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The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers are given.

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

The aim of this paper is to not only provide a definition of a new family of special numbers and polynomials of higher-order with their generating functions, but also to investigate their fundamental properties in the spirit of probabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities and formulas involving the Chu--Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions, and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm which returns special values of these numbers and polynomials. In addition, we show that our new identities and formulas are connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present some theoretical and applied details on probabilistic distributions arising from the aforementioned Chu-Vandermonde-type convolution formulas.

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah numbers.

Mediterranean International Conference of Pure and Applied Mathematics and Related Areas -- OCT 26-29, 2018 -- Antalya, TURKEY WOS: 000503720000009 The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given. Scientific Research Project Administration of Akdeniz UniversityAkdeniz University The second author was supported by the Scientific Research Project Administration of Akdeniz University.

The aim of this paper is to investigate some classes of higher-order Apostol-type numbers and Apostol-type polynomials. We construct Lerch-type zeta functions which interpolate these numbers and polynomials at negative integers. Moreover, by combining some well-known identities such as the Chu-Vandermonde identity with the Lerch-type zeta functions and generating functions for the higher-order Apostol-type numbers and Apostol-type polynomials, we derive some relations and identities including functional equation for these Lerch-type zeta functions with other zeta type functions, Raabe-type multiplication formula for the higher-order Apostol-type polynomials and the Stirling numbers. Finally, we give some remarks and observations on Lerch-type zeta functions and their functional equations. Mathematics Subject Classification (2010): Primary 11B68, 11S40, 11S80, 33E20; Secondary 26C05, 30B40, 44A20. Keywords: Bernoulli numbers and Bernoulli polynomials, Euler numbers and Euler polynomials, Riemann and Hurwitz (or generalized) zeta functions, Hurwitz-Lerch zeta function, Lerch zeta function, polylogarithm function, multiplication formula, functional equation, Mellin transformation

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Y n ( λ ) and the polynomials Y n ( x ; λ ) , which were recently introduced by Simsek [30] . We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol–Bernoulli numbers, the Apostol–Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz–Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Y n ( λ ) , the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials.

In this paper, by applying the $p$-adic $q$-integrals to a family of continuous differentiable functions on the ring of $p$-adic integers, we construct new generating functions for generalized Apostol-type numbers and polynomials attached to the Dirichlet character of a finite abelian group. By using these generating functions with their functional equations, we derive various new identities and relations for these numbers and polynomials. These results are generalizations of known identities and relations including some well-known families of special numbers and polynomials such as the generalized Apostol-type Bernoulli, the Apostol-type Euler, the Frobenius-Euler numbers and polynomials, the Stirling numbers, and other families of numbers and polynomials. Moreover, by the help of these generating functions, we also construct other new families of numbers and polynomials with their generating functions. By using these functions, we investigate some fundamental properties of these numbers and polynomials. Finally, we also give explicit formulas for computing the Apostol--Bernoulli and Apostol-Euler numbers.

We construct generating functions for beta-type rational functions and the beta polynomials. By using these generating functions, we derive a collection of functional equations and PDEs. By using these functional equations and PDEs, we give derivative formulas, a recurrence relation and a variety of identities related to these polynomials. We also give a relation between the beta-type rational functions and the Bernstein basis functions. Integrating these identities and relations, we derive various combinatorial sums involving binomial coefficients, some old and some new, for the beta-type rational functions and the Bernstein basis functions. Finally, by applying the Laplace transform to these generating functions, we obtain two series representations for the beta-type rational functions.

The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind.

The main objective in this paper is first to establish new identities for the λ -Stirling type numbers of the second kind, the λ -array type polynomials, the Apostol–Bernoulli polynomials and the Apostol–Bernoulli numbers. We then construct a λ -delta operator and investigate various generating functions for the λ -Bell type numbers and for some new polynomials associated with the λ -array type polynomials. We also derive several other identities and relations for these polynomials and numbers.

By using partial differential equations (PDEs) of the generating functions for the unification of the Bernoulli, Euler and Genocchi polynomials and numbers, we derive many new identities and recurrence relations for these polynomials and numbers. In [33], Srivastava et al. defined a unified presentation of certain meromorphic functions related to the families of the partial zeta type functions. By using these functions, we construct p-adic functions which are related to the partial zeta type functions. By applying these p-adic function, we construct unified presentation of p-adic L-functions. These functions give us generalization of the Kubota–Leopoldt p-adic L-functions, which are related to the Bernoulli numbers and the other p-adic L-functions, which are related to the Euler numbers and polynomials. We also give some remarks and comments on these functions.

The aim of this paper is to introduce and investigate several new identities related to a unification and generalization of the three families of generalized Apostol type polynomials such as the Apostol–Bernoulli polynomials, the Apostol–Euler polynomials and the Apostol–Genocchi polynomials. The results presented here are based upon the theory of the Umbral Calculus and the Umbral Algebra. We also introduce some operators. By using a unified generating function for these Apostol type polynomials, which was constructed recently by Ozden et al. (2010) [42] , we derive many new properties of these polynomials. Moreover, we give relations between these polynomials and the Stirling numbers of the first and second kind.