We introduce two kinds of q-Euler polynomials and numbers, and investigate many of their interesting properties. In particular, we establish q-symmetry properties of these q-Euler polynomials, from which we recover the so-called Kaneko-Momiyama identity for the ordinary Euler polynomials, discovered recently by Wu, Sun, and Pan. Indeed, a q-symmetry and q-recurrence formulas among sum of produc...

in this study, the bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. to this aim, the operational matrices of integration and the product for bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. some examples are presented to illustrate the efficien...

In this paper we study a number of conjectures on the behavior value distribution eigenfunctions. On two dimensional torus, observe that symmetry conjecture holds in strongest possible sense. other hand, provide counterexample for higher tori, which relies computer-assisted argument. Moreover prove theorem certain class trigonometric polynomials might be independent interest.

Kupershmidt and Tuenter have introduced reflection symmetries for the q-Bernoulli numbers and the Bernoulli polynomials in 2005 , 2001 , respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the cl...

The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate p-adic fermionic integral on Zp. From these symmetries, we can derive some recurrence identities for the ζ-Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the ζEuler polyn...

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of exponential and product nonlinearities. Also, we consider singular case. We construct a kind of spectral collocation method based on shifted Jacobi polynomials to implement this method. A number of specific numerical examples demonstrate the accuracy and the efficiency of the propos...