Advanced Mathematical Methods for Scientists and Engineers

On the q-extension of Euler and Genocchi numbers

The present paper deals with Bernstein polynomials and Frobenius-Euler numbers and polynomials. We apply the method of generating function and fermionic p-adic integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and Frobenius-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between Frobenius-Euler numbers and polynomials. Furthermore, we derive an integral representation of Bernstein polynomials of degree n on Zp . Also we deduce a fermionic p-adic integral representation of product Bernstein polynomials of different degrees n1, n2,...on Zp and show that it can be written with Frobenius-Euler numbers which yields a deeper insight into the effectiveness of this type of generalizations. Our applications possess a number of interesting properties which we state in this paper.

A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials

On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q=−1

The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, we derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

/pdf/generating-functions-for-generalized-stirling-type-numbers-3ysfs5qd0s.pdf

Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

In this article, we will discuss the behaviour of the roots of the multiple Changhee q-Bernoulli polynomials for values of the index n by using computer.

Exploring the multiple Changhee q-Bernoulli polynomials

We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet
 -Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted
 -Euler polynomials. Finally, we give a table for the solutions of the twisted
 -Euler polynomials.

https://link.springer.com/content/pdf/10.1155%2F2007%2F86052.pdf

Some Relationships between the Analogs of Euler Numbers and Polynomials

The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.

/pdf/on-the-fermionic-adic-integral-representation-of-bernstein-3cdu802i5e.pdf

On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

In this paper we explore the shapes of the q-numbers K n, q and the q-polynomials K n, q (x). Finally, we describe the structure of the roots of the q-polynomials K n, q (x) for values of the index n using a computer.

A numerical investigation of the roots of q-polynomials

In this paper, we give a fermionic p-adic integral representions of Benstein polynomials associated with Euler numbers and polynomials. Finally, we give some interesting identities for the Euler numbers by using the properties of our integral represention.

C. S. Ryoo

Papers

Exploring the multiple Changhee q-Bernoulli polynomials

Some Relationships between the Analogs of Euler Numbers and Polynomials

On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

A numerical investigation of the roots of q-polynomials

On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials