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A study on a class of \(q\)-Euler polynomials under the symmetric group of degree \(n\). (English) Zbl 1348.11017

Summary: Motivated by the paper of T. Kim et al. [J. Nonlinear Sci. Appl. 9, No. 3, 1077–1082 (2016; Zbl 1405.11023)], we study a class of \(q\)-Euler polynomials earlier given by Kim et al. in [Proc. Jangjeon Math. Soc. 12, No. 1, 77–92 (2009; Zbl 1208.11131)]. We derive some new symmetric identities for \(q\)-extension of \(\lambda\)-Euler polynomials, using fermionic \(p\)-adic invariant integral over the \(p\)-adic number field originally introduced by Kim in [Russ. J. Math. Phys. 16, No. 4, 484–491 (2009; Zbl 1192.05011)], under symmetric group of degree \(n\) denoted by \(S_n\).

MSC:

11B68 Bernoulli and Euler numbers and polynomials
05A19 Combinatorial identities, bijective combinatorics
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
05A30 \(q\)-calculus and related topics
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