CYCLIC q-MZSV SUM

We present a family of identities ‘cyclic sum formula’ and ‘sum formula’ for a version of multiple q-zeta star values. We also discuss a problem of q-generalization of shuffle products. Introduction and notation The classical idea of introducing an additional parameter to an expression or formula we wish to deal with, is quite fruitful in many situations. This may simplify a proof of the corresponding identity or lead to a more general identity which have several other useful specializations of the introduced parameter. The story of introducing the parameter q (or, the ‘quantum’ parameter) often has a different flavor. Our motivation to study q-analogues of multiple zeta values (MZVs) (1) ζ(k) = ζ(k1, . . . , kr) = ∑ n1>···>nr≥1 1 n1 1 · · ·n kr r , k1, . . . , kr ∈ {1, 2, . . . }, k1 ≥ 2, and multiple zeta star values (MZSVs) (2) ζ(k) = ζ(k1, . . . , kr) = ∑ n1≥···≥nr≥1 1 n1 1 · · ·n kr r , k1, . . . , kr ∈ {1, 2, . . . }, k1 ≥ 2, is to a better understanding the structure of linear and algebraic relations between the numbers (1) (or (2)). An important advantage of the q-model is that proving the absence of such relations is a much easier task (cf. [17]): the functional case is normally not as hard as the numerical one. On the other hand, showing that some relations hold is normally easier for numbers than for functions. The main problem here is finding an appropriate q-analogue which is often dictated by already existing proofs of the corresponding original identities. In this paper we hope to convince the reader that there is no uniform q-generalization of the multiple zeta (star) values, but having several q-analogues in mind and a simple way to pass from one q-model to another gives one a very natural parallel between the numbers and their q-analogues. Throughout the article we assume that q ∈ C satisfies |q| < 1. Let us first recall the definition of the q-MZVs and q-MZSVs which is already accepted to be Date: March 16, 2008. 2000 Mathematics Subject Classification. Primary 11M06; Secondary 11G55, 16W25. The work of Yasuo Ohno was supported by a special programme of the Kinki University (Osaka). The work of Wadim Zudilin was supported by a fellowship of the Max Planck Institute for Mathematics (Bonn). 1 2 YASUO OHNO, JUN-ICHI OKUDA, AND WADIM ZUDILIN dominating [1], [2], [12], [16]: (3) ζq(k1, k2, . . . , kr) = ∑ n1≥n2≥···≥nr≥1 qn1(k1−1)+n2(k2−1)+···+nr(kr−1) [n1]1 [n2]2 · · · [nr]r and (4) ζ q (k1, k2, . . . , kr) = ∑ n1≥n2≥···≥nr≥1 qn1(k1−1)+n2(k2−1)+···+nr(kr−1) [n1]1 [n2]2 · · · [nr]r , where [n] = [n]q = (1 − q)/(1 − q) is a q-analogue of the positive integer n and conditions for the multi-index k = (k1, . . . , kr) are exactly the same as in (1) and (2) (such multi-indices are called admissible). The corresponding q-analogues of the values of Riemann’s zeta function are as follows: ζq(k) = ζ q (k) = ∑ n≥1 qn(k−1) [n]k . We add one more notation for our convenience: ζ? q (k1, k2, . . . , kr) = (1− q)−(k1+k2+···+kr)ζ? q (k1, k2, . . . , kr) (5) = ∑ n1≥n2≥···≥nr≥1 qn1(k1−1)+n2(k2−1)+···+nr(kr−1) (1− qn1)k1(1− qn2)k2 · · · (1− qnr )kr ; the same convention is used for ζq(k1, k2, . . . , kr). A different version of q-analogues for the numbers (1) and (2) is given by the formulae (6) zq(k1, k2, . . . , kr) = ∑ n1>n2>···>nr≥1 q1 (1− qn1)k1(1− qn2)k2 · · · (1− qnr )kr and (7) zq(k1, k2, . . . , kr) = ∑ n1≥n2≥···≥nr≥1 q1 (1− qn1)k1(1− qn2)k2 · · · (1− qnr )kr ; this time we even do not require the condition k1 > 1. Several relations for the MZSVs have very simple q-analogues in terms of (7). The examples are zq(2, 1) = 2z ? q(3)− zq(2) ( = ∑ n≥1 q(1 + q) (1− qn)3 ) , zq(2, 1, 1) = 3z ? q(4)− 2zq(3) ( = ∑ n≥1 q(1 + 2q) (1− qn)4 ) , zq(2, 2, 1) = 2z ? q(5)− zq(3) ( = ∑ n≥1 q(1 + 2q − q) (1− qn)5 ) , zq(2, 1, 1, 1) = 4z ? q(5)− 3zq(4) ( = ∑ n≥1 q(1 + 3q) (1− qn)5 ) , zq(2, 1, 2, 1) + z ? q(2, 2, 1, 1) = 5z ? q(6)− 3zq(4) ( = ∑ q(2 + 6q − 3q) (1− qn)6 ) , zq(2, 2, 2, 1) = 2z ? q(7)− zq(4) ( = ∑ n≥1 q(1 + 3q − 3q + q) (1− qn)7 ) .