arXiv subjects

Subuhi Khan

Publications and source records attributed to Subuhi Khan.

13 recordsLinked to original sources

arXiv2025

A novel advancement in the study of Appell polynomials via Pad\`e rational approximants

The use of approximants of Pad\`e type are employed to develop a method aimed at opening new perspectives in the theory of Appell polynomials $a_n(x)$, specified by the generating function \sum_{n=0}^{\infty} \frac{t^n}{n!} a_n(x) = A(t) e^{xt}. In this article, the expansion of amplitude $A(t)$ of the Appell polynomials family in terms of rational approximants yields the possibility of determining the approximation of the $a_n(x)$ in terms of other special polynomials. Application of this approach to Hermite polynomials yields highly accurate approximations in terms of truncated exponential polynomials. Further, monomiality conditions are explored and formalism is extended to consider the Pad\'e approximants within the context of umbral notation.

math.CA↗

arXiv2025

Operational-umbral approach to bivariate degenerate Hermite polynomials and their partial orthogonality

The operational calculus associated with special polynomials has proven to be a powerful tool for analyzing and simplifying their properties. This article examines the bivariate degenerate Hermite polynomials with a focus on their differential equations, monomiality properties, operational identities, and partial orthogonality conditions. These polynomials are redeveloped within the framework of umbral formalism, which is further extended to derive certain new results. The study concludes with key observations and insights that highlight the significance of this approach in advancing the understanding of degenerate Hermite polynomials of negative order.

math.CA↗

arXiv2025

Bivariate degenerate Hermite polynomials in the framework of Lie algebra K5

In this article, the matrix elements of a representation of the 5-dimensional Lie algebra K5 are obtained for the first time. The bivariate degenerate Hermite polynomials Hm(z1, z2|{\tau} ) are considered within the context of this representation. Further, employing the Lie algebraic techniques, certain specific results concerning these polynomials are established.Some examples providing the implicit formulas for the polynomials related to the polynomials Hm(z1, z2|{\tau} ) are considered. Integral equations for these polynomials are also explored.

math.CA↗

arXiv2025

Gauss-Appell polynomials: An umbral calculus approach

This article aims to reinforce the broad applicability of the umbral approach to address complex mathematical challenges and contribute to various scientific and engineering endeavors. The umbral methods are used to reformulate the theoretical framework of special functions and provide powerful techniques for uncovering new extensions and relationships among these functions. This research article introduces an innovative class of special polynomials, specifically the Gauss-Appell polynomials. The fundamental attributes of this versatile family of special polynomials are outlined, including generating relations, explicit representations, and differential recurrence relations. Certain examples of the particular members that belong to the class of Gauss-Appell polynomials are also considered.

math.CA↗

arXiv2025

Higher-order Hermite numbers: Properties and applications to evolution problems

The operational calculus associated with Hermite numbers has been shown to be an effective tool for simplifying the study of special functions. Within this context, Hermite polynomials have been viewed as Newton binomials, with the consequent possibility of establishing previously unknown properties. In this article, this method is extended to study the lacunary Hermite polynomials and obtain novel results concerning their generating functions, recurrence relations, differential equations and certain integral transforms. The proposed method is systematically applied to a variety of evolution equations. Furthermore, this idea is extended to combinatorial interpretation of these polynomials, broadening their applicability in mathematical analysis and discrete structures.

math.NT↗

arXiv2024

Umbral insights into a hybrid family of hypergeometric and Mittag-Leffler functions

The umbral approach provides methods for comprehending and redefining special functions. This approach is employed efficiently in order to uncover intricacies and introduce new families of special functions. In this article, the umbral perspective is adopted to introduce a hybrid family of hypergeometric and Mittag-Leffler functions. The umbral-operational procedures are used to derive the generating functions, explicit representations, differential recurrence formulae, and specific integral formulae. Further, the Laplace and Sumudu transforms for the hypergeometric-Mittag-Leffler functions are established. The graphical representation and pattern for distribution of zeros for suitable values of parameters are also presented.

math.CA↗

arXiv2024

Unveiling new perspectives of hypergeometric functions using umbral techniques

The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of Gauss hypergeometric functions and extend the formalism to certain generalized forms of these functions. It is shown that suggested approach is particularly efficient for evaluating integrals involving hypergeometric functions and their combination with other special functions.

math.CA↗

arXiv2020

Certain results for unified Apostol type-truncated exponential-Gould-Hopper polynomials and their relatives

The present article aims to introduce a unified family of the Apostol type-truncated exponential-Gould-Hopper polynomials and to characterize its properties via generating functions. A unified presentation of the generating function for the Apostol type-truncated exponential-Gould-Hopper polynomials is established and its applications are given. By the use of operational techniques, the quasi-monomial properties for the unified family are proved. Several explicit representations and multiplication formulas related to these polynomials are obtained. Some general symmetric identities involving multiple power sums and Hurwitz-Lerch zeta functions are established by applying different analytical means on generating functions.

math.GM↗

arXiv2017

A linear algebra approach to hybrid polynomial sequences

In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and relationships between the Pascal functional matrices and the Wronskian matrices. The corresponding results for some mixed type Sheffer polynomials are also obtained.

math.CA↗

arXiv2016

Determinant approach to the 2-Iterated q-Appell and mixed type q-Appell polynomials

In this article, the 2-iterated q-Appell family is introduced. Certain 2-iterated q-Appell and mixed type q-special polynomials are considered as members of this family. The numbers related to these polynomials are obtained. The determinant definitions for the 2-iterated q-Appell family and for the 2-iterated and mixed type q-Appell polynomials are established. The graphs of some 2-iterated q-Appell and mixed type q-Appell polynomials are drawn for different values of indices and the roots of these polynomials are also investigated for certain values of index n by using Matlab. Finally, the approximate solutions of the real zeros of these polynomials are given.

math.CA↗

arXiv2016

Finding mixed families of special polynomials associated with Appell sequences

In this paper, certain mixed special polynomial families associated with Appell sequences are introduced and their properties are established. Further, operational rules providing connections between these families and the known special polynomials are established, which are used to derive the identities and results for the members of these new families. Determinantal definitions of the polynomials associated with Appell family is also derived. The approach presented is general.

math.CA↗

arXiv2015

2-iterated Sheffer polynomials

In this article, the 2-iterated Sheffer polynomials are introduced by means of generating function and operational representation. Using the theory of Riordan arrays and relations between the Sheffer sequences and Riordan arrays, a determinantal definition for these polynomials is established. The quasi-monomial and other properties of these polynomials are derived. The generating function, determinantal definition, quasi-monomial and other properties for some new members belonging to this family are also considered.

math.CA↗

arXiv2015

Symmetry identities for 2-variable Apostol type and related polynomials

In this article, certain symmetry identities for the 2-variable Apostol type polynomials are derived. By taking suitable values of parameters and indices, the symmetry identities for the special cases of the 2-variable Apostol type polynomials are established. Further, the symmetry identities for certain members belonging to the 2-variable Apostol type polynomials are also considered.

math.CA↗