Introducing a novel family of \(\Delta_h\)-Sheffer polynomials and their interconnected hybrid variants
Authors
T. Nahid
- Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India.
Sh. A. Wani
- Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India.
P. Alam
- Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India.
W. Ramirez
- Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58 N 55-66, 080002 Barranquilla, Colombia.
- Section of Mathematics International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy.
Abstract
The main objective of this work is to investigate a novel class of polynomials, called the \(\Delta_h\)-Sheffer polynomials and to explore their various properties. The generating function, explicit representations, quasi-monomiality, and certain novel identities involving \(\Delta_h\)-Sheffer polynomials are obtained. Also, the \(\Delta_h\)-Sheffer polynomials are explored via determinant representation. Further, the \({\Delta_h}\) Gould-Hopper-Sheffer polynomials are introduced with the help of \(\Delta_h\)-Sheffer and \({\Delta_h}\) Gould-Hopper polynomials. Certain fascinating results, such as the generating function, determinant form, multiplicative, and derivative operators and many more results for these hybrid form of the \(\Delta_h\)-Sheffer polynomials are also obtained. Certain examples are considered as the special cases of \({\Delta_h}\) Gould-Hopper-Sheffer polynomials.
Share and Cite
ISRP Style
T. Nahid, Sh. A. Wani, P. Alam, W. Ramirez, Introducing a novel family of \(\Delta_h\)-Sheffer polynomials and their interconnected hybrid variants, Journal of Mathematics and Computer Science, 36 (2025), no. 3, 317--325
AMA Style
Nahid T., Wani Sh. A., Alam P., Ramirez W., Introducing a novel family of \(\Delta_h\)-Sheffer polynomials and their interconnected hybrid variants. J Math Comput SCI-JM. (2025); 36(3):317--325
Chicago/Turabian Style
Nahid, T., Wani, Sh. A., Alam, P., Ramirez, W.. "Introducing a novel family of \(\Delta_h\)-Sheffer polynomials and their interconnected hybrid variants." Journal of Mathematics and Computer Science, 36, no. 3 (2025): 317--325
Keywords
- \(\Delta_h\) special polynomials
- \(\Delta_h\) Sheffer polynomials
- monomiality principle
- explicit forms
- determinant form
MSC
- 11B83
- 15A15
- 33E20
- 33B10
- 33E99
References
-
[1]
P. Appell, Sur une classe de polynˆomes, Ann. Sci. ´ Ecole Norm. Sup. (2), 9 (1880), 119–144
-
[2]
F. A. Costabile, E. Longo, h-Appell sequences and related interpolation problem, Numer. Algorithms, 63 (2013), 165–186
-
[3]
G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, In: Advanced special functions and applications (Melfi, 1999), Aracne, Rome, 1 (2000), 147–164
-
[4]
A. Di Bucchianico, D. Loeb, A selected survey of umbral calculus, Electron. J. Combin., 2 (1995), 28 pages
-
[5]
T.-X. He, L. C. Hsu, P. J.-S. Shiue, The Sheffer group and the Riordan group, Discrete Appl. Math., 155 (2007), 1895–1909
-
[6]
C. Jordan, Calculus of finite differences, Chelsea Publishing Co., New York (1965)
-
[7]
A. Muhyi, A new class of Gould-Hopper-Eulerian-type polynomials, Appl. Math. Sci. Eng., 30 (2022), 283–306
-
[8]
A. Muhyi, S. Araci, A note on q-Fubini-Appell polynomials and related properties, J. Funct. Spaces, 2021 (2021), 9 pages
-
[9]
M. A. O¨ zarslan, B. Y. Yas¸ar, h-Gould-Hopper Appell polynomials, Acta Math. Sci. Ser. B (Engl. Ed.), 41 (2021), 1196–1222
-
[10]
W. Ram´ırez, C. Cesarano, S. D´ıaz, New Results for Degenerated Generalized Apostol–bernoulli, Apostol–euler and Apostol– genocchi Polynomials, WSEAS Trans. Math., 21 (2022), 604–608
-
[11]
W. Ram´ırez, C. Cesarano, Some new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol- Genocchi polynomials, Carpathian Math. Publ., 14 (2022), 354–363
-
[12]
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke. Math. J., 5 (1939), 590–622
-
[13]
H. M. Srivastava, M. A. O¨ zarslan, B. Y. Yas¸ar, Difference equations for a class of twice-iterated h-Appell sequences of polynomials, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113 (2019), 1851–1871
-
[14]
J. F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta. Math., 73 (1941), 333–366
-
[15]
W. Wang, A determinantal approach to Sheffer sequences, Linear Algebra Appl., 436 (2014), 228–254
-
[16]
S. A. Wani, S. Khan, Properties and applications of the Gould-Hopper-Frobenius-Euler polynomials, Tbilisi Math. J., 12 (2019), 93–104
-
[17]
S. A. Wani, S. Khan, T. Nahid, Gould-Hopper based Frobenius-Genocchi polynomials and their generalized form, Afr. Mat., 31 (2020), 1397–1408
-
[18]
S. A.Wani, T. Nahid, Finding determinant and integral forms of the 2-iterated 2 D Appell polynomials, J. Anal., 30 (2022), 731–747
-
[19]
S. A. Wani, T. Nahid, K. Hussain, M. B. Jeelani, A unified matrix approach to the Legendre-Sheffer and certain hybrid polynomial sequences, J. Anal., 32 (2024), 843–867
-
[20]
G. Yasmin, A. Muhyi, Extended forms of Legendre-Gould-Hopper-Appell polynomials, Adv. Stud. Contemp. Math., 29 (2019), 489–504
-
[21]
M. Zayed, S. A. Wani, G. I. Oros, W. Ram´ırez, Unraveling multivariable Hermite-Apostol-type Frobenius-Genocchi polynomials via fractional operators, AIMS Math., 9 (2024), 17291–17304
