Exploring Gould–Hopper Sheffer–based Appell polynomials via operational approach and Riordan arrays

Mohamed Rhaima; Waseem Ahmad Khan; Shahid Ahmad Wani; Georgia Irina Oros; Mohamed Rhaima; Waseem Ahmad Khan; Shahid Ahmad Wani; Georgia Irina Oros

doi:10.3934/math.2026578

AIMS Mathematics

2026, Volume 11, Issue 5: 14075-14095. doi: 10.3934/math.2026578

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Research article

Exploring Gould–Hopper Sheffer–based Appell polynomials via operational approach and Riordan arrays

1.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
3.
Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Pune, India
4.
Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania

Received: 24 December 2025 Revised: 25 February 2026 Accepted: 26 February 2026 Published: 19 May 2026
MSC : 11B83, 11T23, 12E10 33C45, 33E20

The operational and algebraic framework offers a powerful and systematic approach for investigating the structural properties of hybrid special polynomial families. In this work, we introduce a new class of Gould–Hopper Sheffer-based Appell polynomials (GHSbAP) by combining the Gould–Hopper polynomial structure with the general theory of Sheffer and Appell sequences. The offered construction is implemented by means of exponential generating functions and operational methods based upon the principle of monomiality. Basic properties of the GHSbAP family are constructed including generating functions, series representations, operational identities, quasi-monomial behavior, and differential equations. Further, the computationally efficient characterization of determinant representation is derived through the relation between Sheffer sequences and generalized Riordan arrays. A number of illustrative cases such as Gould–Hopper-Sheffer based Bernoulli and Euler polynomials are demonstrated.
- Sheffer sequences,
- Gould–Hopper–Sheffer sequences,
- monomiality principle,
- operational rules,
- determinant approach,
- Riordan arrays
Citation: Mohamed Rhaima, Waseem Ahmad Khan, Shahid Ahmad Wani, Georgia Irina Oros. Exploring Gould–Hopper Sheffer–based Appell polynomials via operational approach and Riordan arrays[J]. AIMS Mathematics, 2026, 11(5): 14075-14095. doi: 10.3934/math.2026578

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Abstract

The operational and algebraic framework offers a powerful and systematic approach for investigating the structural properties of hybrid special polynomial families. In this work, we introduce a new class of Gould–Hopper Sheffer-based Appell polynomials (GHSbAP) by combining the Gould–Hopper polynomial structure with the general theory of Sheffer and Appell sequences. The offered construction is implemented by means of exponential generating functions and operational methods based upon the principle of monomiality. Basic properties of the GHSbAP family are constructed including generating functions, series representations, operational identities, quasi-monomial behavior, and differential equations. Further, the computationally efficient characterization of determinant representation is derived through the relation between Sheffer sequences and generalized Riordan arrays. A number of illustrative cases such as Gould–Hopper-Sheffer based Bernoulli and Euler polynomials are demonstrated.

References

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