The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obta...

The Ito equation (a coupled nonlinear wave equation which generalizes the KdV equation) has previously been shown to admit a reduction to a single nonlinear Casimir equation governing the wave solutions. Some analytical properties of the solutions to this equation in certain parameter regimes have been studied recently. However, for general parameter regimes where the analytical approach is not...

Vector-valued L p-convergence of orthogonal series and Lagrange interpolation. Abstract We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the vector-valued expansion problem of L p-functions in terms of Jacobi polynomi...

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms coefficients these The first approach is built on establishing a new formula in which moments shifted polynomials are expressed terms other derived involves hypergeometric function type 4F3(1), cannot be summed general, but special choices involved param...

We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices

This paper presents discrete Galerkin method for obtaining the numerical solution of higher even-order integro-differential equations with variable coefficients. We use the generalized Jacobi polynomials with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. Numerical results are presented to demonstrate the effectiven...

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations generalized Caputo derivatives that appear in the study astronomical objects. The offered solution is approximated as truncated series normalized shifted Jacobi polynomials under assumption exact an element L2. method used solver obtain unknow...

The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same ...

We prove an addition formula for Jacobi functions r ~' ~) (~17_~-~ ) analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structur...

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When f (x) is singular...