**Title:** Identities for the multiparametric higher-order Hermite-based Peters-type Simsek polynomials of the first kind

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-23-00002; **Volume 5 / Issue 1 / Year 2023**, Pages 102-123

**Document Type:** Research Paper

**Author(s):** Irem Kucukoglu ^{a}

^{a}Department of Engineering Fundamental Sciences, Alanya Alaaddin Keykubat University TR-07425 Antalya, Turkey

Received: 14 March 2023, Accepted: 13 August 2023, Published: 13 September 2023

**Corresponding Author:** Irem Kucukoglu (Email address: irem.kucukoglu@alanya.edu.tr)

**Full Text:** PDF

**Abstract**

The main aim of this study is to investigate the multiparametric higher-order Hermite-based Peters-type Simsek numbers and polynomials of the first kind, which were introduced by the author in her recent paper [18]. To achieve this aim, we first provide pseudocodes for symbolic computation of these numbers and polynomials. Moreover, we implement these pseudocodes in the Wolfram language. By these implementations, we provide some tables and plots regarding these numbers and polynomials in some arbitrarily chosen special cases. By using their generating functions with their functional equations, we derive some finite sums, identities and derivative formulas concerning these numbers and polynomials. We also investigate the first order multiparametric Hermite-based Peters-type Simsek polynomials and we provide some remarks and observations about their some reductions. Finally, we conclude the paper by providing some remarks and open problems on the potential applications that could emerge from the Sheffer-type sequences, the heat-type equations, the orthogonality and the analytic continuation.

**Keywords:** Binomial coefficients, combinatorial numbers and polynomials, computational implementations, finite sums, generating functions, heat-type equations, Hermite polynomials, orthogonal polynomials, Peters polynomials, Sheffer sequences, special numbers and polynomials, two variable Simsek polynomials

**References:**

- G. Bretti and P. E. Ricci,
*Multidimensional extensions of the Bernoulli and Appell polynomials*, Taiwanese J. Math.**8**, 415–428, 2004. - G. Dattoli, C. Chiccoli, S. Lorenzutta, G. Maino and A. Torre,
*Theory of generalized Hermite polynomials*, Computers Math. Applic.**28 (4)**, 71–83, 1994. - G. Dattoli, B. Germano and P. E. Ricci,
*Hermite polynomials with more than two variables and associated bi-orthogonal functions*, Integral Transforms Spec. Funct.**20 (1)**, 17–22, 2009. - G. Dattoli, S. Lorenzutta, G. Maino, A. Torre and C. Cesarano,
*Generalized Hermite polynomials and supergaussian forms*, J. Math. Anal. Appl.**203**, 597–609, 1996. - R. Dere and Y. Simsek,
*Hermite base Bernoulli type polynomials on the umbral algebra*, Russ J. Math. Phys.**22 (1)**, 1–5, 2015. - H. W. Gould and A. T. Hopper,
*Operational formulas connected with two generalizations of Hermite polynomials*, Duke. Math. J.**29**, 51–63, 1962. - D. Gun and Y. Simsek,
*Formulas associated with combinatorial polynomials and two parametric Apostol-type polynomials*, AIP Conf. Proc.**2293 (1)**, 2020; Article ID: 180007, https://doi.org/10.1063/5.0026759. - C. Jordan,
*Calculus of finite differences*(Second Edition), Chelsea Publishing Company, New York, 1950. - S. Khan, M. W. Al-Saad and G. Yasmin,
*Some properties of Hermite-based Sheffer polynomials*, Appl. Math. Comput.**217**, 2169–2183, 2010. - S. Khan, T. Nahid and M. Riyasat,
*Partial derivative formulas and identities involving 2-variable Simsek polynomials*, Bol. Soc. Mat. Mex.**26**, 1–13, 2020. - S. Khan, T. Nahid and M. Riyasat,
*Properties and graphical representations of the 2-variable form of the Simsek polynomials*, Vietnam J. Math.**50**, 95–109, 2022. - N. Kilar,
*Generating functions of Hermite type Milne-Thomson polynomials and their applications in computational sciences*, PhD Thesis, Akdeniz University, Institute of Natural and Applied Sciences, Antalya, 2021. - N. Kilar and Y. Simsek,
*Computational formulas and identities for new classes of Hermite-based Milne-Thomson type polynomials: Analysis of generating functions with Euler’s formula*, Math. Meth. Appl. Sci.**44 (8)**, 6731–6762, 2021. - D. S. Kim and T. Kim,
*A note on Boole polynomials*, Integral Transforms Spec. Funct.**25 (8)**, 627–633, 2014. - D. S. Kim, T. Kim and J. Seo,
*A note on Changhee numbers and polynomials*, Adv. Stud. Theoret. Phys.**7**, 993–1003, 2013. - I. Kucukoglu,
*Derivative formulas related to unification of generating functions for Sheffer type sequences*, AIP Conf. Proc.**2116 (1)**, 2019; Article ID: 100016, https://doi.org/10.1063/1.5114092. - I. Kucukoglu,
*Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters*, Math. Meth. Appl. Sci.**45**, 5043–5066, 2022; https://doi.org/10.1002/mma.8092. - I. Kucukoglu,
*Multiparametric Hermite-based Simsek polynomials*, In: Proceedings Book of the 13th Symposium on Generating Functions of Special Numbers and Polynomials and their Applications (GFSNP 2023) (Ed. by M. Alkan, I. Kucukoglu and O. Ones), Antalya, Turkey, March 11-13, 2023, pp. 213–218; ISBN: 978-625-00-1128-7. - I Kucukoglu and Y Simsek,
*Matrix representations for a certain class of combinatorial numbers associated with Bernstein basis functions and cyclic derangements and their probabilistic and asymptotic analyses*, Appl. Anal. Discrete Math.**15 (1)**, 45–68, 2021. - I. Kucukoglu, B. Simsek and Y. Simsek,
*An approach to negative hypergeometric distribution by generating function for special numbers and polynomials*, Turkish J. Math.**43**, 2337–2353, 2019. - I. Kucukoglu, B. Simsek and Y. Simsek,
*Generating functions for new families of combinatorial numbers and polynomials: Approach to Poisson–Charlier polynomials and probability distribution function*, Axioms**8 (4)**, 2019; Article ID: 112, https://doi.org/10.3390/axioms8040112. - N. J. A. Sloane,
*The on-line encyclopedia of integer sequences*, Sequence: OEIS: A059344. - S. Roman,
*The umbral calculus*, Academic Press, New York, 1984. - Y. Simsek,
*Analysis of the*, Cogent Math.*p*-adic*q*-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications**3 (1)**, 2016; Article ID: 1269393. - Y. Simsek,
*Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and*, Turkish J. Math.*p*-adic*q*-integrals**42**, 557–577, 2018. - Y. Simsek and I. Kucukoglu,
*Some certain classes of combinatorial numbers and polynomials attached to Dirichlet characters: Their construction by*. In: Mathematical Analysis in Interdisciplinary Research (Ed. by I.N. Parasidis, E. Providas, T.M. Rassias), Springer Optimization and Its Applications (Volume 179), Springer, 795–857, 2021.*p*-adic integration and applications to probability distribution functions - H. M. Srivastava, I. Kucukoglu and Y. Simsek,
*Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials*, J. Number Theory**181**, 117–146, 2017. - E. Yuluklu,
*Identities for Hermite base combinatorial polynomials and numbers*, AIP Conf. Proc.**2293**, 2020; Article ID: 180015, https://doi.org/10.1063/5.0031017. - E. Yuluklu,
*A note on Hermite base combinatorial polynomials and numbers*, In: Proceedings Book of the 13th Symposium on Generating Functions of Special Numbers and Polynomials and their Applications (GFSNP 2023) (Ed. by M. Alkan, I. Kucukoglu and O. Ones), Antalya, Turkey, March 11-13, 2023, pp. 175–177; ISBN: 978-625-00-1128-7. - Wolfram Research Inc., Mathematica Online (Wolfram Cloud), Champaign, IL, 2020; https://www.wolframcloud.com.