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arXiv · 2211.04858

Multiple orthogonal polynomials associated with the exponential integral

Walter Van Assche·Thomas Wolfs
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Abstract

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^\alpha e^{-x}$ the gamma density and $w_2(x) = x^\alpha E_{\nu+1}(x)$ a density related to the exponential integral $E_{\nu+1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider a related family of mixed type multiple orthogonal polynomials.

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Walter Van Assche, Thomas Wolfs. 2022-11-09. Multiple orthogonal polynomials associated with the exponential integral. https://doi.org/10.1111/sapm.12608

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