**Title:** Interpolation functions for new classes special numbers and polynomials via applications of p-adic integrals and derivative operator

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

**Article ID:** MTJPAM-D-20-00000; **Volume 3 / Issue 1 / Year 2021**, Pages 38-61

**Document Type:** Research Paper

^{a}Department of Mathematics, Faculty of Science University of Akdeniz TR-07058, Antalya-TURKEY

Received: 28 January 2020, Accepted: 16 December 2020, Available online: 7 January 2021.

**Corresponding Author:** Yilmaz Simsek (Email address: ysimsek@akdeniz.edu.tr)

**Full Text: **PDF

**Abstract**

The main purpose of this paper is to not only define Apostol type new classes of numbers and polynomials, but also construct generating function for two new classes of special combinatorial numbers and polynomials by applications of p-adic integrals including the Volkenborn integral and the fermionic integral. By using these generating functions, we introduce not only fundamental properties of these combinatorial numbers and polynomials, but also new identities and formulas. In general, identities and formulas obtained in this paper include the newly introduced combinatorial numbers and polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials, Apostol-Bernoulli numbers and polynomials, Apostol-Euler numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Changhee numbers, the generalized Eulerian type numbers, Eulerian polynomials, Fubini numbers, Dobinski numbers. Moreover, by applying derivative operator to the generating functions for twonew classes of special combinatorial numbers, we construct interpolation functions for these numbers. We also introduce another zeta-type function which interpolates a special case of one of the newly introduced combinatorial numbers at negative integers. Very interesting results are obtained from these interpolation functions, especially a new combinatorial numbers derived. So, 4 open problems are raised involving these new numbers. Finally, we give conclusions for the results of this paper with somecomments and observations.

**Keywords:** Generating functions, special numbers and polynomials, Bernoulli-type numbers and polynomials, Euler-type numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Fubini numbers, p-adic integral

**References:**

- M. Alkan and Y. Simsek,
*Generating function for**q**-Eulerian polynomials and their decomposition and applications*, Fixed Point Theory Appl.**2013 (72)**, 1–14, 2013. - H. Alzer, J. Choi,
*The Riemann zeta function and classes of infinite series*, Appl. Anal. Discrete Math.**11**, 386–398, 2017. - T. M. Apostol,
*On the Lerch zeta function*, Pacific J. Math.**1**, 161–167, 1951. - A. A. Aygunes and Y. Simsek,
*Unification of multiple Lerch-Zeta type functions*, Adv. Studies Contemp. Math,**21**, 367–373, 2011. - T.-T. Bai and Q.-M. Luo, A Simple proof of a binomial identity with applications, Montes Taurus J. Pure Appl. Math. Article ID: MTJPAM-D-19-00008
**1 (2)**, 13–20, 2019. - A. Bayad, and Y. Simsek,
*Values of twisted Barnes zeta functions at negative integers*, Russ. J. Math. Phys.**139 (20)**, 129–137, 2013. - A. Bayad and Y. Simsek,
*Note on the Hurwitz Zeta function of higher order*, AIP Conference Proceedings**1389**, 389, 389–391, 2011, https://doi.org/10.1063/1.3636744. - L. Carlitz,
*Eulerian numbers and polynomials*, Math. Mag.**32**, 247–260, 1959. - L. Carlitz,
*Generating functions*, Fibonacci Q.**7**, 359–393, 1969. - L. Carlitz,
*Some numbers related to the Stirling numbers of the first and second kind*,*Publ. Elektroteh. Fak. Univ. Beogr., Mat.***(544-576)**, 49–55, 1976. - L. Carlitz,
*A note on the multiplication formulas for the Bernoulli and Euler polynomials*, Proc. Am. Math. Soc.**4**, 184–188, 1953. - J. Choi,
*Remark on the Hurwitz-Lerch zeta function*, Fixed Point Theory and Appl.,**2013 (70)**, 2013. - J. Choi and H. M. Srivastava,
*Certain families of series associated with the Hurwitz-Lerch Zeta function*, Appl. Math. Comput.**170 (1)**, 399–409, 2005. - J. Choi and H. M. Srivastava,
*The multiple Hurwitz zeta function and the multiple Hurwitz-Euler eta function*, Taiwanese J. Math.**15(2)**, 501–522, 2011. - J. Choi,
*Note on Apostol-Daehee polynomials and numbers*, Far East J. Math. Sci.,**101 (8)**, 1845–1857, 2017. - L. Comtet,
*Advanced Combinatorics*. Dordrecht-Holland/ Boston-U.S.A.: D. Reidel Publication Company, 1974. - B.S. El-Desouky and A. Mustafa,
*New results and matrix representation for Daehee and Bernoulli numbers and polynomials,*Appl. Math. Sci.**9 (73)**, 3593–3610, 2015 - I. J. Good,
*The number of ordering of**n**candidates when ties are permitted*, Fibonacci Quart.**13**, 11–18, 1975. - H. W. Gould,
*Combinatorial Numbers and Associated Identities: Table 1: Stirling Numbers*, Edited and Compiled by Jocelyn Quaintance May 3, 2010, https://math.wvu.edu/hgould/Vol.7.PDF - N. Kilar and Y. Simsek,
*A new family of Fubini numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials*, J. Korean Math. Soc.**54 (5)**, 1605–1621, 2017. - N. Kilar and Y. Simsek,
*Identities and relations for Fubini type numbers and polynomials via generating functions and**p**-adic integral approach*, Publ. Inst. Math., Nouv. Sér.**106 (120)**, 113–123, 2019. - D. Kim, H. Ozden Ayna, Y. Simsek and A. Yardimci ,
*New families of special numbers and polynomials arising from applications of**p**-adic**q**-integrals*, Adv. Difference Equ.**2017 (207)**, 1–11, 2017, DOI 10.1186/s13662- 017-1273-4 - D.S. Kim and T. Kim,
*Daehee numbers and polynomials*, Appl. Math. Sci. (Ruse),**7 (120)**, 5969–5976, 2013. - D.S. Kim, T. Kim, S.-H. Lee, J.-J. Seo,
*A Note on the lambda-Daehee polynomials*, Int. J. Math. Anal,**7 (62)**, 3069–3080, 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. Kim,
*On a**q**-analogue of the**p**-adic*log*gamma functions and related integrals*, J. Number Theory**76**, 320–329, 1999. - T. Kim,
*q**-Volkenborn integration*, Russ. J. Math. Phys.**19**, 288–299, 2002. - T. Kim,
*A note on**q**-Volkenborn integration*, Proc. Jangjeon Math. Soc.**8 (1)**, 13–17, 2005. - T. Kim,
*On the analogs of Euler numbers and polynomials associated with**p**-adic**q**-integral on**Z*_{p}*at**q*= −1, J. Math. Anal. Appl.**331 (2)**, 779–792, 2007. - T. Kim,
*q**-Euler numbers and polynomials associated with**p**-adic**q**-integrals*, J. Nonlinear Math. Phys.**14 (1)**, 15–27, 2007. - T. Kim and D. S. Kim,
*A Note on central Bell numbers and polynomials*, Russian J. Math. Phy.**27**, 76–81, 2020. - T. Kim, M.S. Kim and L.C. Jang,
*New**q**-Euler numbers and polynomials associated with**p**-adic**q**-integrals*, Adv. Stud. Contemp. Math.**15**, 140–153, 2007. - T. Kim, D. V. Dolgy, D. S. Kim and J. J. Seo,
*Differential equations for Changhee polynomials and their applications*, J. Nonlinear Sci. Appl.**9**, 2857–2864, 2016. - T. Kim, S.H. Rim, Y. Simsek and D. Kim,
*On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions*, J. Korean Math. Soc.**45**, 435–453, 2008. - I. Kucukoglu,
*Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials*, arXiv:2012.09208v1. - I. Kucukoglu, B. Simsek and Y. Simsek,
*An approach to negative hypergeometric distribution by generating function for special numbers and polynomials*, Turk. J. Math.**43**, 2337–2353, 2019. - I. Kucukoglu and Y. Simsek,
*On a family of special numbers and polynomials associated with Apostol-type numbers and poynomials and combinatorial numbers*, Appl. Anal. Discrete Math.,**13**, 478–494, 2019. - I. Kucukoglu and Y. Simsek,
*Identities and relations on the**q**-Apostol type Frobenius-Euler numbers and polynomials*, J. Korean Math. Soc.**56 (1)**, 265–284, 2019. - B. Kurt and Y. Simsek,
*Notes on generalization of the Bernoulli type polynomials*, Appl. Math. Comput.**218**, 906–911, 2011. - B. Kurt and Y. Simsek,
*On the generalized Apostol-type Frobenius-Euler polynomials,*Adv. Differ. Equ.**1 (2013)**, 2013, https://doi.org/10.1186/1687-1847-2013- 1 - Q-M. Luo,
*Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions*, Taiwanese J. Math.**10**, 917–925, 2006. - Q.-M. Luo, H.M. Srivastava,
*Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind*, Appl. Math. Comput.**217**, 5702–5728, 2011. - 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. - T. Mansour,
*Combinatoral identities and inverse binomial coefficients*, Adv. Appl. Math.**28**, 196–202, 2002. - M.A. Özarslan,
*Unified Apostol–Bernoulli, Euler and Genocchi polynomials*, Comput. Math. Appl.**62**, 2452–2462, 2011. - H. Ozden,
*Unification of generating function of the Bernoulli, Euler and Genocchi numbers and polynomials*, AIP Conference Proceedings**1281**, 1125 (2010); https://doi.org/10.1063/1.3497848. - H. Ozden, Y. Simsek, H.M. Srivastava,
*A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials*, Comput. Math. Appl.**60**, 2010, 2779–2787. - J.-W. Park,
*On the*, J. Comput. Anal. Appl.*λ*-Daehee polynomials with*q*-parameter**20 (1)**, 11–20, 2016. - S.-H. Rim, T. Kim and S.S. Pyo,
*Identities between harmonic, hyperharmonic and Daehee numbers*, J. Inequal Appl.**1**, 168, 2018. - E.D. Rainville,
*Special Functions,*New York, The Macmillan Company, 1960. - S. Roman,
*The Umbral Calculus,*New York, Dover Publications, 2005. - G.-C. Rota,
*The number of partitions of a set*, American Math. Monthly**71 (5)**, 498–504, 1964. - W.H. Schikhof,
*Ultrametric Calculus: An Introduction to**p**-adic Analysis*, Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge, 1984. - Y. Simsek,
*Special functions related to Dedekind-type DC-sums and their applications*, Russ. J. Math. Phys.**17 (4)**, 495–508, 2010. - Y. Simsek,
*On twisted generalized Euler numbers,*Bull. Korean Math. Soc.**41**, 299–306, 2004. - Y. Simsek,
*Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications*. Fixed Point Theory Appl.**87**, 1–28, 2013. - Y. Simsek,
*Generating functions for*, Axioms*q*-Apostol type Frobenius-Euler numbers and polynomials**1**, 395–403, 2012, doi:10.3390/axioms1030395 - Y. Simsek,
*On**q**-deformed Stirling numbers*, Int. J. Math. Comput.**17 (2)**, 70–80, 2012 - Y. Simsek,
*Special numbers on analytic functions,*Appl. Math.**5**, 1091–1098, 2014. - Y. Simsek,
*A new combinatorial approach to analysis: Bernstein basis functions, combinatorial identities and Catalan numbers*, Math. Meth. Appl. Sci.**38 (14)**, 3007–3021, 2015. - Y. Simsek,
*Beta-Type Polynomials And Their Generating Functions*, Appl. Math. Comput.**254**, 172–182, 2015. - Y. Simsek,
*Combinatorial sums and binomial identities associated with the Beta-type polynomials*, Hacet. J. Math. Stat.**47 (5)**, 1144–1155, 2018. - Y. Simsek,
*Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers*, Math. Meth. Appl. Sci.**40 (7)**, 2347–2361, 2017. - Y. Simsek,
*Analysis of the*, Cogent Math. Stat.*p*-adic*q*-Volkenborn integrals: An approach to generalized Apostol- type special numbers and polynomials and their applications**2016**, 1269393, 2016, https: //dx.doi.org/10.1080/23311835.2016.1269393. - Y. Simsek,
*Apostol type Daehee numbers and polynomials*, Adv. Stud. Contemp. Math. (Kyungshang),**26 (3)**, 555–566, 2016. - Y. Simsek,
*Identities on the Changhee numbers and Apostol-type Daehee polynomials*, Adv. Stud. Contemp. Math. (Kyungshang),**27 (2)**, 199–212, 2017. - Y. Simsek,
*On generating functions for the special polynomials*, Filomat**31 (1)**, 9–16, 2017. - Y. Simsek,
*New families of special numbers for computing negative order Euler numbers and related numbers and polynomials*, Appl. Anal. Discrete Math.**12**, 1– 35, 2018, https://doi.org/10.2298/AADM1801001S. - Y. Simsek,
*Combinatorial identities and sums for special numbers and polynomials*, Filomat**32 (20)**, 6869–6877, 2018. - Y. Simsek,
*Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic**q**-integrals*. Turk. J. Math.**42**, 557–577, 2018. - Y. Simsek,
*Explicit formulas for**p**-adic integrals: Approach to**p**-adic distributions and some families of special numbers and polynomials*, Montes Taurus J. Pure Appl. Math. Article ID: MTJPAM-D-19-00005,**1 (1)**, 1–76, 2019. - Y. Simsek and A. Yardimci,
*Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and**p**-adic integrals*, Adv. Difference Equ.**308**, 2016, https://dx.doi.org/10.1186/s13662-016-1041-x. - Y. Simsek, T. Kim, D.W. Park, Y.S. Ro, L.J. Jang and S.H. Rim,
*An explicit formula for the multiple Frobenius-Euler numbers and polynomials,*JP J. Algebra Number Theory Appl.**4**, 519–529, 2004. - 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, H. Ozden, I. N. Cangul, Y. Simsek, A
*unified presentation of certain meromorphic functions related to the families of the partial zeta type functions and the**L**-functions*, Appl. Math. Comput.**219**, 3903–3913, 2012. - H.M. Srivastava, B. Kurt and Y. Simsek,
*Some families of Genocchi type polynomials and their interpolation functions*, Integr. Transf. Spec. F.**23**(**12**), 2012, 919–938. - H.M. Srivastava, B. Kurt and Y. Simsek,
*CORRIGENDUM:**Some families of Genocchi type polynomials and their interpolation functions*, Integr. Transf. Spec. F.**23**(**12**), 2012, 939–940, DOI: 10.1080/10652469.2012.690950. - H.M. Srivastava and J. Choi,
*Zeta and**q**-Zeta Functions and Associated Series and Integrals*, Amsterdam, Elsevier Science Publishers, 2012. - H.M. Srivastava, J. Choi, S
*eries Associated with the Zeta and Related Functions*, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. - B. Sury, T. Wang, and F.-Z. Zhao,
*Some identities involving reciprocals of binomial coefficients*. J. Integer Sequences**7**, Article 04.2.8, 2004. - R. Tremblay, S. Gaboury, and B.-J. Fugère,
*A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem*, Appl. Math. Lett.**24**, 1888–1893, 2011.