The connection between semi-classical orthogonal polynomials and discrete integrable systems is well established. The earliest example of a discrete integrable system in semi-classical orthogonal polynomials can be attributed first to Shohat in 1939 [16], then second by Freud [10] in 1976. However it wasn’t until the 1990’s, when the focus within integrable systems shifted from continuous to di...

We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrödinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formu...

Big q-Jacobi polynomials {Pn(·; a, b, c; q)}∞n=0 are classically defined for 0 < a < q −1, 0 < b < q−1 and c < 0. For the family of little q-Jacobi polynomials {pn(·; a, b|q)}∞n=0, classical considerations restrict the parameters imposing 0 < a < q−1 and b < q−1. In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditio...

Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

In this paper, we obtain the extended Wright generalized Hypergeometric function using extended Beta function. We also obtain certain integral representations, Mellin transform and some derivative properties of extended Wright generalized Hypergeometric function. Further, we represent extended Wright generalized Hypergeometric function in the form of Laguerre polynomials and Whittaker function.

A new algorithm for finding line spectral frequencies, LSF, is introduced, based on Laguerre method of root approximation. The method allows to assuredly find all roots one by one without recourse to polynomial deflation, which allows approximation to a high precision. Error bounds can be estimated by approximating from two sides with added margin. An improved variant of Laguerre recursion sche...

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using the matrix models we described in [5], in the limit as β → ∞. We find that the fluctuations are described by Gaussians of variance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for sma...