We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an L2 space weighted by the weight function of the continuous q-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a q-analog of the expansion of eixy in Jacobi polynomials of argument x. We also outline a general procedure of finding integral repre...

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager’s algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z3−case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closel...

Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stir...

An asymptotic expansion for the Jacobi polynomials and for the functions of the second kind is extended to the case of a weight function that is the product of the Jacobi weight function with an arbitrary positive analytic function.

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...

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler–Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev or...

First we give a compact treatment of the Jacobi polynomials on a simplex in IR which exploits and emphasizes the symmetries that exist. This includes the various ways that they can be defined: via orthogonality conditions, as a hypergeometric series, as eigenfunctions of an elliptic pde, as eigenfunctions of a positive linear operator, and through conditions on the Bernstein–Bézier form. We the...

We construct Jacobi-weighted orthogonal polynomials (α,β,γ) n,r (u,v,w), α,β,γ > −1, α+ β + γ = 0, on the triangular domain T . We show that these polynomials (α,β,γ) n,r (u, v,w) over the triangular domain T satisfy the following properties: (α,β,γ) n,r (u,v,w) ∈ n, n≥ 1, r = 0,1, . . . ,n, and (α,β,γ) n,r (u,v,w) ⊥ (α,β,γ) n,s (u,v,w) for r =s. Hence, (α,β,γ) n,r (u,v,w), n= 0,1,2, . . ., r =...

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, fea...

In this paper we determine the complete coset weight distributions of the second order Reed-Muller code RM(2, 6) of length 64. Our method fully uses the interaction between the Jacobi polynomials for the code RM(2, 6) and those of the dual code RM(3, 6). The method also employs the group theoretic reduction processes to diminish the runtimes of computing the Jacobi polynomials for the code RM(2...