Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications
Abstract
:1. Introduction
2. Identities and Integral Formulas for the Hermite-Type Combinatorial Simsek Polynomials
3. Derivative Formulas for the Hermite-Type Combinatorial Simsek Polynomials
4. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Gould, H.W.; Hopper, A.T. Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 1962, 29, 51–63. [Google Scholar] [CrossRef]
- Dattoli, G. Generalized polynomials, operational identities and their applications. J. Comput. Appl. Math. 2000, 118, 111–123. [Google Scholar] [CrossRef]
- Bayad, A.; Simsek, Y.; Srivastava, H.M. Some array type polynomials associated with special numbers and polynomials. Appl. Math. Comput. 2014, 244, 149–157. [Google Scholar] [CrossRef]
- Dere, R.; Simsek, Y. Hermite base Bernoulli type polynomials on the umbral analysis. Russ. J. Math. Phys. 2015, 22, 1–5. [Google Scholar] [CrossRef]
- Gun, D. Relations among trigonometric functions, Apostol-type numbers and Peters-type Simsek polynomials. Montes Taurus J. Pure Appl. Math. 2023, 5, 90–101. [Google Scholar]
- Gun, D.; Simsek, Y. Formulas associated with combinatorial polynomials and two parametric Apostol-type polynomials. AIP Conf. Proc. 2020, 2293, 180007. [Google Scholar] [CrossRef]
- Kilar, N. Building generating functions for degenerate Simsek-type numbers and polynomials of higher order. Montes Taurus J. Pure Appl. Math. 2024, 6, 186–198. [Google Scholar]
- Kilar, N.; Simsek, Y. A special approach to derive new formulas for some special numbers and polynomials. Turk. J. Math. 2020, 44, 2217–2240. [Google Scholar] [CrossRef]
- Kilar, N.; Simsek, Y. Computational formulas and identities for new classes of Hermite-based Milne-Thomson type polynomials: Analysis of generating functions with Euler’s formula. Math. Methods Appl. Sci. 2021, 44, 6731–6762. [Google Scholar] [CrossRef]
- Kim, D.S.; Kim, T. A note on Boole polynomials. Integral Transform. Spec. Funct. 2014, 25, 627–633. [Google Scholar] [CrossRef]
- Kim, D.S.; Kim, T.; Rim, S.H.; Lee, S.H. Hermite polynomials and their applications associated with Bernoulli and Euler numbers. Discret. Dyn. Nat. Soc. 2012, 2012, 974632. [Google Scholar] [CrossRef]
- Khan, S.; Al-Saad, M.W.; Yasmin, G. Some properties of Hermite-based Sheffer polynomials. Appl. Math. Comput. 2010, 217, 2169–2183. [Google Scholar] [CrossRef]
- Khan, S.; Nahid, T.; Riyasat, M. Partial derivative formulas and identities involving 2-variable Simsek polynomials. Bol. Soc. Mat. Mex. 2020, 26, 1–13. [Google Scholar] [CrossRef]
- Khan, S.; Nahid, T.; Riyasat, M. Properties and graphical representations of the 2-variable form of the Simsek polynomials. Vietnam J. Math. 2022, 50, 95–109. [Google Scholar] [CrossRef]
- Kucukoglu, I. Identities for the multiparametric higher-order Hermite-based Peters-type Simsek polynomials of the first kind. Montes Taurus J. Pure Appl. Math. 2023, 5, 102–123. [Google Scholar]
- Kucukoglu, I. Unification of the generating functions for Sheffer type sequences and their applications. Montes Taurus J. Pure Appl. Math. 2023, 5, 71–88. [Google Scholar]
- Kucukoglu, I.; Simsek, Y. On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers. Appl. Anal. Discret. Math. 2019, 13, 478–494. [Google Scholar] [CrossRef]
- Kucukoglu, I.; Simsek, Y. Computational identities for extensions of some families of special numbers and polynomials. Turk. J. Math. 2021, 45, 2341–2365. [Google Scholar] [CrossRef]
- Kucukoglu, I.; Simsek, Y. New formulas and numbers arising from analyzing combinatorial numbers and polynomials. Montes Taurus J. Pure Appl. Math. 2021, 3, 238–259. [Google Scholar]
- Kucukoglu, I.; Simsek, Y. 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. Discret. Math. 2021, 15, 045–068. [Google Scholar] [CrossRef]
- Kucukoglu, I.; Simsek, B.; Simsek, Y. An approach to negative hypergeometric distribution by generating function for special numbers and polynomials. Turk. J. Math. 2019, 43, 2337–2353. [Google Scholar] [CrossRef]
- Kucukoglu, I.; Simsek, B.; Simsek, Y. Generating functions for new families of combinatorial numbers and polynomials: Approach to Poisson–Charlier polynomials and probability distribution function. Axioms 2019, 8, 112. [Google Scholar] [CrossRef]
- Simsek, Y. Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their alications. Fixed Point Theory Appl. 2013, 2013, 87. [Google Scholar] [CrossRef]
- Simsek, Y. Identities associated with generalized Stirling type numbers and Eulerian type polynomials. Math. Comput. Appl. 2013, 18, 251–263. [Google Scholar] [CrossRef]
- Simsek, Y. Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications. Cogent Math. 2016, 3, 1269393. [Google Scholar] [CrossRef]
- Simsek, Y. Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals. Turk. J. Math. 2018, 42, 557–577. [Google Scholar] [CrossRef]
- Simsek, Y. Remarks and some formulas associated with combinatorial numbers. AIP Conf. Proc. 2019, 2116, 100002. [Google Scholar]
- Simsek, Y. Formulas for Poisson–Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 2019, 113, 931–948. [Google Scholar] [CrossRef]
- Simsek, Y. Peters type polynomials and numbers and their generating functions: Approach with p-adic integral method. Math. Methods Appl. Sci. 2019, 42, 7030–7046. [Google Scholar] [CrossRef]
- Simsek, Y. On Boole-type combinatorial numbers and polynomials. Filomat 2020, 34, 559–565. [Google Scholar] [CrossRef]
- Simsek, Y. A new family of combinatorial numbers and polynomials associated with Peters numbers and polynomials. Appl. Anal. Discret. Math. 2020, 14, 627–640. [Google Scholar] [CrossRef]
- Simsek, Y.; Kucukoglu, I. Some certain classes of combinatorial numbers and polynomials attached to Dirichlet characters: Their construction by p-adic integration and applications to probability distribution functions. In Mathematical Analysis in Interdisciplinary Research; Parasidis, I.N., Providas, E., Rassias, T.M., Eds.; Springer Optimization and Its Applications; Springer: Cham, Switzerland, 2021; Volume 179, pp. 795–857. [Google Scholar]
- Simsek, Y.; So, J.S. On generating functions for Boole type polynomials and numbers of higher order and their applications. Symmetry 2019, 11, 352. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Choi, J. Zeta and q-Zeta Functions and Associated Series and Integrals; Elseveir: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Srivastava, H.M.; Kucukoglu, I.; Simsek, Y. Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials. J. Number Theory 2017, 181, 117–146. [Google Scholar] [CrossRef]
- Yuluklu, E. On the Hermite base Apostol-Euler type polynomials and numbers. AIP Conf. Proc. 2018, 1978, 040005. [Google Scholar] [CrossRef]
- Yuluklu, E. A note on the Milne-Thomson type polynomials. AIP Conf. Proc. 2018, 1978, 040004. [Google Scholar] [CrossRef]
- Yuluklu, E. Identities for Hermite base combinatorial polynomials and numbers. AIP Conf. Proc. 2020, 2293, 180015. [Google Scholar] [CrossRef]
- Yuluklu, E. A note on Hermite base combinatorial polynomials and numbers. In Proceedings of the 13th Symposium on Generating Functions of Special Numbers and Polynomials and their Applications (GFSNP 2023), Antalya, Turkey, 11–13 March 2023; pp. 175–177, ISBN 978-625-00-1128-7. [Google Scholar]
- Yuluklu, E.; Simsek, Y.; Komatsu, T. Identities related to special polynomials and combinatorial numbers. Filomat 2017, 31, 4833–4844. [Google Scholar] [CrossRef]
- Ayman-Mursaleen, M.; Heshamuddin, M.; Rao, N.; Sinha, B.K.; Yadav, A.K. Hermite polynomials linking Szász–Durrmeyer operators. Comput. Appl. Math. 2024, 43, 223. [Google Scholar] [CrossRef]
- Rao, N.; Avinash Kumar Yadav, A.K.; Mursaleen, M.; Sinha, B.K.; Jha, N.K. Szász-Beta operators via Hermite polynomial. J. King Saud Univ. Sci. 2024, 36, 103120. [Google Scholar] [CrossRef]
- Comtet, L. Advanced Combinatorics; D. Reidel Publishing Company: Dordrecht, The Netherlands, 1974. [Google Scholar]
- Jordan, C. Calculus of Finite Differences, 2nd ed.; Chelsea Publishing Company: New York, NY, USA, 1950. [Google Scholar]
- Roman, S. The Umbral Calculus; Academic Press: New York, NY, USA, 1984. [Google Scholar]
- Appell, P.; De Fériet, J.K. Fonctions Hypergeométriques et Hypersphériques: Polynomes d’Hermite; Gauthier-Villars: Paris, France, 1926. [Google Scholar]
- Yuluklu, E. Series representation for Milne-Thomson type polynomials with approach of Mellin transformation. AIP Conf. Proc. 2019, 2116, 100013. [Google Scholar] [CrossRef]
- Merlini, D.; Sprugnoli, R.; Verri, M.C. The Cauchy numbers. Discret. Math. 2006, 306, 1906–1920. [Google Scholar] [CrossRef]
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Yuluklu, E. Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications. Mathematics 2024, 12, 2505. https://doi.org/10.3390/math12162505
Yuluklu E. Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications. Mathematics. 2024; 12(16):2505. https://doi.org/10.3390/math12162505
Chicago/Turabian StyleYuluklu, Eda. 2024. "Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications" Mathematics 12, no. 16: 2505. https://doi.org/10.3390/math12162505