On some properties of special functions involving $k$-gamma and $k$-digamma functions

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.