The authors establish a new analytical identity relating infinite series to derivatives of the q -digamma function ψ q (z). Specifically, they prove that, for all q ∈ R + ∖ 1 and complex r with ℜ (r) > 0, the following exact equality holds true: ∑ k = 0 ∞ q k + r (q k + r − 1) 2 = { − log (1 / q) + ψ 1 / q ′ (r) (log (1 / q) ) 2 (0 1). A comprehensive convergence analysis and investigation of special cases for specific parameter values is also presented. This identity creates new research opportunities in q -analysis with potential applications to partial differential equations, number theory and statistical physics. Moreover, several connections with fundamental mathematical constants and special functions are established.
Srivastava et al. (Wed,) studied this question.