On certain two-parameter deformations of multiple zeta values
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RESEARCH
On certain two-parameter deformations of multiple zeta values Masaki Kato* * Correspondence:
[email protected] Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe 657-8501, Japan
Abstract In a previous paper, we showed that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. The integrals appearing in this formula can be considered to be one-parameter deformations of q-double zeta values and thus two-parameter deformations of double zeta values. In this paper, we introduce certain integrals, regarded as two-parameter deformations of multiple zeta values, and investigate their properties. In particular, we consider two-parameter generalizations of the harmonic and shuffle product formulas, which are fundamental relations for multiple zeta values. Keywords: Multiple zeta values, q-Analogues of multiple zeta values, Elliptic gamma function, Harmonic and Shuffle products Mathematics Subject Classification: 33E30
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Functions ψk and Kronecker’s double series . . . . . . . . . . . . . . . . . 3 q-Analogues of multiple zeta values . . . . . . . . . . . . . . . . . . . . . 4 Two-parameter deformations of multiple zeta values . . . . . . . . . . . 5 Two-parameter extensions of the harmonic and shuffle product formulas 6 Proofs of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: An alternative proof of the main theorem of [8] . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Nowadays research on elliptic analogues of various special functions is conducted intensively. Ruijsenaars’s elliptic gamma function ([14]) is a fundamental and important special function in this area. For p, q ∈ C satisfying 0 < |p|, |q| < 1 and z ∈ C, the elliptic gamma function (z; p, q) is defined by
(z; p, q) :=
∞ 1 − z −1 pm+1 q n+1 . 1 − zpm q n
m,n=0
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We now define the q-shifted factorial (z; q)∞ by (z; q)∞
∞ := (1 − q i z). i=0
Then it is easy to see that the limit of (z; p, q) as p → 0 is given by lim (z; p, q) =
p→0
1 . (z; q)∞
Furthermore, the following formula holds (For example, see Sect. 1.10 of [5]): lim (1 − q)1−x
q→1
(q; q)∞ = (x), (q x ; q)∞
where (x) denotes the usual gamma function. The above formulas show that the elliptic gamma function can be consider to be a two-parameter deformation of the gamma function. For other basic properties and applications of the elliptic gamma function, we refer to [4,12–14] and references therein. In [8], we showed the elliptic digamma function, defined by ψ0,0 (z; p, q) := z
d l
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