Duality for Finite Multiple Harmonic q-Series

Abstract

We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities--for example, with all parameters equal to 1--have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function.

12 pages AMSLaTeX. Submitted for publication October 26, 2003; revised September 14, 2004. New title reflects change in emphasis and new section devoted to connections with inverse pairs and Hoffman duality. References added and typos corrected

References in corpus (8)

• Multiple $q$-Zeta Values
• q-Multiple Zeta Functions and q-Multiple Polylogarithms
• Parametric Euler Sum Identities
• The Algebra and Combinatorics of Shuffles and Multiple Zeta Values
• A q-analog of Euler's decomposition formula for the double zeta function
• Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth
• Partition Identities for the Multiple Zeta Function
• Some remarks on q-deformed multiple polylogarithms

Cited by in corpus (9)

• Multiple $q$-Zeta Values
• Thirty-two Goldbach Variations
• Cyclotomic analogues of finite multiple zeta values
• Signed q-Analogs of Tornheim's Double Series
• Evaluations of multiple Dirichlet $L$-values via symmetric functions
• Positivity of the Hypergeometric Coon Amplitude
• Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers
• Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values
• Sum of interpolated finite multiple harmonic $q$-series