We show that limit transitions from Askey-Wilson polynomials to q-Racah, little and big q-Jacobi polynomials can be made rigorous on the level of their orthogonality measures in a suitable weak sense. This allows us to derive the orthogonality relations and norm evaluations for the q-Racah polynomials, little and big q-Jacobi polynomials by taking limits in the orthogonality relations and norm ...

Potential problems for thin elliptical plates are solved exactly with emphasis on computation of the electrostatic energy. Expansions in terms of Jacobi polynomials are used.

X iv :0 90 5. 25 57 v2 [ m at h. R T ] 1 0 Ju n 20 09 JACOBI–TRUDY FORMULA FOR GENERALISED SCHUR POLYNOMIALS A.N. SERGEEV AND A.P. VESELOV Abstract. Jacobi–Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.

Let {Rn} and {Pn} be two polynomial systems which induce signed polynomial hypergroup structures on N0. We investigate when the Banach algebra l(N0, h) can be continuously embedded into or is isomorphic to l(N0, h ). We find sufficient conditions on the connection coefficients cnk given by Rn = ∑n k=0 cnkPk, for the existence of such an embedding or isomorphism. Finally we apply these results t...

It is shown that if α, β ≥ − 12 , then the orthonormal Jacobi polynomials p (α,β) n fulfill the local estimate |p n (t)| ≤ C(α, β) ( √ 1− x+ 1 n ) α+ 2 ( √ 1 + x+ 1 n ) β+ 2 for all t ∈ Un(x) and each x ∈ [−1, 1], where Un(x) are subintervals of [−1, 1] defined by Un(x) = [x− φn(x) n , x+ φn(x) n ]∩[−1, 1] for n ∈ N and x ∈ [−1, 1] with φn(x) = √ 1− x2+ 1 n . Applications of the local estimate ...

It is shown that a conjecture concerning the derivatives of orthogonal polynomials, proved by Nevai in 1990 for generalized Jacobi weights, holds for doubling weights as well.

We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q| = 1. In this framework, we prove the Heine’s relations and a connection formula for the q-hypergeometric function of the Barnes type. We also prove an orthogonality relation of the q-little Jacobi polynomials at |q| = 1.

Abstract. There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The ...

For arbitrary β > 0, we use the orthogonal polynomials techniques developed in [10, 11] to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these...

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight slr+1 irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the...