Abstract
In this paper, we consider the twisted q-Bernoulli numbers and polynomials with weight α by using the bosonic q-integral on . From the construction of the twisted q-Bernoulli numbers with weight α, we derive some identities and relations. MSC: 11B68, 11S40, 11S80.
Highlights
In [ ], for α ∈ Q and n ∈ Z+, q-Bernoulli numbers with weight α is defined by Kim as follows:
The nth q-Bernoulli polynomials with weight α are defined by βnα,ξ,q(x) = ξ y[y + x]nqα dμq(y)
When x =, we can obtain some identity on the twisted q-Bernoulli numbers with weight α as follows
Summary
For f ∈ UD(Zp), the p-adic q-integral on Zp, which is called the bosonic q-integral, defined by Kim as follows: Iq(f ) = From ( ), we note that f (x) dμq(x) ≤ p f see [ – ] , ( ) In [ ], Carlitz considered the expansion of q-Bernoulli numbers as follows: In [ ], for α ∈ Q and n ∈ Z+, q-Bernoulli numbers with weight α is defined by Kim as follows:
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