In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.

In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials.

We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for 3-variable degenerate Hermite Kampé de Fériet polynomials,. Finally, we study the structure and symmetry of pattern about the zeros of the 3-variable degenerate Hermite Kampé de Fériet equations.

In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations.

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions ${}_1{F}_1$ and ${}_2{F}_1$ .

In this paper, we study differential equations arising from the generating function of the $(r,\beta )$ -Bell polynomials. We give explicit identities for the $(r,\beta )$ -Bell polynomials. Finally, we find the zeros of the $(r,\beta )$ -Bell equations with numerical experiments.