Subuhi Khan

Laguerre-based Appell polynomials: Properties and applications

By using certain operational methods, the authors introduce a family of Laguerre-based Appell polynomials. Some properties of Laguerre-Appell polynomials are considered. Further, a correspondence between the Appell family and the Laguerre-Appell family is established, which is proved to be useful for the derivation of results involving Laguerre-Appell polynomials from the results of the corresponding Appell polynomials. Furthermore, several identities for Laguerre-based Bernoulli and Euler polynomials are derived as applications.

On Crofton–Glaisher type relations and derivation of generating functions for Hermite polynomials including the multi-index case

Abstract The Glaisher rule is an operational identity involving the action of an exponential operator containing the second-order derivatives acting on an exponential function. We use the Crofton and monomiality formalism to derive generalized forms to the multi-dimensional case and show its usefulness in the derivation of old and new forms of generating functions for a wealth of Hermite polynomials families.

Lie-theoretic generating relations of two variable Laguerre polynomials

Abstract Generating relations involving two variable laguerre polynomials L n ( x , y ) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl (2) with the help of Weisners method by giving suitable interpretations to the index n of the polynomials L n ( x , y ).

Lie-theoretic generating relations of Hermite 2D polynomials

In this paper, we derive some generating relations involving Hermite 2D polynomials (H2DP) Hm,n(U;x,y), of two variables with an arbitrary 2D matrix U as parameter using Lie-theoretic approach. Certain (known or new) generating relations for the polynomials related to H2DP are also obtained as special cases.

Some properties of Hermite-based Sheffer polynomials

Abstract In this paper, the concepts and the formalism associated with monomiality principle and Sheffer sequences are used to introduce family of Hermite-based Sheffer polynomials. Some properties of Hermite–Sheffer polynomials are considered. Further, an operational formalism providing a correspondence between Sheffer and Hermite–Sheffer polynomials is developed. Furthermore, this correspondence is used to derive several new identities and results for members of Hermite–Sheffer family.

General-Appell Polynomials within the Context of Monomiality Principle

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.

2-Variable generalized hermite matrix polynomials and lie algebra representation**

In this paper, we introduce the 2-variable generalized Hermite matrix polynomials (2VGHMaP) H n γ ( x, y; A ) and discuss their special properties. Further we use the representation theory of the harmonic oscillator Lie algebra G (0,1) to derive certain results involving these polynomials. Furthermore, as applications of our main results, we derive the generating relations for the ordinary as well as matrix polynomials related to 2VGHMaP H n γ ( x, y; A ).

Operational versus Lie-algebraic methods and the theory of multi-variable Hermite polynomials

In this article, we consider the problem of developing a unified point of view on the theory of multi-variable and multi-index Hermite polynomials. We combine the principle of monomiality with the methods of operational nature and show that this approach provides a more flexible tool than a full Lie-algebraic treatment. We also provide several examples illustrating this unified point of view.

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.

Operational methods and Laguerre–Gould Hopper polynomials ☆

Abstract In this paper, the authors introduce the Laguerre–Gould Hopper polynomials by combining the operational methods with the principle of monomiality. Generating function, series definition, differential equation, and certain other properties of Laguerre–Gould Hopper polynomials are derived. Further, operational representations of these polynomials are established, which are used to get integral representations and expansion formulae for these polynomials.