The Wilson function transform. (English) Zbl 1079.33005

Groenevelt, Wolter

Int. Math. Res. Not. 2003, No. 52, 2779-2817 (2003).

It is well-known that the Jacobi polynomials can be expressed in terms of the \(_2F_1\)-hypergeometric series. Wilson gave a generalization of Jacobi polynomials as orthogonal \(_4F_3\)-polynomials. These polynomials are called Wilson polynomials, and they are eigenfunctions of a second-order difference operator \(\Lambda \). Certain type of nonpolynomial eigenfunctions of a difference operator \(L\) closely related to \(\Lambda \) are called Wilson functions. The author considers two Hilbert spaces, and based on the asymptotic behaviour of the Wilson function, show that a truncated inner product of two Wilson functions approximates reproducing kernels. This allows him to define two transforms, called Wilson function transforms. The Wilson function transforms of a Jacobi function, and of a Wilson polynomial, are calculated explicitly by using an integral representation of the Wilson function.

Reviewer: Nicolae Cotfas (Bucureşti)

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C20	Generalized hypergeometric series, \({}_pF_q\)
39A05	General theory of difference equations
81R12	Groups and algebras in quantum theory and relations with integrable systems

Keywords:

Wilson plynomials; Jacobi polynomials; second-order difference equations; Wilson function transform