SPECIAL VALUES OF THE SPECTRAL ZETA FUNCTION OF THE NON-COMMUTATIVE HARMONIC OSCILLATOR AND CONFLUENT HEUN EQUATIONS

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.