The non-symmetric Lommel function transform

Abstract

In this work, we introduce a differential-difference operator $T_{\alpha ,\nu }$, which includes as particular cases the derivative operator and the Dunkl operator associated with root system of type $A_1$. We show that the symmetrization of the eigenfunction of this operator is a solution of an inhomogeneous Bessel differential equation. Also we construct an intertwining operator between $T_{\alpha ,\nu }$ and the Dunkl operator. As application, we study a polynomials set of Boas and Buck type by using the quasi-monomiality method related to this operator.

Keywords:

• special functions
• differential-difference operator
• intertwining operator

2000 Mathematics Subject Classifications:

• 33E30
• 33C47

Acknowledgements

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No (RG-1435-026).

Disclosure statement

No potential conflict of interest was reported by the authors.