 04312 Takashi ICHINOSE and Masato WAKAYAMA
 Special values of the spectral zeta function of
the noncommutative harmonic oscillator
and confluent Heun equations
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Sep 30, 04

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Abstract. We study
the special values at $s=2$ and $3$ of the spectral zeta function
$\zeta_Q(s)$ of
the noncommutative 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 byproduct 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,
noncommutative harmonic oscillators, $\zeta(3)$, Heun's differential equation.
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