On the modified q -Euler polynomials with weight
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RESEARCH
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On the modified q-Euler polynomials with weight Seog-Hoon Rim1 , Jin-Woo Park1* , Jongkyum Kwon2 and Sung-Soo Pyo1 * Correspondence: [email protected] 1 Department of Mathematics Education, Kyungpook National University, Taegu, 702-701, Republic of Korea Full list of author information is available at the end of the article
Abstract In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on Zp and give new explicit formulas related to these numbers and polynomials. Keywords: modified q-Euler polynomials; modified q-Euler polynomials with weight; fermionic p-adic q-integral on Zp
Throughout this paper Zp , Qp and Cp will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p–νp (p) = p . –
In this paper, we assume that q ∈ Cp with | – q|p < p p– so that qx = exp(x log q) for x x ∈ Zp . The q-number of x is denoted by [x]q = –q . Note that limq→ [x]q = x. Let d be a –q fixed integer bigger than , and let p be a fixed prime number and (d, p) = . We set
X∗ =
Xd = lim Z/dpN Z, ← – N
(a + dpZp ),
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