A New Identity for the Derivative of the q-Digamma Function and Its Applications

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.