satisfy)g Fo(\017)i Fl(2)f Fo(R)2231 2478 y Fn(b)2266 2463 y Fp(,)h Fo(\017)54 b(=)-55 b Fl(2)43 b(\000)p Fp(Re\()p Fo(c)2772 2477 y Fn(q)2811 2463 y Fp(\))28 b Fl(\000)f Fg(N)3038 2477 y Fm(0)3077 2463 y Fp(,)44 b(1)h Fl(\024)e Fo(q)j Fl(\024)e Fo(k)3599 2477 y Fm(0)3639 2463 y Fp(,)g(and)75 2576 y Fo(\017)35 b(=)-55 b Fl(2)25 b(\000)p Fp(Re\()p Fo(b)475 2590 y Fn(q)513 2576 y Fp(\))c Fl(\000)f Fg(Z)p Fp(,)30 b(1)c Fl(\024)f Fo(q)j Fl(\024)d Fo(k)s Fp(.)41 b(The)30 b(corresp)s(onding)f (path)h Fo(\015)2254 2590 y Fn(\017)2317 2576 y Fp(is)g(describ)s(ed)g (as)g(ab)s(o)m(v)m(e)i(in)e(Lemma)g(1.)75 2733 y(In)g(the)g(next)h (lemma)g(w)m(e)g(study)e(the)i(analytic)h(con)m(tin)m(uation)g(of)e Fo(v)s Fp(\()p Fo(t;)15 b(z)t Fp(\))32 b(with)e(resp)s(ect)h(to)g Fo(t)p Fp(.)75 2905 y Fd(Lemma)k(2)45 b Fc(Ther)-5 b(e)42 b(exists)g Fo(r)i(>)c Fp(0)p Fc(,)k(such)d(that)i(the)f(series)f Fo(v)s Fp(\()p Fo(t;)15 b(z)t Fp(\))43 b Fc(is)e(c)-5 b(onver)g(gent)42 b(for)g Fo(t)f Fl(2)f Fo(D)s Fp(\(0)p Fo(;)15 b(r)s Fp(\))42 b Fc(and)75 3018 y Fo(z)f Fl(2)c Fo(D)s Fp(\(0)p Fo(;)15 b(\032)p Fp(\))p Fc(.)63 b(The)39 b(function)h Fo(v)s Fp(\()p Fo(t;)15 b(z)t Fp(\))40 b Fc(has)g(an)g(analytic)g(c)-5 b(ontinuation)41 b(with)f(r)-5 b(esp)g(e)g(ct)41 b(to)f Fo(t)d Fl(2)g Fo(S)3497 3032 y Fn(\032)3537 3018 y Fc(,)j(for)g(al)5 b(l)75 3131 y Fo(z)30 b Fl(2)24 b Fo(D)s Fp(\(0)p Fo(;)15 b(\032)p Fp(\))p Fc(,)34 b(wher)-5 b(e)34 b Fo(S)887 3145 y Fn(\032)959 3131 y Fc(is)f(any)g(se)-5 b(ctor)34 b(c)-5 b(enter)g(e)g(d)34 b(at)f(0)g(such)g(that)1058 3317 y Fo(S)1114 3331 y Fn(\032)1180 3317 y Fl(\032)25 b(f)p Fo(t)g Fl(2)g Fg(C)g Fp(:)g Fo(e)1648 3279 y Fn(z)1689 3317 y Fo(t)35 b(=)-55 b Fl(2)25 b Fp([1)p Fo(;)15 b Fp(+)p Fl(1)p Fp(\))66 b 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Fo(u)p Fp(\))28 b(=)2190 3969 y Fh(1)2160 3996 y Fi(X)2164 4191 y Fn(j)t Fm(=0)2537 4020 y Fp(\()2569 3996 y(~)2572 4020 y Fo(b)2611 4034 y Fm(1)2671 4020 y Fp(+)20 b Fo(w)r Fp(\))2864 4034 y Fn(j)2916 4020 y Fl(\001)15 b(\001)g(\001)h Fp(\()3069 3996 y(~)3072 4020 y Fo(b)3111 4035 y Fn(k)3175 4020 y Fp(+)j Fo(w)r Fp(\))3367 4034 y Fn(j)p 2316 4061 1310 4 v 2316 4159 a Fo(j)5 b Fp(!\(1\))2498 4127 y Fn(A)2498 4186 y(j)2557 4159 y Fp(\()r(~)-47 b Fo(c)2631 4173 y Fm(1)2692 4159 y Fp(+)19 b Fo(w)r Fp(\))2884 4118 y Fn(m)2946 4127 y Fk(1)2884 4186 y Fn(j)3002 4159 y Fl(\001)c(\001)g(\001)h Fp(\()r(~)-47 b Fo(c)3197 4174 y Fn(k)3234 4183 y Fk(0)3293 4159 y Fp(+)20 b Fo(w)r Fp(\))3486 4101 y Fn(m)3548 4113 y Fj(k)3581 4128 y Fk(0)3486 4186 y Fn(j)3635 4082 y Fo(u)3687 4045 y Fn(j)75 4363 y Fp(whic)m(h)30 b(con)m(v)m(erge)j (absolutely)e(for)f Fl(j)p Fo(u)p Fl(j)c Fo(<)f Fp(1)31 b(and)e Fo(w)f Fl(2)d Fo(\015)1980 4377 y Fn(\017)2043 4363 y Fp(\(see)31 b(for)f(instance)i([7],)f([9]\).)216 4476 y(Under)37 b(the)i(upp)s(er)d(men)m(tioned)i(conditions)h Fd(i)p Fp(\))f(and)f Fd(ii)p Fp(\))q(,)j(w)m(e)e(get)i(from)d([12],)42 b(page)c(108,)k(an)c(in)m(tegral)75 4589 y(represen)m(tation)31 b(for)g(this)f(generalized)i(h)m(yp)s(ergeometric)f(function.)41 b(W)-8 b(e)31 b(de\014ne)f(the)g(functions)459 4858 y Fo(B)5 b Fp(\()p Fo(t)601 4872 y Fm(2)640 4858 y Fo(;)15 b(:)g(:)g(:)h(;)f(t)874 4873 y Fn(k)917 4858 y Fo(;)g(w)r Fp(\))27 b(=)1182 4744 y Fn(A)p Fm(+1)1195 4771 y Fi(Y)1191 4965 y Fn(q)r Fm(=2)1340 4858 y Fo(t)1370 4791 y Fm(~)1373 4808 y Fn(b)1403 4816 y Fj(q)1437 4808 y Fh(\000)p Fm(1+)p Fn(w)1373 4869 y(q)1639 4858 y Fp(\(1)21 b Fl(\000)f Fo(t)1864 4872 y Fn(q)1902 4858 y Fp(\))1937 4820 y Fh(\000)p Fm(\()2016 4802 y(~)2019 4820 y Fn(b)2049 4828 y Fj(q)2084 4820 y Fm(+)p Fn(w)r Fm(\))2264 4743 y Fn(k)2301 4752 y Fk(0)2242 4771 y Fi(Y)2238 4965 y Fn(q)r Fm(=1)2401 4771 y Fi(Y)2377 4968 y Fn(\034)8 b Fh(2S)2506 4976 y Fj(q)2556 4858 y Fo(t)2586 4802 y Fm(~)2589 4820 y Fn(b)2619 4828 y Fj(\034)2658 4820 y Fm(+)p Fn(w)r Fh(\000)p Fm(1)2589 4880 y Fn(\034)2859 4858 y Fp(\(1)21 b Fl(\000)f Fo(t)3084 4872 y Fn(\034)3127 4858 y Fp(\))3164 4820 y Fm(~)-37 b Fn(c)3193 4828 y Fj(q)3227 4820 y Fh(\000)3279 4802 y Fm(~)3282 4820 y Fn(b)3312 4828 y Fj(\034)3351 4820 y Fh(\000)p Fm(1)3446 4858 y Fo(;)1013 5238 y(A)p Fp(\()p Fo(w)r Fp(\))27 b(=)1351 5163 y(\005)1419 5125 y Fn(A)p Fm(+1)1419 5189 y Fn(q)r Fm(=2)1581 5163 y Fp(sin)15 b Fo(\031)s Fp(\()1795 5139 y(~)1798 5163 y Fo(b)1837 5177 y Fn(q)1896 5163 y Fp(+)20 b Fo(w)r Fp(\))p 1351 5217 739 4 v 1664 5300 a Fo(\031)1719 5274 y Fn(A)2109 5163 y Fp(\005)2177 5123 y Fn(k)2214 5132 y Fk(0)2177 5189 y Fn(q)r Fm(=1)2306 5163 y Fl(f)p Fp(\000\()r(~)-47 b Fo(c)2482 5177 y Fn(q)2541 5163 y Fp(+)20 b Fo(w)r Fp(\))p Fl(g)2779 5130 y Fn(m)2841 5138 y Fj(q)p 2109 5217 772 4 v 2152 5312 a Fp(\005)2220 5281 y Fn(k)2220 5341 y(q)r Fm(=)p Fn(A)p Fm(+2)2456 5312 y Fp(\000\()2545 5288 y(~)2548 5312 y Fo(b)2587 5326 y Fn(q)2645 5312 y Fp(+)g Fo(w)r Fp(\))2891 5238 y Fo(;)75 5470 y Fp(and)30 b(the)g(constan)m(t)1452 5648 y(\000)25 b(=)g(1)p Fo(=)1763 5534 y Fn(k)1800 5543 y Fk(0)1740 5562 y Fi(Y)1735 5756 y Fn(q)r Fm(=1)1899 5562 y Fi(Y)1876 5759 y Fn(\034)8 b Fh(2S)2005 5767 y Fj(q)2054 5648 y Fp(\000\()r(~)-47 b Fo(c)2185 5662 y Fn(q)2244 5648 y Fl(\000)2331 5624 y Fp(~)2335 5648 y Fo(b)2374 5662 y Fn(\034)2417 5648 y Fp(\))p Fo(;)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fp(5)75 399 y(where)32 b Fl(S)395 413 y Fn(q)463 399 y Fp(=)d([)p Fo(A)23 b Fp(+)e(2)i(+)e Fo(\033)983 413 y Fn(q)r Fh(\000)p Fm(1)1111 399 y Fo(;)15 b(A)23 b Fp(+)f(1)g(+)g Fo(\033)1547 413 y Fn(q)1585 399 y Fp(].)48 b(W)-8 b(e)34 b(ha)m(v)m(e)h(the)e(follo)m (wing)h(in)m(tegral)h(represen)m(tation)f(whic)m(h)e(is)75 511 y(an)e(analytic)i(con)m(tin)m(uation)g(of)f(the)g(generalized)h(h)m (yp)s(ergeometric)f(function)722 720 y Fo(F)13 b Fp(\()825 696 y(~)828 720 y Fo(b)867 734 y Fm(1)927 720 y Fp(+)20 b Fo(w)r(;)15 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Fn(b)1449 1153 y Fp(and)30 b Fo(u)35 b(=)-55 b Fl(2)25 b Fp([1)p Fo(;)15 b Fp(+)p Fl(1)p Fp(\).)216 1268 y(In)28 b(the)g(follo)m(wing,)i(w)m(e)f (assume)f(that)h Fo(w)f Fl(2)c Fo(\015)1753 1282 y Fn(\017)1814 1268 y Fp(and)k(is)g(written)g(in)g(the)h(form)e Fo(w)h Fp(=)d Fo(\017)16 b Fp(+)f Fo(iy)3269 1282 y Fn(w)3326 1268 y Fp(,)29 b Fo(y)3425 1282 y Fn(w)3506 1268 y Fl(2)c Fp([0)p Fo(;)15 b Fl(1)p Fp(\).)75 1381 y(F)-8 b(or)31 b Fo(y)283 1395 y Fn(w)370 1381 y Fp(su\016cien)m(tly)g(large,)g(w)m(e) g(ha)m(v)m(e)1341 1604 y Fl(j)15 b Fp(sin)g Fo(\031)s Fp(\()1595 1580 y(~)1598 1604 y Fo(b)1637 1618 y Fn(q)1696 1604 y Fp(+)20 b Fo(w)r Fp(\))p Fl(j)26 b(\024)f Fo(e)2078 1566 y Fn(\031)r Fm(\()p Fn(y)2183 1574 y Fj(w)2234 1566 y Fm(+)p Fh(j)p Fm(Im\()2417 1549 y(~)2420 1566 y Fn(b)2450 1574 y Fj(q)2485 1566 y Fm(\))p Fh(j)p Fm(\))2563 1604 y Fo(:)75 1815 y Fp(F)-8 b(or)31 b Fo(u)36 b(=)-56 b Fl(2)25 b Fp([1)p Fo(;)15 b Fl(1)p Fp(\),)32 b(w)m(e)f(ha)m(v)m(e)543 2038 y Fl(j)p Fp(\(1)21 b Fl(\000)f Fo(t)793 2052 y Fm(2)847 2038 y Fl(\001)15 b(\001)g(\001)i Fo(t)1002 2053 y Fn(k)1044 2038 y Fo(u)p Fp(\))1131 2005 y Fh(\000)p Fm(\()1210 1988 y(~)1213 2005 y Fn(b)1243 2014 y Fk(1)1279 2005 y Fm(+)p Fn(w)r Fm(\))1418 2038 y Fl(j)543 2157 y Fp(=)25 b(exp\()p Fl(\000)p Fp(\(Re\()1058 2133 y(~)1061 2157 y Fo(b)1100 2171 y Fm(1)1140 2157 y Fp(\))c(+)f Fo(\017)p Fp(\))15 b(ln)g Fl(j)p Fp(1)21 b Fl(\000)f Fo(t)1680 2171 y Fm(2)1734 2157 y Fl(\001)15 b(\001)g(\001)h Fo(t)1888 2172 y Fn(k)1931 2157 y Fo(u)p Fl(j)k Fp(+)g(\()p Fo(y)2199 2171 y Fn(w)2276 2157 y Fp(+)g(Im\()2508 2133 y(~)2511 2157 y Fo(b)2550 2171 y Fm(1)2589 2157 y Fp(\)\)arg)s(\(1)h Fl(\000)f Fo(t)3013 2171 y Fm(2)3067 2157 y Fl(\001)15 b(\001)g(\001)h Fo(t)3221 2172 y Fn(k)3264 2157 y Fo(u)p Fp(\)\))75 2365 y(with)30 b Fl(\000)p Fo(\031)e(<)d Fp(arg)r(\(1)c Fl(\000)f Fo(t)882 2379 y Fm(2)936 2365 y Fl(\001)15 b(\001)g(\001)h Fo(t)1090 2380 y Fn(k)1133 2365 y Fo(u)p Fp(\))26 b Fo(<)e(\031)s Fp(.)41 b(Moreo)m(v)m(er,)814 2605 y Fl(j)p Fp(\(1)21 b Fl(\000)f Fo(t)1064 2619 y Fn(q)1102 2605 y Fp(\))1137 2567 y Fh(\000)p Fm(\()1216 2550 y(~)1219 2567 y Fn(b)1249 2575 y Fj(q)1284 2567 y Fm(+)p Fn(w)r Fm(\))1423 2605 y Fo(t)1456 2556 y Fm(\()1480 2538 y(~)1483 2556 y Fn(b)1513 2564 y Fj(q)1548 2556 y Fh(\000)p Fm(1+)p Fn(w)r Fm(\))1456 2617 y Fn(q)1777 2605 y Fl(j)26 b Fp(=)f(\(1)c Fl(\000)f Fo(t)2149 2619 y Fn(q)2187 2605 y Fp(\))2222 2567 y Fh(\000)p Fm(\(Re\()2411 2550 y(~)2414 2567 y Fn(b)2444 2575 y Fj(q)2479 2567 y Fm(\)+)p Fn(\017)p Fm(\))2622 2605 y Fo(t)2655 2556 y Fm(\(Re\()2789 2538 y(~)2792 2556 y Fn(b)2822 2564 y Fj(q)2857 2556 y Fm(\))p Fh(\000)p Fm(1+)p Fn(\017)p Fm(\))2655 2617 y Fn(q)3090 2605 y Fo(;)75 2816 y Fp(and)671 2935 y Fl(j)p Fp(\(1)h Fl(\000)f Fo(t)921 2949 y Fn(\034)964 2935 y Fp(\))999 2897 y Fm(\()r(~)-37 b Fn(c)1057 2905 y Fj(q)1092 2897 y Fh(\000)1144 2880 y Fm(~)1147 2897 y Fn(b)1177 2905 y Fj(\034)1216 2897 y Fh(\000)p Fm(1\))1338 2935 y Fo(t)1371 2897 y Fm(\()1395 2880 y(~)1398 2897 y Fn(b)1428 2905 y Fj(\034)1467 2897 y Fm(+)p Fn(w)r Fh(\000)p Fm(1\))1371 2957 y Fn(\034)1696 2935 y Fl(j)26 b Fp(=)f(\(1)c Fl(\000)f Fo(t)2068 2949 y Fn(\034)2111 2935 y Fp(\))2146 2897 y Fm(\(Re)q(\()r(~)-37 b Fn(c)2315 2905 y Fj(q)2349 2897 y Fm(\))p Fh(\000)p Fm(Re)q(\()2539 2880 y(~)2542 2897 y Fn(b)2572 2905 y Fj(\034)2611 2897 y Fm(\))p Fh(\000)p Fm(1\))2760 2935 y Fo(t)2793 2897 y Fm(\(Re)q(\()2928 2880 y(~)2931 2897 y Fn(b)2961 2905 y Fj(\034)3000 2897 y Fm(\))p Fh(\000)p Fm(1+)p Fn(\017)p Fm(\))2793 2957 y Fn(\034)3233 2935 y Fo(:)75 3109 y Fp(Using)31 b(the)f(Stirling)h(form)m(ula)f(\(see)i([2]\),)1447 3320 y(\000\()p Fo(z)t Fp(\))26 b Fl(\030)f Fp(\(2)p Fo(\031)s Fp(\))1912 3282 y Fm(1)p Fn(=)p Fm(2)2023 3320 y Fo(z)2069 3282 y Fn(z)2109 3320 y Fo(z)2155 3282 y Fh(\000)p Fm(1)p Fn(=)p Fm(2)2320 3320 y Fo(e)2362 3282 y Fh(\000)p Fn(z)2457 3320 y Fo(;)75 3530 y Fp(when)k Fl(j)p Fo(z)t Fl(j)d(!)g(1)k Fp(with)g Fl(\000)p Fo(\031)23 b Fp(+)d Fo(\016)29 b Fl(\024)c Fp(arg)r(\()p Fo(z)t Fp(\))h Fl(\024)f Fo(\031)e Fl(\000)d Fo(\016)s Fp(,)31 b(0)26 b Fo(<)f(\016)k(<)c(\031)s Fp(,)31 b(and)e(for)i(all)g Fo(f)j Fl(2)25 b Fg(C)p Fp(,)1520 3741 y(\()p Fo(f)30 b Fp(+)20 b Fo(z)t Fp(\))1802 3703 y Fm(\()p Fn(f)7 b Fm(+)p Fn(z)s Fm(\))2018 3741 y Fl(\030)25 b Fo(z)2160 3703 y Fn(f)7 b Fm(+)p Fn(z)2297 3741 y Fo(e)2339 3703 y Fn(f)2384 3741 y Fo(;)75 3951 y Fp(when)23 b Fl(j)p Fo(z)t Fl(j)j(!)f(1)p Fp(,)g(w)m(e)g(get)g(the)f(follo)m(wing)h (estimates.)40 b(There)24 b(exist)g Fo(M)5 b(;)2541 3928 y Fp(~)2516 3951 y Fo(K)2593 3965 y Fm(1)2659 3951 y Fo(>)25 b Fp(0,)g(suc)m(h)f(that)h(for)e(all)i Fl(j)p Fo(w)r Fl(j)h Fo(>)f(M)10 b Fp(,)933 4167 y(\005)1001 4126 y Fn(k)1038 4135 y Fk(0)1001 4193 y Fn(q)r Fm(=1)1129 4167 y Fl(f)p Fp(\000\()r(~)-47 b Fo(c)1305 4181 y Fn(q)1364 4167 y Fp(+)20 b Fo(w)r Fp(\))p Fl(g)1602 4134 y Fn(m)1664 4142 y Fj(q)p 933 4220 772 4 v 975 4316 a Fp(\005)1043 4285 y Fn(k)1043 4345 y(q)r Fm(=)p Fn(A)p Fm(+2)1279 4316 y Fp(\000\()1368 4292 y(~)1371 4316 y Fo(b)1410 4330 y Fn(q)1468 4316 y Fp(+)g Fo(w)r Fp(\))1740 4241 y Fl(\024)1860 4218 y Fp(~)1836 4241 y Fo(K)1913 4255 y Fm(1)1952 4241 y Fl(j)p Fo(w)r Fl(j)2069 4201 y Fm(Re)r(\()2181 4151 y Fa(P)2255 4163 y Fj(k)2288 4178 y Fk(0)2255 4222 y Fj(q)r Fk(=1)2380 4201 y Fm(~)-36 b Fn(c)2410 4209 y Fj(q)2444 4201 y Fn(m)2506 4209 y Fj(q)2540 4201 y Fh(\000)2595 4151 y Fa(P)2670 4171 y Fj(k)2670 4221 y(q)r Fk(=)p Fj(A)p Fk(+2)2883 4183 y Fm(~)2885 4201 y Fn(b)2915 4209 y Fj(q)2950 4201 y Fm(\))2982 4241 y Fo(:)75 4532 y Fc(In)47 b(the)g(fol)5 b(lowing)47 b(estimates,)k(we)c(always)h (assume)f(that)h Fo(w)53 b Fl(2)d Fo(\015)2478 4546 y Fn(\017)2511 4532 y Fc(,)g Fl(j)p Fo(w)r Fl(j)h Fo(>)g(M)10 b Fc(,)50 b(and)d Fo(u)k Fp(=)f Fo(e)3511 4499 y Fn(z)3551 4532 y Fo(t)c Fc(wher)-5 b(e)75 4645 y Fo(z)30 b Fl(2)24 b Fo(D)s Fp(\(0)p Fo(;)15 b(\032)p Fp(\))34 b Fc(and)g Fo(t)25 b Fl(2)g Fo(S)923 4659 y Fn(\032)983 4645 y Fl([)20 b Fo(D)s Fp(\(0)p Fo(;)15 b(r)s Fp(\))p Fc(,)33 b Fo(r)28 b(<)d Fp(1)33 b Fc(b)-5 b(eing)32 b(smal)5 b(l)34 b(enough.)216 4760 y Fp(F)-8 b(rom)39 b(the)f(latter)h(estimates,)j(w)m(e)d(deduce)f (that)g(there)g(exist)h(constan)m(ts)2872 4737 y(^)2847 4760 y Fo(K)7 b(;)2992 4737 y Fp(^)2971 4760 y Fo(C)45 b Fl(2)38 b Fg(R)3246 4774 y Fm(+)3305 4760 y Fp(,)i(indep)s(enden)m(t) 75 4872 y(from)30 b Fo(u)p Fp(,)g Fo(w)r Fp(,)i(suc)m(h)e(that)75 5097 y(\()p Fo(E)177 5111 y Fm(1)217 5097 y Fp(\))418 b Fl(j)p Fo(F)13 b Fp(\()798 5073 y(~)801 5097 y Fo(b)840 5111 y Fm(1)900 5097 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)1257 5073 y Fp(~)1260 5097 y Fo(b)1299 5112 y Fn(k)1363 5097 y Fp(+)j Fo(w)r Fp(;)d(~)-47 b Fo(c)1600 5111 y Fm(1)1660 5097 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)j(;)f Fp(~)-47 b Fo(c)2060 5112 y Fn(k)2097 5121 y Fk(0)2156 5097 y Fp(+)20 b Fo(w)r Fp(;)15 b(1;)g Fo(u)p Fp(\))p Fl(j)28 b(\024)2695 5074 y Fp(^)2674 5097 y Fo(C)7 b(y)2812 5042 y Fm(^)2794 5059 y Fn(K)2791 5119 y(w)2862 5097 y Fo(e)2904 5059 y Fn(y)2939 5067 y Fj(w)2989 5059 y Fm(\()p Fn(A)p Fm(+1\))p Fn(\031)3234 5097 y Fo(:)75 5307 y Fp(W)-8 b(e)32 b(recall)g(no)m(w)e(one)h(con)m(tiguit)m(y)i(relation)f(b)s(et)m (w)m(een)f(generalized)h(h)m(yp)s(ergeometric)f(functions)g(\(see)g ([9],)h(p.)75 5420 y(45)f(and)f([1]\),)75 5690 y(\()p Fo(C)7 b(r)s Fp(\))911 5633 y(\()p Fo(ud=du)22 b Fp(+)d(\()1333 5609 y(~)1336 5633 y Fo(b)1375 5647 y Fn(j)1433 5633 y Fp(+)g Fo(w)r Fp(\)\))p Fo(F)13 b Fp(\()p Fo(:)i(:)g(:)j(;)1926 5609 y Fp(~)1929 5633 y Fo(b)1968 5647 y Fn(j)2026 5633 y Fp(+)h Fo(w)r(;)c(:)g(:)g(:)j Fp(;)f(~)-47 b Fo(c)2425 5647 y Fm(1)2485 5633 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)e Fp(1;)g Fo(u)p Fp(\))911 5752 y(=)25 b(\()1039 5728 y(~)1042 5752 y Fo(b)1081 5766 y Fn(j)1138 5752 y Fp(+)20 b Fo(w)r Fp(\))p Fo(F)13 b Fp(\()p Fo(:)i(:)g(:)i(;)e Fp(\()1631 5728 y(~)1634 5752 y Fo(b)1673 5766 y Fn(j)1731 5752 y Fp(+)20 b(1\))h(+)f Fo(w)r(;)15 b(:)g(:)g(:)i Fp(;)g(~)-47 b Fo(c)2322 5766 y Fm(1)2382 5752 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)e Fp(1;)g Fo(u)p Fp(\))p Fo(:)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 75 100 a Fp(6)75 399 y(W)-8 b(e)32 b(write)309 592 y Fo(d)p 283 633 100 4 v 283 716 a(du)392 654 y(F)13 b Fp(\()p Fo(:)i(:)g(:)i(;)657 630 y Fp(~)660 654 y Fo(b)699 668 y Fn(j)756 654 y Fp(+)j Fo(w)r(;)15 b(:)g(:)g(:)i Fp(;)g(~)-47 b Fo(c)1155 668 y Fm(1)1215 654 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)j(;)d Fp(1;)g Fo(u)p Fp(\))26 b(=)1924 592 y(1)p 1880 633 132 4 v 1880 716 a(2)p Fo(i\031)2037 530 y Fi(Z)2088 736 y Fn(C)2138 748 y Fj(\016)2201 592 y Fo(F)13 b Fp(\()p Fo(:)i(:)g(:)i(;)2466 568 y Fp(~)2469 592 y Fo(b)2508 606 y Fn(j)2565 592 y Fp(+)j Fo(w)r(;)15 b(:)g(:)g(:)i Fp(;)h(~)-48 b Fo(c)2964 606 y Fm(1)3025 592 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)e Fp(1;)g Fo(\020)7 b Fp(\))p 2201 633 1352 4 v 2717 716 a(\()p Fo(\020)27 b Fl(\000)20 b Fo(u)p Fp(\))2997 690 y Fm(2)3563 654 y Fo(d\020)75 919 y Fp(where)1381 1041 y Fo(C)1446 1056 y Fn(\016)1509 1041 y Fp(=)25 b Fl(f)p Fo(\020)33 b Fl(2)24 b Fg(C)h Fp(:)h Fl(j)p Fo(\020)h Fl(\000)20 b Fo(u)p Fl(j)25 b Fp(=)g Fl(j)p Fo(u)p Fl(j)p Fo(\016)s Fl(g)p Fo(;)75 1218 y Fp(and)30 b Fo(\016)k Fp(is)c(su\016cien)m(tly)h(small.)42 b(F)-8 b(rom)31 b(\()p Fo(E)5 b Fp(1\),)32 b(w)m(e)e(deduce)g(that)552 1474 y Fl(j)p Fo(u)665 1413 y(d)p 639 1453 100 4 v 639 1536 a(du)748 1474 y(F)13 b Fp(\()851 1450 y(~)854 1474 y Fo(b)893 1488 y Fm(1)953 1474 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)1310 1450 y Fp(~)1313 1474 y Fo(b)1352 1489 y Fn(k)1416 1474 y Fp(+)j Fo(w)r Fp(;)d(~)-47 b Fo(c)1653 1488 y Fm(1)1713 1474 y Fp(+)20 b Fo(w)r(;)15 b(:)g(:)g(:)i(;)h Fp(~)-48 b Fo(c)2112 1489 y Fn(k)2149 1498 y Fk(0)2209 1474 y Fp(+)20 b Fo(w)r Fp(;)15 b(1;)g Fo(u)p Fp(\))p Fl(j)28 b(\024)2748 1451 y Fp(^)2727 1474 y Fo(C)2809 1413 y Fp(1)p 2809 1453 46 4 v 2810 1536 a Fo(\016)2864 1474 y(y)2931 1420 y Fm(^)2912 1436 y Fn(K)2909 1496 y(w)2981 1474 y Fo(e)3023 1436 y Fn(y)3058 1444 y Fj(w)3108 1436 y Fm(\()p Fn(A)p Fm(+1\))p Fn(\031)3353 1474 y 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