**Title:** Relations among trigonometric functions, Apostol-type numbers and Peters-type Simsek polynomials

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

**Article ID:** MTJPAM-D-23-00001; **Volume 5 / Issue 1 / Year 2023**, Pages 90-101

**Document Type:** Research Paper

^{a}Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey

Received: 10 February 2023, Accepted: 27 July 2023, Published: 6 September 2023

**Corresponding Author:** Damla Gun (Email address: damlagun@akdeniz.edu.tr)

**Full Text:** PDF

**Abstract**

The main purpose of this paper is to derive some new identities and finite sums involving some trigonometric functions, Apostol-type numbers, the Stirling numbers, and two variable Fibonacci polynomials with the aid of generating functions for the Peters-type Simsek numbers and polynomials. We give finite and infinite series representations containing the Peters-type Simsek numbers and polynomials of the first and second kinds. Using trigonometric functions and generating functions, we obtain some formulas relations among the Apostol Bernoulli numbers and polynomials, the Stirling numbers, and the Peters-type Simsek numbers of the first kind. Finally, we introduce some infinite series computational formulas associated with trigonometric functions, the Peters-type Simsek numbers and polynomials, Fibonacci numbers and polynomials, and the Chebyshev polynomials of the second kind.

**Keywords:** Apostol type numbers and polynomials, combinatorial numbers and polynomials, Changhee numbers, Humbert polynomials, generating function, Peters-type Simsek numbers of the first kind, special numbers

**References:**

- T. M. Apostol,
*On the Lerch zeta function*, Pac. Asian J. Math.**1 (2)**, 161–167, 1951. - J. P. Boyd,
*Chebyshev and Fourier spectral methods*(Second Edition), Dover Publication, Inc., New York, 2000. - L. Carlitz,
*Some theorems on Bernoulli numbers of higher order*, Pac. Math.**2 (2)**, 127–139, 1952. - B. S. El-Desouky and M. Abdelfattah,
*New results and matrix representation for Daehee and Bernoulli numbers and polynomials*, Appl. Math. Sci. (Ruse)**9 (73)**, 3593–3610, 2015. - D. Gun and Y. Simsek,
*Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers*, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM**114 (167)**, 2020; https://doi.org/10.1007/s13398-020-00899-z. - P. Humbert,
*Some extensions of Pincherle’s polynomials*, Proc. Edinb. Math. Soc.**39**, 21–24, 1920. - N. Kilar and Y. Simsek,
*Identities for special numbers and polynomials involving Fibonacci-type polynomials and Chebyshev polynomials*, Adv. Stud. Contemp. Math.**30 (4)**, 493–502, 2020. - D. S. Kim and T. Kim,
*Daehee numbers and polynomials*, Appl. Math. Sci. (Ruse)**7**, 5969–5976, 2013. - D. S. Kim, T. Kim and J. Seo,
*A note on Changhee numbers and polynomials*, Adv. Stud. Theor. Phys.**7**, 993–1003, 2013. - T. Koshy,
*Fibonacci and Lucas numbers with applications*, John Wiley & Sons, Inc., New York, 2001. - Q. M. Luo and H. M. Srivastava,
*Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials*, J. Math. Anal. Appl.**308**, 290–302, 2005. - J. C. Mason and D. C. Handscomb,
*Chebyshev polynomials*, Chapman and Hall/CRC, New York, 2003. - L. M. Milne-Thomson,
*Calculus of Finite Differences*, Chelsea, American Mathematical Society, 1980. - G. Ozdemir and Y. Simsek,
*Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers*, Filomat**30 (4)**, 969–975, 2016. - G. Ozdemir, Y. Simsek and G. V. Milovanović,
*Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials*, Mediterr. J. Math.**14**, 2017; Article ID: 117. - E. D. Rainville,
*Special functions*, The Macmillan Company, New York, 1960. - Y. Simsek,
*On*, Int. J. Math. Comput.*q*-deformed Stirling numbers**15 (2)**, 1–11, 2012. - Y. Simsek,
*Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications*, Fixed Point Theory Appl.**2013**, 2013; Article ID: 87. - Y. Simsek,
*Apostol type Daehee numbers and polynomials*, Adv. Stud. Contemp. Math.**26 (3)**, 555–566, 2016. - Y. Simsek,
*Identities on the Changhee numbers and Apostol-type Daehee polynomials*, Adv. Stud. Contemp. Math.**27 (2)**, 199–212, 2017. - Y. Simsek,
*New families of special numbers for computing negative order Euler numbers and related numbers and polynomials*, Appl. Anal. Discret. Math.**12**, 1–35, 2018. - 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,
*Remarks and some formulas associated with combinatorial numbers*, AIP Conference Proceedings**2116 (1)**, 2019; Article ID: 100002, https://doi.org/10.1063/1.5114078. - Y. Simsek,
*A new family of combinatorial numbers and polynomials associated with Peters numbers and polynomials*, Appl. Anal. Discrete Math.**14**, 627–640, 2020. - Y. Simsek,
*Interpolation functions for new classes special numbers and polynomials via applications of**p**-adic integrals and derivative operator*, Montes Taurus J. Pure Appl. Math.**3 (1)**, 38–61, 2021. - Y. Simsek,
*Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from**p**-adic integrals and special functions*, Math. Methods Appl. Sci.**45 (16)**, 9520–9544, 2022. - Y. Simsek,
*Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials*, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM**117 (130)**, 2023; https://doi.org/10.1007/s13398-023-01464-0. - H. M. Srivastava,
*Some generalizations and basic (or**q**-) extensions of the Bernoulli, Euler and Genocchi polynomials*, Appl. Math. Inf. Sci.**5 (3)**, 390–444, 2011. - H. M. Srivastava and J. Choi,
*Zeta and**q**-zeta functions and associated series and integrals*, Elsevier, Amsterdam, 2012. - 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. - H. M. Srivastava and A. Pinter,
*Remarks on some relationships between the Bernoulli and Euler polynomials*, Appl. Math. Lett.**17**, 375–380, 2004.