Computing powers of two generalizations of the logarithm. (English) Zbl 1088.11015

Zudilin, Wadim

Sémin. Lothar. Comb. 53, B53c, 6 p. (2005).

Summary: We prove multiple-series representations for positive integer powers of the series \[ L(z;\alpha)=\sum_{n=1}^\infty\frac{z^n}{n+\alpha},\;\; |z|<1, \; \alpha\geq0, \quad\hbox{and}\quad \ell_q(z)=\sum_{n=1}^\infty\frac{z^nq^n}{1-q^n},\;\; |z|\leq1, \; |q|<1. \] The results generalize a known formula for powers of the series for the ordinary logarithm \(-\log(1-z) = L(z;0)\).

MSC:

11B65	Binomial coefficients; factorials; \(q\)-identities
05E05	Symmetric functions and generalizations
33B10	Exponential and trigonometric functions
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)