We study the special values at s = 2 and 3 of the spectral zeta function ζQ(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 ζQ(S) converges absolutely for Re s>1 and further the respective values ζQ(2) and ζQ(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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