Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality
arxiv(2024)
Abstract
We address the problem of the weak asymptotic behavior of zeros of families
of generalized hypergeometric polynomials as their degree tends to infinity.
The main tool is the representation of such polynomials as a finite free
convolution of simpler elements; this representation is preserved in the
asymptotic regime, so we can formally write the limit zero distribution of
these polynomials as a free convolution of explicitly computable measures. We
derive a simple expression for the S-transform of the limit distribution, which
turns out to be a rational function, and a representation of the Kampé de
Fériet polynomials in terms of finite free convolutions. We apply these
tools, as well as those from [arXiv:2309.10970], to the study of some
well-known families of multiple orthogonal polynomials (Jacobi-Piñeiro and
multiple Laguerre of the first and second kinds), obtaining results on their
zeros, such as interlacing, monotonicity, and asymptotics.
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