We dedicate this paper to Sir John Kingman on his 70th Birthday. In modern mathematical population genetics the ancestral history of a population of genes back in time is described by John Kingman’s coalescent tree. Classical and modern approaches model gene frequencies by diffusion processes. This paper, which is partly a review, discusses how coalescent processes are dual to diffusion process...

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials introduced by Cariñena et al., [3]. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical...

After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss different extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approach, which are new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numeri...

Abstract. We consider a Lagrange–Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r−1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L-spaces, 1 < p < ∞, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ...

Marcinkiewicz-Zygmund Inequalities involving the zeros {xkn} of Jacobi polynomials for the weight wα,β can take the form

In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished throu...

The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial...

The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpr...

In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expre...

We develop a class of stochastic numerical schemes for Hamilton-Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the HamiltonJacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the JinXin relaxation approximation in physical space...