In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

Abstract. For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal expansions with respect to L∞ norm, which generalize analogous results obtained for little qLegendre, little q-Jacobi and little q-Laguerre polynomials, b...

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. [3] in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or more generally Jacobi) polynomials in a study by Nagel [4]. For this reason, they do not deserve any special study since their properties...

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

A commuting family of symmetric matrices are called the block Jacobi matrices, if they are block tridiagonal. They are related to multivariate orthogonal polynomials. We study their eigenvalues and joint eigenvectors. The joint eigenvalues of the truncated block Jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials.

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...