In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provid...

An analysis is presented of the torsional vibration of a rigid disc embedded to the surface of a transversely isotropic elastic half space. By using Hankel transform the mixed boundary value problem transform into a dual integral equation, which then transform to a Fredholm integral equation. After reduction to a Fredholm’s integral equation of second kind, solutions to the problem are obtained...

This paper deals with the Catalan, Hankel, binomial transforms of generalized bivariate Fibonacci and Lucas polynomials. Also, some useful results such as generating functions, Binet formulas, summations defined by using recurrence relations these special polynomials are presented. Furthermore, certain important among deduced obtained new formulas. Finally, Catalan Cassini formulas for also der...

In our previous works “Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants” (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263–1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Sel...