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    A Note on Two Parametric Kinds of Eulerian-type Polynomials Related to Some Special Numbers and Polynomials
    (American Institute of Physics Inc., 2023) Kilar, Neslihan; Simsek, Yilmaz
    The main idea of this paper is to give some identities and applications of the two parametric kinds of Eulerian-type polynomials. By using generating functions and functional equations of these special polynomials, we derive some identities and relations associated with the two parametric kinds of Eulerian-type polynomials, the Hermite polynomials, and well-known combinatorial numbers and polynomials. Moreover, we give a remark on these special polynomials and functions. © 2023 American Institute of Physics Inc.. All rights reserved.
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    Families of unified and modified presentation of Fubini numbers and polynomials
    (MTJPAM Turkey, 2023) Kilar, Neslihan; Simsek, Yilmaz
    The goal of this paper is to define new families of unified and modified presentation of the Fubini numbers and polynomials with their generating functions. Using generating functions and their functional equations, many properties of these polynomials and numbers are presented. Relations among unified and modified presentation of the Fubini numbers and polynomials, Stirling type numbers, combinatorial type polynomials, and unified presentation of the generalized Bernoulli, Euler and Genocchi polynomials are given. Many novel identities and relations including these polynomials and numbers are also given. Moreover, new Hurwitz-Lerch type zeta functions, which interpolate unified and modified presentation of the Fubini numbers and polynomials at negative integers, are defined. Furthermore, suitable links of identities and relations, which are found in this paper, with those in earlier and future studies are indicated. © 2023, MTJPAM Turkey. All rights reserved.
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    Formulae bringing to light from certain classes of numbers and polynomials
    (Springer-Verlag Italia Srl, 2023) Kilar, Neslihan; Kim, Daeyeoul; Simsek, Yilmaz
    With aid of generating functions and their functional equation methods and special functions involving trigonometric functions, the motivation of this paper is to study by blending certain families polynomials associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Hermite type polynomials, the Stirling numbers, the telephone numbers, the Chebyshev polynomials. Therefore, the purpose of this paper is to examine certain families of numbers and functions related to generalized Hermite-Kampe de Feriet polynomials and trigonometric functions. By using functional equations of generating functions, we derive numerous new formulae and relations involving parametric Hermite type polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the generalized Hermite-Kampe de Feriet polynomials and the telephone numbers. Moreover, applying derivative operator to these generating functions, we give many recurrence relations and computational formulae, and certain finite sums. Finally, some special cases of these results are reduced to not only the well-known Chebyshev polynomials, which have applications in a wide variety of different areas, but also special trigonometric functions and finite sums.
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    Generating Functions for the Fubini Type Polynomials and Their Applications
    (Springer, 2023) Simsek, Yilmaz; Kilar, Neslihan
    One of the aims of this chapter is to give Fubini type numbers and polynomials discovered with the help of generating functions or defined by combinatorial methods and also their general properties with known methods or techniques that we have found. The second purpose of this chapter is to give formulas and relations that we have just found, besides the known ones, using generating functions and their functional equations. The third purpose of this chapter is to give the relations between Fubini-type numbers and polynomials and other special numbers and polynomials. The fourth of the purposes of this chapter will be to give tables with Fubini-type numbers and polynomials, as well as other special numbers and special polynomials. In addition, by using Wolfram Mathematica version 12.0, graphs of Fubini type polynomials and their generating functions, surface graphs and mathematical codes will be given. The fifth purpose of this chapter, some known applications in the theory of approximation with Fubini-type numbers and polynomials are summarized. The sixth of the purposes of this chapter is to give zeta-type functions that interpolate Fubini-type numbers and polynomials at negative integers. Moreover, throughout this chapter, we are tried diligently to present the results obtained in comparison with other known results and their reductions, taking into account the relevant sources. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
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    On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry
    (Mdpi, 2021) Battaloglu, Rifat; Simsek, Yilmaz
    The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.
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    Recurrence relations, associated formulas, and combinatorial sums for some parametrically generalized polynomials arising from an analysis of the Laplace transform and generating functions
    (Springer, 2023) Kilar, Neslihan; Simsek, Yilmaz; Srivastava, H. M.
    The aim of this paper is to obtain some interesting infinite series representations for the Apostol-type parametrically generalized polynomials with the aid of the Laplace transform and generating functions. In particular, by using the method of generating functions, we derive not only recurrence relations, but also several other formulas, identities, and relations as well as combinatorial sums for these parametrically generalized numbers and polynomials and for other known special numbers and polynomials. These identities, relations and combinatorial sums are related to the two-parameter types of the Apostol-Bernoulli polynomials of higher order, the two-parameter types of Apostol-Euler polynomials of higher order, the two-parameter types of Apostol-Genocchi polynomials of higher order, the Apostol-Bernoulli polynomials of higher order, the Apostol-Euler polynomials of higher order, the Apostol-Genocchi polynomials of higher order, the cosine- and sine-Bernoulli polynomials, the cosine- and sine-Euler polynomials, the lambda-array-type polynomials, the lambda-Stirling numbers, the polynomials C-n(x,y)Cn(x,y), and the polynomials S-n(x,y)Sn(x,y). Finally, we present several new recurrence relations for these special polynomials and numbers.