TY - JOUR

T1 - Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent heun equations

AU - Ichinose, Takashi

AU - Wakayama, Masato

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85010143531&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010143531&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.59.39

DO - 10.2206/kyushujm.59.39

M3 - Article

AN - SCOPUS:85010143531

VL - 59

SP - 39

EP - 100

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 1

ER -