Some formulae for the q-Bernoulli and Euler polynomials of higher order

The purpose of this paper is to give a proof of Kummer type congruence for the q-Bernoulli numbers of higher order, which is an answer to a part of the problem in a previous publication (see Indian J. Pure Appl. Math. 32 (2001) 1565-1570).     

On the q-extension of Euler and Genocchi numbers

Carlitz has introduced an interesting q-analogue of Frobenius-Euler numbers in [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987-1000; L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the q-Euler numbers. A recent author's study of more general q-Euler and Genocchi numbers can be found in previous publication [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139-141]. In this paper we give a new construction of q-Euler numbers, which are different from Carlitz's q-extension and author's q-extension in previous publication (see [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139-141]). By using our q-extension of Euler numbers, we can also consider a new q-extension of Genocchi numbers and obtain some interesting relations between q-extension of Euler numbers and q-extension of Genocchi numbers.     

On the q-analog of the Laplace transform

In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform.     

On the twisted q-Euler numbers and polynomials associated with basic q-l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.     

Some <Emphasis Type="Italic">p</Emphasis>-Adic Integrals on ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Associated with Trigonometric Functions

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ?_p associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.     

On p-adic interpolating function for q-Euler numbers and its derivatives

In this paper we study a two-variable p-adic q-l-function lp,q(s,t|χ) for Dirchlet's character χ, with the property that

     

A numerical computation of the structure of the roots of q-Bernoulli polynomials

Over the years, there has been increasing interest in solving mathematical problems with the aid of computers. The main purpose of this paper is to construct new generating functions of q  -Bernoulli numbers _βn,qr${\beta }_{n,q^r}$ and q  -Bernoulli polynomials _βn,qr(x)${\beta }_{n,q^r}(x)$. We study the q  -Bernoulli polynomials _βn,qr(x)${\beta }_{n,q^r}(x)$ and investigate the roots of the q  -Bernoulli polynomials _βn,qr(x)${\beta }_{n,q^r}(x)$ for values of the index n by using computer. Finally, we consider the reflection symmetries of the q-Bernoulli polynomials.     

Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind

In this paper, we derive five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices. Then we apply twelve basic identities for Sheffer polynomials, seven from previous results, to degenerate Bernoulli polynomials and Korobov polynomials of the first kind and get some new identities. In addition, letting λ→ 0 in such identities gives us those for Bernoulli polynomials and Bernoulli polynomials of the second kind.     

Identities involving Frobenius–Euler polynomials arising from non-linear differential equations

In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order.