On certain properties of Hybrid Sheffer–$ \lambda $-type special polynomials and their applications

Shahid Ahmad Wani; Waseem Ahmad Khan; William Ramírez; Dixon Salcedo; Mohamed Rhaima; Shahid Ahmad Wani; Waseem Ahmad Khan; William Ramírez; Dixon Salcedo; Mohamed Rhaima

doi:10.3934/math.2026396

AIMS Mathematics

2026, Volume 11, Issue 4: 9563-9586. doi: 10.3934/math.2026396

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Research article

On certain properties of Hybrid Sheffer–$ \lambda $-type special polynomials and their applications

1.
Symbiosis Institute of Technology Pune, Symbiosis International (Deemed) University, Pune, India
2.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
3.
Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58, 55-66, 080002 Barranquilla, Colombia
4.
Computer Science and Electronics Department, Universidad de la Costa, Calle 58, 55-66, 080002 Barranquilla, Colombia
5.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi Arabia

Received: 16 January 2026 Revised: 21 February 2026 Accepted: 04 March 2026 Published: 10 April 2026
MSC : 11B83, 33C45, 12E10, 33E20, 11T23

In this work, a new class of Gould-Hopper Sheffer-$ \lambda $ polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer-$ \lambda $ sequences. The proposed family was defined through an appropriate exponential generating function involving trigonometric factors and was shown to possess rich algebraic and operational properties. Explicit series representations and determinant forms were derived using Riordan array techniques and Cramer's rule. By employing the monomiality principle, the associated multiplicative and derivative operators were constructed, establishing the quasi-monomial character of the introduced polynomials and leading to the corresponding differential equations. Furthermore, several important subclasses, including Gould-Hopper-Bernoulli-$ \lambda $, Euler-$ \lambda $, Genocchi-$ \lambda $, and Laguerre-$ \lambda $ polynomials, were obtained as illustrative examples. These examples demonstrated the unifying nature of the proposed framework and highlighted its potential applicability in operational calculus, special functions, and mathematical physics.
- Sheffer sequences,
- $ \lambda $-polynomials,
- Gould-Hopper polynomials,
- monomiality principle,
- operational rules,
- determinant approach,
- applications
Citation: Shahid Ahmad Wani, Waseem Ahmad Khan, William Ramírez, Dixon Salcedo, Mohamed Rhaima. On certain properties of Hybrid Sheffer–$ \lambda $-type special polynomials and their applications[J]. AIMS Mathematics, 2026, 11(4): 9563-9586. doi: 10.3934/math.2026396

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Abstract

In this work, a new class of Gould-Hopper Sheffer-$ \lambda $ polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer-$ \lambda $ sequences. The proposed family was defined through an appropriate exponential generating function involving trigonometric factors and was shown to possess rich algebraic and operational properties. Explicit series representations and determinant forms were derived using Riordan array techniques and Cramer's rule. By employing the monomiality principle, the associated multiplicative and derivative operators were constructed, establishing the quasi-monomial character of the introduced polynomials and leading to the corresponding differential equations. Furthermore, several important subclasses, including Gould-Hopper-Bernoulli-$ \lambda $, Euler-$ \lambda $, Genocchi-$ \lambda $, and Laguerre-$ \lambda $ polynomials, were obtained as illustrative examples. These examples demonstrated the unifying nature of the proposed framework and highlighted its potential applicability in operational calculus, special functions, and mathematical physics.

References

[1]	L. C. Andrews, Special functions for engineers and applied mathematicians, New York: Macmillan Publishing Company, 1985.
[2]	F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. Appl. Math., 234 (2010), 1528–1542. https://doi.org/10.1016/j.cam.2010.02.033 doi: 10.1016/j.cam.2010.02.033
[3]	F. A. Costabile, E. Longo, An algebraic approach to Sheffer polynomial sequences, Integr. Transf. Spec. F., 25 (2014), 295–311. https://doi.org/10.1080/10652469.2013.842234 doi: 10.1080/10652469.2013.842234
[4]	C. Cesarano, Y. Quintana, W. Ramírez, Degenerate versions of hypergeometric Bernoulli–Euler polynomials, Lobachevskii J. Math., 45 (2024), 3509–3521. https://doi.org/10.1134/S1995080224604235 doi: 10.1134/S1995080224604235
[5]	C. Cesarano, Y. Quintana, W. Ramírez, A survey on orthogonal polynomials from a monomiality principle point of view, Encyclopedia, 4 (2024), 1355–1366. https://doi.org/10.3390/encyclopedia4030088 doi: 10.3390/encyclopedia4030088
[6]	G. Dattoli, E. D. Palma, S. Licciardi, E. Sabia, From circular to Bessel functions: A transition through the umbral method, Fractal Fract., 1 (2017), 9. https://doi.org/10.3390/fractalfract1010009 doi: 10.3390/fractalfract1010009
[7]	G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, Adv. Spec. Funct. Appl., 2000.
[8]	H. W. Gould, A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29 (1962), 51–63. https://doi.org/10.1215/S0012-7094-62-02907-1 doi: 10.1215/S0012-7094-62-02907-1
[9]	N. B. Hepsisler, B. Cekim, M. A. Ozarslan, Generalization of Sheffer-$\lambda$ polynomials, Demonstr. Math., 58 (2025), 20250154. https://doi.org/10.1515/dema-2025-0154 doi: 10.1515/dema-2025-0154
[10]	S. Khan, M. Riyasat, Determinantal approach to certain mixed special polynomials related to Gould-Hopper polynomials, Appl. Math. Comput., 251 (2015), 599–614. https://doi.org/10.1016/j.amc.2014.11.081 doi: 10.1016/j.amc.2014.11.081
[11]	S. Khan, M. Haneef, Properties and applications of Sheffer-based $\lambda$-polynomials, Bol. Soc. Mat. Mex., 30 (2024), 9. https://doi.org/10.1007/s40590-023-00584-2 doi: 10.1007/s40590-023-00584-2
[12]	S. Roman, The Umbral calculus, New York: Academic Press, 1984, 59–63.
[13]	S. Roman, G. C. Rota, The umbral calculus, Adv. Math., 27 (1978), 95–188. https://doi.org/10.1016/0001-8708(78)90087-7
[14]	Y. Quintana, W. Ramírez, A degenerate version of hypergeometric Bernoulli polynomials: Announcement of results, Commun. Appl. Ind. Math., 15 (2024), 36–43. https://doi.org/10.2478/caim-2024-0011 doi: 10.2478/caim-2024-0011
[15]	I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J., 5 (1939), 590–622. https://doi.org/10.1215/S0012-7094-39-00549-1 doi: 10.1215/S0012-7094-39-00549-1
[16]	J. F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math., 73 (1941), 333–366. https://doi.org/10.1007/BF02392231 doi: 10.1007/BF02392231
[17]	W. Wang, A determinantal approach to Sheffer sequences, Linear Algebra Appl., 163 (2014), 228–254. https://doi.org/10.1016/j.laa.2014.09.009 doi: 10.1016/j.laa.2014.09.009
[18]	W. Wang, T. Wang, Generalized Riordan arrays, Discrete Math., 308 (2008), 6466–6500. https://doi.org/10.1016/j.disc.2007.12.037
[19]	S. A. Wani, I. Ahmad, J. A. Ganie, Fractional calculus and families of generalized Legendre-Laguerre-Appell polynomials, Fract. Differ. Calc., 12 (2022), 133–145. https://doi.org/10.7153/fdc-2022-12-09 doi: 10.7153/fdc-2022-12-09
[20]	S. A. Wani, T. Nahid, K. Hussain, M. Begum, A unified matrix approach to the Legendre-Sheffer and certain hybrid polynomial sequences, J. Anal., 32 (2024), 843–867. https://doi.org/10.1007/s41478-023-00648-6 doi: 10.1007/s41478-023-00648-6
[21]	S. A. Wani, J. Choi, Truncated exponential based Frobenius–Genocchi and Apostol-type Frobenius–Genocchi polynomials, Montes Taurus J. Pure Appl. Math., 4 (2022), 85–96.

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