04-312 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) 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 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.

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