Multiple orthogonal polynomials associated with the exponential integral

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.

arXiv comment38 pages, 3 figures. Some corrections and extra references

References in corpus (7)

• Some classical multiple orthogonal polynomials
• Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• The nearest neighbor recurrence coefficients for multiple orthogonal polynomials
• Singular value statistics of matrix products with truncated unitary matrices
• Mehler-Heine asymptotics for multiple orthogonal polynomials
• Laguerre-Angelesco multiple orthogonal polynomials on an $r$-star

Cited by in corpus (1)

• Rational approximation of Euler's constant using multiple orthogonal polynomials