New $q$-identities Via $q$-Derivative of Basic Hypergeometric Series with Respect to Parameters
Ronald Orozco López
TL;DR
This work develops an operator-based framework to generate new $q$-identities by applying $q$-differential and deformed $q$-exponential operators to classical basic hypergeometric sums. It introduces and leverages the deformed series ${}_r\Phi_s$ along with parameter-derivative formulas to produce structured, higher-order identities. The main contributions include explicit derivative and transform relations that connect differentiated ${}_2\phi_s$ sums to ${}_3\phi_2$, ${}_4\phi_3$, and ${}_4\phi_4$ forms, demonstrated for the $q$-Gauss sum, $q$-Chu-Vandermonde sum, and Jackson's transformation. These results provide a systematic method to generate a broad class of $q$-identities and deepen connections between $q$-difference operators and basic hypergeometric theory.$
Abstract
In this paper, we use the effect of the $q$-differential and deformed $q$-exponential operators on basic hypergeometric series to find new $q$-identities from the $q$-Gauss sum, the $q$-Chu-Vandermonde's sum, and Jackson's transformation formula.