Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we introduce the generalized twisted q-Euler numbers E (α) Keywords: Euler numbers and polynomials, twisted q-Euler numbers and q-Euler polynomials, generalized twisted q-Euler numbers and polynomials, q-Euler ze...

and Applied Analysis 3 where n, k ∈ Z see 1, 9, 10 . For n, k ∈ Z , the p-adic Bernstein polynomials of degree n are defined by Bk,n x k x k 1 − x n−k for x ∈ Zp, see 1, 10, 11 . In this paper, we consider Bernstein polynomials to express the p-adic q-integral on Zp and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with ...

The purpose of this paper is to construct new q-Euler numbers and polynomials. Finally we will consider the Witt's type formula associated with these q-Euler numbers and polynomials, and construct q-partial zeta functions and p-adic q-l-functions which interpolate new q-Euler numbers and polynomials at negative integers.

The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials. In particular, by using the q-Volkenborn integration on Zp, we construct p-adic q-Euler numbers and polynomials of higher order. We also define new generating functions of multiple q-Euler numbers and polynomials. Furthermore, we construct Euler q-Zeta function.

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

and Applied Analysis 3 The Hermite polynomials are given by Hn x H 2x n n ∑ l 0 ( n l ) 2xHn−l, 1.11 see 23, 24 , with the usual convention about replacing H by Hn. In the special case, x 0, Hn 0 Hn are called the nth Hermite numbers. From 1.11 , we note that d dx Hn x 2n H 2x n−1 2nHn−1 x , 1.12 see 23, 24 , and Hn x is a solution of Hermite differential equation which is given by y′′ − 2xy′ n...

In this article we prove some relations between two-variable q-Bernoulli polynomials and two-variable q-Euler polynomials. By using the equality eq (z)Eq (−z) = 1, we give an identity for the two-variable qGenocchi polynomials. Also, we obtain an identity for the two-variable q-Bernoulli polynomials. Furthermore, we prove two theorems which are analogues of the q-extension Srivastava-Pinter add...

Abstract For each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ [ x ] of smallest d...

In this paper, we exhibit two methods to numerically solve the fractional integro differential equations and then proceed to compare the results of their applications on different problems. For this purpose, at first shifted Jacobi polynomials are introduced and then operational matrices of the shifted Jacobi polynomials are stated. Then these equations are solved by two methods: Caputo fractio...