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 < q < 1 ) − log ⁡ q + ψ q ′ ( r ) ( log ⁡ q ) 2 ( q > 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.