**Title:** Formulas and Relations of Special Numbers and Polynomials arising from Functional Equations of Generating Functions

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

**Article ID:** MTJPAM-D-20-00035; **Volume 3 / Issue 1 / Year 2021**, Pages 106-123

**Document Type:** Research Paper

**Author(s):** Neslihan Kilar ^{a} , Yilmaz Simsek ^{b}

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

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

Received: 21 October 2020, Accepted: 8 November 2020, Available online: 7 January 2021.

**Corresponding Author:** Neslihan Kilar (Email address: neslihankilar@gmail.com)

**Full Text:** PDF

**Abstract**

The aim of this paper is to introduce and investigate some new identities and formulas involving many kinds of special numbers and polynomials with help of the some known results derived from blending special formulas, generating functions and their functional equations. By using functional equations of generating functions for special numbers and polynomials, we give some relations and identities including the Genocchi polynomials of negative order, the Euler numbers and polynomials of negative order, the Changhee numbers and polynomials of negative order, the Lah numbers, the Hermite polynomials, the central factorial numbers, the Bernoulli numbers of higher order, the Daehee numbers, the Bernstein basis functions, the Stirling numbers, and also the combinatorial numbers and polynomials. Moreover, we also give several combinatorial sums and identities associated with aforementioned numbers and polynomials. Finally, we derive some finite and infinite series representations that include the incomplete gamma function and aforementioned numbers. In addition, convenient links of identities, formulas, relations and results appointed in this paper with those in earlier and future studies come to attention in detail for readers.

**Keywords:** Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Central factorial numbers, Hermite polynomials, Stirling numbers, Lah numbers, Combinatorial numbers, Generating functions, Incomplete gamma function

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