Dae San Kim

Some identities of Frobenius-Euler polynomials arising from umbral calculus

In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.

Gauss sums for symplectic groups over a finite field

For a nontrivial additive character λ and a multiplicative character χ of the finite field withq elements, the ‘Gauss’ sums Σλ(trg) overg∈Sp(2n,q) and Σχ(detg)λ(trg) overg∈GSp(2n, q) are considered. We show that it can be expressed as a polynomial inq with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain ‘generalized Kloosterman sums over nonsingular matrices’ and ‘generalized Kloosterman sums over nonsingular alternating matrices’, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

Some new identities of Frobenius-Euler numbers and polynomials

In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, several authors have studied some identities of Frobenius-Euler polynomials. From the methods of our paper, we can also derive many interesting identities of Frobenius-Euler numbers and polynomials.

Some identities of q-Euler polynomials arising from q-umbral calculus

Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (see (Araci et al. in J. Number Theory 133(10):3348-3361, 2013)). In this paper, we develop the new method of q-umbral calculus due to Roman, and we study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.

Umbral calculus and Sheffer sequences of polynomials

In this paper, we investigate some properties of Sheffer sequences of polynomials arising from umbral calculus. From these properties, we derive new and interesting identities between Sheffer sequences of polynomials. An application to normal ordering is presented.

A note on nonlinear Changhee differential equations

In this paper, we study nonlinear Changhee differential equations and derive some new and explicit identities of Changhee and Euler numbers from those nonlinear differential equations.

A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials

In this paper, we consider poly-Bernoulli and higher-order poly-Bernoulli polynomials and derive some new and interesting identities of those polynomials by using umbral calculus.

Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S3

In this paper, we derive eight basic identities of symmetry in three variables related to Bernoulli polynomials and power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the p-adic integral expression of the generating function for the Bernoulli polynomials and the quotient of integrals that can be expressed as the exponential generating function for the power sums.

Some identities for the product of two Bernoulli and Euler polynomials

Let ℙn be the space of polynomials of degree less than or equal to n. In this article, using the Bernoulli basis {B0(x), . . . , Bn(x)} for ℙn consisting of Bernoulli polynomials, we investigate some new and interesting identities and formulae for the product of two Bernoulli and Euler polynomials like Carlitz did.

Frobenius-Euler polynomials and umbral calculus in the p-adic case

In this paper, we study some p-adic Frobenius-Euler measure related to umbral calculus in the p-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.MSC:05A10, 05A19.