**Title:** New Formulas and Numbers Arising from Analyzing Combinatorial Numbers and Polynomials

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

**Article ID:** MTJPAM-D-20-00038; **Volume 3 / Issue 3 / Year 2021 (Special Issue)**, Pages 238-259

**Document Type:** Research Paper

**Author(s):** Irem Kucukoglu ^{a} , Yilmaz Simsek ^{b}

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

^{b}Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey

Received: 30 October 2020, Accepted: 15 December 2020, Published: 25 April 2021.

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

**Full Text:** PDF

**Abstract**

In this paper, we derive various identities involving the negative higher-order combinatorial numbers and polynomials and other kinds of special numbers and polynomials such as the Stirling numbers, the Lah numbers, the negative higher-order Changhee numbers and polynomials, and the positive higher-order Bernoulli numbers and polynomials. Furthermore, by using the integral formulas of not only the negative higher-order combinatorial numbers and polynomials but also their generating functions, we obtain some identities and combinatorial sums. We give some infinite series, involving the negative higher-order combinatorial numbers, with their values in terms of the falling factorials, the Catalan numbers, the Daehee numbers (linear combination of the Stirling numbers and the Bernoulli numbers) and the Changhee numbers (linear combination of the Stirling numbers and the Euler numbers). As application of these infinite series, we also set two new sequences of special numbers with their generating functions, and investigate their properties. We pose an open question related to one of these number sequences. By using an infinite series arising from the integral of the generating functions for the negative higher-order combinatorial numbers and polynomials, we also introduce a new family of polynomials associated with the Bernstein basis functions. In addition, we derive symmetry property, integral formulas and derivative formula for these newly introduced polynomials. Moreover, by implementing an explicit formula of these newly introduced polynomials in Mathematica with the aid of the Wolfram programming language, we present some plots of these newly introduced polynomial functions for some of their randomly selected special cases. We also give some further results including series representations, combinatorial sums, integral formulas and relations for some of combinatorial numbers and poynomials. Finally, we present some observations and comments on our results.

**Keywords:** Generating functions, Special numbers and polynomials, Bernoulli numbers and polynomials, Stirling numbers, Bell numbers, Lah numbers, Catalan numbers, Daehee and Changhee numbers, Bernstein basis functions

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