Differential equations for p, q-Touchard polynomials

Fractional differential equations with Legendre polynomials

q-DIFFERENTIAL EQUATIONS FOR q-CLASSICAL POLYNOMIALS AND q-JACOBI-STIRLING NUMBERS

The q-classical polynomials are orthogonal polynomial sequences that are eigenfunctions of a second order q-differential operator of a certain type. We explicitly construct q-differential equations of arbitrary even order fulfilled by these polynomials, while giving explicit expressions for the integer composite powers of the aforementioned second order q-differential operator. The latter is ac...

The generalized Touchard polynomials revisited

We discuss the generalized Touchard polynomials introduced recently by Dattoli et al. as well as their extension to negative order introduced by the authors with operationial methods. The connection to generalized Stirling and Bell numbers is elucidated and analogs to Burchnall’s identity are derived. A recursion relation for the generalized Touchard polynomials is established and it is shown t...

fractional differential equations with legendre polynomials

Application of Laguerre Polynomials for Solving Infinite Boundary Integro-Differential Equations

In this study‎, ‎an efficient method is presented for solving infinite boundary integro-differential equations (IBI-DE) of the second kind with degenerate kernel in terms of Laguerre polynomials‎. ‎Properties of these polynomials and operational matrix of integration are first presented‎. ‎These properties are then used to transform the integral equation to a matrix equation which corresponds t...