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,
Dx) 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.
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