Takashi ICHINOSE and Masato WAKAYAMA
Special values of the spectral zeta function of 
the non-commutative harmonic oscillator 
and confluent Heun equations
(895K, ps.file)

ABSTRACT.  We study 
the special values at $s=2$ and $3$ of the spectral zeta function 
$\zeta_Q(s)$ of 
the non-commutative harmonic oscillator $Q(x,D_x)$ introduced in \cite{PW1, 2}. 
It is shown that the series defining $\zeta_Q(s)$ converges 
absolutely for Re $s>1$ and further the respective 
values $\zeta_Q(2)$ and $\zeta_Q(3)$ are represented essentially 
by contour integrals of the solutions, respectively, of 
a singly confluent Heun's ordinary differential equation and of 
exactly the same but an inhomogeneous equation. 
As a by-product of these results, we obtain integral representations of the 
solutions of these equations by rational functions. 
\par\noindent\text{{2000 Mathematics Subject Classification}} : 
11M36, 81Q10. 
\par\noindent\text{Key Words} : spectral zeta functions, 
Riemann's zeta function, harmonic oscillators, 
non-commutative harmonic oscillators, $\zeta(3)$, Heun's differential equation.