On some properties of special functions involving $k$-gamma and $k$-digamma functions
Li Yin, Jumei Zhang
TL;DR
This work extends classical gamma and digamma theory to k-parameterized analogues by developing four Furdui-type integral series expansions tied to the Riemann zeta function and hypergeometric functions, and by establishing new identities and inequalities for the Hadamard k-gamma function and Nielsen k-beta function. It provides explicit series expressions for Furdui-type integrals, notably a closed form for ∫_0^k x^m ψ_k(x) dx in terms of k, ln k, γ, and ζ(s), and connects these to representations involving ln Γ_k. The paper also introduces a k-generalization of Hadamard's gamma function with a functional equation and related inequalities, and delivers comprehensive identities and monotonicity results for the Nielsen k-beta function, including various series expansions and connections to ψ_k and Γ_k. Together, these results extend classical special-function identities to a flexible k-parameter setting and link them to fundamental constants and hypergeometric structures, with open problems inviting further exploration of higher-order monotonicity properties.
Abstract
Based on $k$-gamma and $k$-digamma functions, we show four series expansions to the Furdui-type integral related to Riemann zeta function and hypergeometric function, and also present some new identities, series expansions and inequalities on the Hadamard $k$-gamma function and the Nielsen $k$-beta function. Finally, we also pose an open problem.