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The non-symmetric Lommel function transform. (English) Zbl 1321.33024

The paper is devoted to study the differential-difference operator \(T_{\alpha,\nu}\) which is defined as \[ T_{\alpha,\nu}f(x)=f'(x) +(2 \alpha+\nu+\frac{1}{2}) \Delta f(x)+(\nu+\frac{1}{2})\Delta f(-x), \] with \(\Delta\) the backward shift operator.
With the aim of showing its importance in the theory of special functions and integral transforms, first the authors prove that, under some initial conditions, the even and the odd parts of its eigenfunctions can be expressed in terms of the Lommel functions. They also prove that its eigenfunctions, called non-symmetric Lommel functions, have an integral representation of Mehler type in terms of the non-symmetric Bessel function.
The authors also introduce the non-symmetric Hardy transform and establish a relationship between this transformation and Dunkl transforms and they prove the inversion formula for the non-symmetric Hardy transforms. Finally they construct an intertwining operator between \(T_{\alpha,\nu}\) and the Dunkl operator. As application they study a set of polynomials of Boas and Buck type by using the quasi monomiality method related to this operator.

MSC:

33E30 Other functions coming from differential, difference and integral equations
33C47 Other special orthogonal polynomials and functions
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