1. Elementary limit formulas We consider the classical orthogonal polynomials as monic polynomials pn(x) = x + terms of degree less than n. We have • Jacobi polynomials p (α,β) n (x) with respect to weight function (1 − x) (1 + x) on (−1, 1); • Laguerre polynomials ln(x) with respect to weight function e −x x on (0,∞); • Hermite polynomials hn(x) with respect to weight function e −x on (−∞,∞). ...

Inner products of the type 〈f, g〉S = 〈f, g〉ψ0 + 〈f ′, g〉ψ1, where one of the measures ψ0 or ψ1 is the measure associated with the Jacobi polynomials, are usually referred to as Jacobi-Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Jacobi-Sobolev inner products. The inner products are such that the associated pair...

Let the Sobolev-type inner product f, g = R f gdµ 0 + R f ′ g ′ dµ 1 with µ 0 = w + M δ c , µ 1 = N δ c where w is the Jacobi weight, c is either 1 or −1 and M, N ≥ 0. We obtain estimates and asymptotic properties on [−1, 1] for the polynomials orthonormal with respect to .,. and their kernels. We also compare these polynomials with Jacobi orthonormal polynomials.

Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be ...

We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable z (spectral parameter) and the other a recurrence relation in n (the lattice variable). For the Jacobi weight w(x) = (1− x)(1 + x) , x ∈ [−1, 1], we s...

Abstract. In this work, hierarchical Jacobi-based expansions are explored for the static analysis of multilayered beams, plates, and shells as structural theories well shape functions. Jacobi polynomials, denoted P_p^((γ,θ) ), belong to family classical orthogonal polynomials depend on two scalars parameters

i P (α,β) ni (x) et en déduire une évaluation dans le cas particulier où n1 = n2 + · · ·+ nm. ABSTRACT. — The explicit non-negative representation of the linearization coefficients of the Jacobi polynomials obtained by RAHMAN seems to be difficult to be derived by combinatorial methods. However several combinatorial interpretations can be provided for the integral of the product of Jacobi polyn...

—An explicit formula for the Fourier coef cients of the Legendre polynomials can be found in the Bateman Manuscript Project. However, formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials. The methods described here apply in principle to an...

We study the bispectrality of Jacobi type polynomials, which are eigenfunctions higher-order differential operators and can be defined by taking suitable linear combinations a fixed number consecutive polynomials. polynomials include, as particular cases, Krall-Jacobi As main results we prove that always satisfy recurrence relations (i.e., they bispectral). also families only orthogonal with re...