THE GENERAL INVERSION PAIR AND ITS APPLICATIONS

Abstract

In this paper, a general inversion pair is established which provides deformed versions of the Humbert polynomial, Wilson polynomials and Racah polynomials along with their inverse series. Their several particular cases have been illustrated which include the deformed polynomials of Kinney, Pincherle, Gegenbauer, Legendre etc. The general inverse pair also deforms the Bessel function together with the Neumann series as its inverse series. The $p$-deformed Gauss sum is used together with the inverse series to derive certain summation formulas. Finally, all six classes of inverse series relations involving combinatorial identities due to John Riordan (Combinatorial identities, Wiley, 1968) are extended.