A study of algebraic and operational characteristics of hybrid special polynomials in the context of Sheffer sequences

Mohra Zayed; Shahid Ahmad Wani; Waseem Ahmad Khan; Ketan Kotecha; Mdi Begum Jeelani; Mohra Zayed; Shahid Ahmad Wani; Waseem Ahmad Khan; Ketan Kotecha; Mdi Begum Jeelani

doi:10.3934/nhm.2026056

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1479-1500. doi: 10.3934/nhm.2026056

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A study of algebraic and operational characteristics of hybrid special polynomials in the context of Sheffer sequences

1.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India
3.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
4.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia

Received: 19 March 2026 Revised: 25 June 2026 Accepted: 25 June 2026 Published: 07 July 2026

This paper addresses the problem of constructing and analyzing a new four-variable family of hybrid special polynomials that unifies several classical polynomial systems within the Sheffer sequence framework. Specifically, we introduce the Legendre-Laguerre-Gould-Hopper-based Sheffer polynomials (LeLGHSP) by embedding the Legendre-Laguerre–Gould–Hopper hybrid structure defined via Bessel-Tricomi functions and higher-order exponential terms into the Sheffer generating function governed by the pair $ (\mu(\tau), \sigma(\tau)) $. Using operational calculus, inverse derivative operators, and Riordan array methods, we derive explicit series representations, determinantal forms via Cramer's rule, four operational identities expressing the new family in terms of known Sheffer and Gould–Hopper polynomials, and shift relations of binomial type. The quasi-monomial structure of the family is established by constructing multiplicative and derivative operators satisfying the Weyl algebra commutation relation, from which differential equations governing the polynomials are obtained directly. Three important special cases (generalized Legendre-Laguerre-Gould-Hopper-Hermite, Laguerre, and Pidduck-type Sheffer polynomials) are investigated in a unified manner. Surface plots illustrate the geometric behavior of the family. The framework generalizes numerous classical polynomial families and has potential applications in quantum mechanics, heat-type equations, approximation theory, and combinatorics.
- Sheffer sequences,
- Legendre-Laguerre-Gould-Hopper-Sheffer sequences,
- monomiality principle,
- surface plots,
- operational rules,
- determinant approach
Citation: Mohra Zayed, Shahid Ahmad Wani, Waseem Ahmad Khan, Ketan Kotecha, Mdi Begum Jeelani. A study of algebraic and operational characteristics of hybrid special polynomials in the context of Sheffer sequences[J]. Networks and Heterogeneous Media, 2026, 21(4): 1479-1500. doi: 10.3934/nhm.2026056

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Abstract

This paper addresses the problem of constructing and analyzing a new four-variable family of hybrid special polynomials that unifies several classical polynomial systems within the Sheffer sequence framework. Specifically, we introduce the Legendre-Laguerre-Gould-Hopper-based Sheffer polynomials (LeLGHSP) by embedding the Legendre-Laguerre–Gould–Hopper hybrid structure defined via Bessel-Tricomi functions and higher-order exponential terms into the Sheffer generating function governed by the pair $ (\mu(\tau), \sigma(\tau)) $. Using operational calculus, inverse derivative operators, and Riordan array methods, we derive explicit series representations, determinantal forms via Cramer's rule, four operational identities expressing the new family in terms of known Sheffer and Gould–Hopper polynomials, and shift relations of binomial type. The quasi-monomial structure of the family is established by constructing multiplicative and derivative operators satisfying the Weyl algebra commutation relation, from which differential equations governing the polynomials are obtained directly. Three important special cases (generalized Legendre-Laguerre-Gould-Hopper-Hermite, Laguerre, and Pidduck-type Sheffer polynomials) are investigated in a unified manner. Surface plots illustrate the geometric behavior of the family. The framework generalizes numerous classical polynomial families and has potential applications in quantum mechanics, heat-type equations, approximation theory, and combinatorics.

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