**Title:** Formulas for Fubini type numbers and polynomials of negative higher order

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

**Article ID:** MTJPAM-D-22-00003; **Volume 5 / Issue 3 / Year 2023**, Pages 23-36

**Document Type:** Research Paper

**Author(s):** Neslihan Kilar ^{a}

^{a}Department of Computer Technologies, Bor Vocational School, Niğde Ömer Halisdemir University, Niğde TR-51700 Turkey

Received: 13 February 2022, Accepted: 1 April 2022, Published: 27 May 2022.

**Corresponding Author:** Neslihan Kilar (Email address: neslihankilar@ohu.edu.tr and neslihankilar@gmail.com)

**Full Text:** PDF

**Abstract**

The present paper deals with the Fubini type numbers and polynomials with their generating functions and functional equations. By using these functions, some properties and applications of these polynomials are investigated. Many relations and computation formulas connected with the Stirling type numbers, the Apostol type polynomials and numbers of order *-r*, the Bernoulli polynomials of order *-r*, the Euler polynomials and numbers of order *-r*, the Fubini type numbers and polynomials of order *-r* and combinatorial numbers are given. Applying the derivative operator to the generating functions of these polynomials, some formulas and combinatorial sums including these numbers and polynomials are also given. Moreover, applying the Riemann integral to some formulas, we derive several interesting finite combinatorial sums associated with the Bernstein basis functions, the Cauchy numbers and the Stirling type numbers.

**Keywords:** Cauchy numbers, Stirling numbers, Fubini type numbers and polynomials, Bernstein basis functions, combinatorial numbers, special numbers, generating function

**References:**

- A. Cayley,
*On the analytical forms called trees*, Second part, Philosophical Magazine, Series IV**18 (121)**, 374–378, 1859. - L. Comtet,
*Advanced combinatorics: The art of finite and infinite expansions*, D. Reidel Publications, Dordrecht and Boston, 1974. - J.-M. De Koninck,
*Those fascinating numbers*, American Mathematical Society, 2009. - R. Goldman,
*Pyramid algorithms: A dynamic programming approach to curves and surfaces for geometric modeling*, Morgan Kaufmann Publishers, R. Academic Press, San Diego, 2002. - R. Golombek,
*Aufgabe 1088*, Elem. Math.**49**, 126–127, 1994. - 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,
*Fubini type numbers and their generating functions*, Akdeniz University, MSc Thesis in Mathematics, Antalya, 2017. - 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,
*Some relationships between Fubini type polynomials and other special numbers and polynomials*, AIP Conf. Proc.**2116**, 2019; Article ID: 100017. - N. Kilar and Y. Simsek,
*Identities and relations for Fubini type numbers and polynomials via generating functions and*, Publ. Inst. Math. (Beograd) (N.S.)*p*-adic integral approach**106 (120)**, 113–123, 2019. - N. Kilar and Y. Simsek,
*Formulae to Fubini type numbers emerge from application of*, GU J Sci, Part A*p*-adic integrals**8 (4)**, 402–410, 2021. - I. Kucukoglu,
*Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters*, Math. Methods Appl. Sci.**45 (9)**, 5043–5066, 2022. - I. Kucukoglu and Y. Simsek,
*New formulas and numbers arising from analyzing combinatorial numbers and polynomials*, Montes Taurus J. Pure Appl. Math.**3 (3)**, 238–259, 2021. - 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**, 45–68, 2021. - 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**, 2019; Article ID: 112. - 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. - G. G. Lorentz,
*Bernstein polynomials*, Chelsea Pub. Comp., New York, 1986. - T. Nahid and M. Ali,
*Several characterizations of Bessel functions and their applications*, Georgian Math. J.**29 (1)**, 83–93, 2022. - T. Nahid and C. S. Ryoo,
*2-Variable Fubini-degenerate Apostol-type polynomials*, Asian-Eur. J. Math. 2021; DOI:10.1142/S179355712250

0929. - S. Roman,
*The umbral calculus*, Dover Publications, New York, 2005. - 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,
*Analysis of the Bernstein basis functions: An approach to combinatorial sums involving binomial coefficients and Catalan numbers*, Math. Methods Appl. Sci.**38 (14)**, 3007–3021, 2015. - 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. - Y. Simsek,
*Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and*, Turk. J. Math.*p*-adic*q*-integrals**42**, 557–577, 2018. - Y. Simsek,
*Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers*, J. Math. Anal. Appl.**477**, 1328–1352, 2019. - Y. Simsek,
*Explicit formulas for*, Montes Taurus J. Pure Appl. Math.*p*-adic integrals: Approach to*p*-adic distributions and some families of special numbers and polynomials**1 (1)**, 1–76, 2019. - Y. Simsek,
*Interpolation functions for new classes special numbers and polynomials via applications of*, Montes Taurus J. Pure Appl. Math.*p*-adic integrals and derivative operator**3 (1)**, 38–61, 2021. - 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. - A. Xu,
*On an open problem of Simsek concerning the computation of a family of special numbers*, Appl. Anal. Discrete Math.**13**, 61–72, 2019.