Enumerative proof of a curious congruence for Eulerian numbers

In this paper, by using q-deformed bosonic p-adic integral, we give -Bernoulli numbers and polynomials, we prove Witt's type formula of -Bernoulli polynomials and Gauss multiplicative formula for -Bernoulli polynomials. By using derivative operator to the generating functions of -Bernoulli polynomials and generalized -Bernoulli numbers, we give Hurwitz type -zeta functions and Dirichlet's type -L-functions; which are interpolated -Bernoulli polynomials and generalized -Bernoulli numbers, respectively. We give generating function of -Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and -Bernoulli polynomials and ordinary Bernoulli numbers of order r and -Bernoulli numbers, respectively. We also study on -Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define -partial zeta function and interpolation function.

We give a new construction of the -extensions of Euler numbers and polynomials. We present new generating functions which are related to the -Euler numbers and polynomials. We also consider the generalized -Euler polynomials attached to Dirichlet's character and have the generating functions of them. We obtain distribution relations for the -Euler polynomials and have some identities involving -Euler numbers and polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the -Euler polynomials at negative integers.

We present a new generating function related to the -Bernoulli numbers and -Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and -Bernstein polynomials. We also consider the generalized -Bernoulli polynomials attached to Dirichlet's character and have their generating function . We obtain distribution relations for the -Bernoulli polynomials and have some identities involving -Bernoulli numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of this generating function which interpolates the -Bernoulli polynomials at negative integers and is associated with -Bernstein polynomials.

The main goal of this paper is to study some interesting identities for the multiple twisted ( p , q ) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted ( p , q ) -Euler zeta function and multiple twisted ( p , q ) -L-function, which interpolate the Carlitz-type higher order twisted ( p , q ) -Euler numbers and Carlitz-type higher order twisted ( p , q ) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted ( p , q ) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials by using the symmetric property for the multiple twisted ( p , q ) -L-function.

In this paper we define the degenerate Carlitz-type ( p , q ) -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type ( p , q ) -Euler numbers and polynomials.

We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . We observe an interesting phenomenon of “scattering” of the zeros of the generalized -Euler polynomials in complex plane.

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

<abstract><p>In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.</p></abstract>

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form: where is the twisted (h, q)-Bernoulli polynomials. We also define new identities involving (h, q)-Bernoulli polnomials and numbers.

In this paper, we give some further properties of -adic - -function of two variables, which is recently constructed by Kim (2005) and Cenkci (2006). One of the applications of these properties yields general classes of congruences for generalized -Bernoulli polynomials, which are -extensions of the classes for generalized Bernoulli numbers and polynomials given by Fox (2000), Gunaratne (1995), and Young (1999, 2001).

In this paper, the definitions of both higher-order multivariable Euler's numbers and polynomial, higher-order multivariable Bernoulli's numbers and polynomial are given and some of their important properties are expounded. As a result, the mathematical relationship between higher-order multivariable Euler's polynomial (numbers) and higher-order multivariable Bernoulli's polynomial (numbers) are thus obtained.

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

We define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted -Bernoulli numbers and polynomials and between twisted generalized -Bernoulli numbers and polynomials.

New Changhee [formula omitted]-Euler numbers and polynomials associated with [formula omitted]-adic [formula omitted]-integrals

Simon Newcomb’s Problem