Multiplicative and Additive Finite Free Convolutions for q-Polynomials

Abstract

We study $q$-analogs of finite free convolutions and their interaction with families of $q$-hypergeometric polynomials. First, we revisit the $q$-multiplicative finite free convolution, previously introduced in the literature, and show that it acts naturally on $q$-hypergeometric polynomials: the convolution of two such polynomials remains within the same class, with parameters obtained by concatenation. This observation provides a simple mechanism for constructing large families of $q$-hypergeometric polynomials whose zeros are real and whose logarithmic mesh is controlled. We illustrate it with an example of multiple little $q$-Jacobi polynomials of the first kind. A result of independent interest is also an alternative definition of the $q$-multiplicative convolution in terms of $q$-differential operators. Motivated by the additive finite free convolution, we introduce a $q$-additive finite free convolution and study its algebraic and analytic properties. Although this convolution does not preserve real-rootedness in general, we show that a natural modification involving a $q$-multiplicative convolution restores the preservation of real roots and interlacing for polynomials with bounded logarithmic mesh. Finally, we develop a systematic method to translate product identities of $q$-hypergeometric functions into convolution identities for $q$-hypergeometric polynomials. This approach yields several explicit formulas for $q$-additive convolutions and produces new families of real-rooted $q$-hypergeometric polynomials.