A note on expansions of q-exponential equations and q-Hardy–Hille type formulas

Abstract

By using noncommutative q-analogue of binomial theorem, we construct new q-exponential operators for q-Laguerre polynomials and deduce expansions of q-exponential equations, which lead us to use a systematic method for studying summations and integrals involving q-Laguerre polynomials, such as the Rogers formula, Hardy–Hille formula, mixed-type, $U(n+1)$ type generating functions for q-Laguerre polynomials and transformational identity. We generalize some results of [Sci. China Math. 66(2023), 1199–1216] and [Adv. Math. 131(1997), 93–187].