Connection and linearization coefficients of the Askey–Wilson polynomials

On structure formulas for Wilson polynomials

“…By studying various properties of D and S, we prove, following previous works, [5][6][7] that the Wilson polynomials defined in this paper by (1.2) (and the Continuous Dual Hahn polynomials *Corresponding author. Email: koepf@mathematik.uni-kassel.de as special case of Wilson polynomials defined by (1.3)) are solutions of the divided difference equation φ(x 2 )D 2 y(x 2 ) + ψ(x 2 )SDy(x 2 ) + λy(x 2 ) = 0, (1.1) where φ and ψ are polynomials of degree 2 and 1, respectively, and λ is a constant depending on the degree of the polynomial solution and the four parameters a, b, c and d and are given in (2.16)- (2.18).…”

“…(3) Connection formulas (3.13) and (3.17) were proved in [7] by a limiting process using the connection formulas for the Askey-Wilson polynomials.…”

Connection and inversion coefficients for basic hypergeometric polynomials

“…The Askey-Wilson polynomials are defined by [5] P n (x; a, b, c, d/q) = (ab, ac, dq; q) n a n 4 φ 3 q −n , abcdq n−1 , ae iθ , ae −iθ ab, ac, ad ; q; q , x = cos(θ), Note that, the inversion and connection problems for Askey-Wilson polynomials were already studied by many authors: Askey and Wilson used orthogonality assumption [5], Area et al used Verma Formula [2]. However, Foupouagnigni et al solved this problem recurrently by computer algebra tools [11].…”

A New Linear Inversion Formula for a class of Hypergeometric polynomials

“…Closed analytical formulae for generalized linearization coefficients for Jacobi polynomials and other special polynomials have also been addressed in [8,9]. Writing the latter coefficient…”