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

Based on the importance of q -Laguerre polynomials as the basis polynomials in the q -Askey scheme and Hardy–Hille formulas in generatingfunctionology, we construct a new pair of bivariate q -Laguerre polynomials and deduce expansion of analytic functions via q -exponential equations and q -partial differential equations, which leads to further research q -analysis of the q -Laguerre polynomials, such as the Rogers formula, q -Hardy–Hille formula, mixed-type, 𝑈 ⁢ ( 𝑛 + 1 ) type generating functions for q -Laguerre polynomials and Andrews–Askey integrals, which generalize results of Liu (2023) [40] and Milne (1997) [44] and enhance results of Cao et al. (2024) [15] .