A note on type 2 q-Bernoulli and type 2 q-Euler polynomials

Abstract

Abstract As is well known, power sums of consecutive nonnegative integers can be expressed in terms of Bernoulli polynomials. Also, it is well known that alternating power sums of consecutive nonnegative integers can be represented by Euler polynomials. In this paper, we show that power sums of consecutive positive odd q-integers can be expressed by means of type 2 q-Bernoulli polynomials. Also, we show that alternating power sums of consecutive positive odd q-integers can be represented by virtue of type 2 q-Euler polynomials. The type 2 q-Bernoulli polynomials and type 2 q-Euler polynomials are introduced respectively as the bosonic p-adic q-integrals on Zp $\mathbb{Z}_{p}$ and the fermionic p-adic q-integrals on Zp $\mathbb{Z}_{p}$. Along the way, we will obtain Witt type formulas and explicit expressions for those two newly introduced polynomials.

Keywords

• Type 2 q-Bernoulli polynomials
• Type 2 q-Euler polynomials
• p-adic q-integral
• Power sums of consecutive positive odd q-integers