Multiplicative and Additive Finite Free Convolutions for q-Polynomials
arXiv:2606.15003
Abstract
We study -analogs of finite free convolutions and their interaction with families of -hypergeometric polynomials. First, we revisit the -multiplicative finite free convolution, previously introduced in the literature, and show that it acts naturally on -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 -hypergeometric polynomials whose zeros are real and whose logarithmic mesh is controlled. We illustrate it with an example of multiple little -Jacobi polynomials of the first kind. A result of independent interest is also an alternative definition of the -multiplicative convolution in terms of -differential operators. Motivated by the additive finite free convolution, we introduce a -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 -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 -hypergeometric functions into convolution identities for -hypergeometric polynomials. This approach yields several explicit formulas for -additive convolutions and produces new families of real-rooted -hypergeometric polynomials.