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**The Wilson function transform.**
*(English)*
Zbl 1079.33005

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 |