Mumtaz Riyasat

Differential and integral equations for the 2-iterated Appell polynomials

In this article, a set of differential equations of finite order ( k th order, k ź N ) for the 2-iterated Appell polynomials are derived. Particular cases k = 1 and k = 2 are also considered. The integral equations for the Appell and 2-iterated Appell polynomials are established. Further, as an illustration the differential and integral equations for the 2-iterated generalized Bernoulli polynomials are derived. This article is first attempt in the direction of deriving integral equations for the Appell and 2-iterated Appell families and for some members belonging to these families.

Determinantal approach to certain mixed special polynomials related to Gould-Hopper polynomials

In this article, the Gould-Hopper-Sheffer polynomials are introduced by means of generating function and determinantal definition. Certain properties of these polynomials are established and operational relations between the Sheffer and Gould-Hopper-Sheffer polynomials are derived. Examples of some members belonging to the Gould-Hopper-Sheffer and Gould-Hopper-Appell polynomials families are considered and the corresponding numbers for certain polynomials are also obtained.

Certain results for the 2-variable Apostol type and related polynomials

In this article, the 2-variable general polynomials are taken as base with Apostol type polynomials to introduce a family of 2-variable Apostol type polynomials. These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored.

A Note on q-Bernoulli–Euler Polynomials

In this article, a mixed family of q-Bernoulli–Euler polynomials is introduced by means of generating function, series definition, and determinantal definition. Further, the numbers related to the q-Bernoulli–Euler polynomials are considered, and the graph of the q-Bernoulli–Euler polynomials is also drawn for index n = 3 and \( q = 1/2 \).

On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials

Abstract The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

Finding symmetry identities for the 2-variable Apostol type polynomials

Abstract The main aim of this article is to established certain symmetry identities for the 2-variable Apostol type polynomials. The symmetry identities for some special polynomials related to the 2-variable Apostol type polynomials are deduced as special cases. Certain interesting examples are considered to establish the symmetry identities for the 2-variable Gould-Hopper-Apostol type, 2-variable generalized Laguerre-Apostol type and 2-variable truncated exponential-Apostol type polynomials. The special cases of the symmetry identities associated with these polynomials are also given.

q-difference equations for the composite 2D q-Appell polynomials and their applications

Abstract The main aim of this article is to introduce a new class of composite 2D q-Appell polynomials and to study their properties. The generating function, series definition and some explicit relations for these polynomials are derived. These polynomials are studied from determinantal view point and their q-recurrence relations and q-difference equations are established. The composite 2D q-Bernoulli, q-Euler and q-Genocchi and composite q-Bernoulli–Euler, q-Bernoulli–Genocchi and q-Euler–Genocchi polynomials are studied as particular members of this class. Certain interesting examples are considered in terms of these members to give the applications of main results.

Determinantal Approach to Hermite-Sheffer Polynomials

In this article, the determinantal definition for the Hermite-Sheffer polynomials is established using linear algebra tools. Further, the Hermite-Sheffer matrix polynomials are introduced by means of their generating function.