### Abstract

We study the special values at s = 2 and 3 of the spectral zeta function ζ_{Q}(s) of the non-commutative harmonic oscillator Q(x, D_{x}) introduced in A. Parmeggiani and M. Wakayama (Proc. Natl Acad. Sci. USA 98 (2001), 26-31; Forum Math. 14 (2002), 539-604). It is shown that the series defining ζ_{Q}(S) converges absolutely for Re s>1 and further the respective values ζ_{Q}(2) and ζ_{Q}(3) are represented essentially by contour integrals of the solutions, respectively, of a singly confluent Heun 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.

Original language | English |
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Pages (from-to) | 39-100 |

Number of pages | 62 |

Journal | Kyushu Journal of Mathematics |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2005 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Kyushu Journal of Mathematics*,

*59*(1), 39-100. https://doi.org/10.2206/kyushujm.59.39