Abstract
In this paper, we derive several identities of symmetry in three variables related to Carlitz-type \(q\)-Euler polynomials and alternating \(q\)-power sums. These and most of their identities are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the fermionic \(p\)-adic \(q\)-integral expressions of the generating functions for the Carlitz-type \(q\)-Euler polynomials.
Highlights
In this paper, we derive several identities of symmetry in three variables related to Carlitz-type q-Euler polynomials and alternating q-power sums
The derivations of identities are based on the fermionic p-adic q-integral expressions of the generating functions for the Carlitz-type q-Euler polynomials
We will omit those, as this requires much space
Summary
Let p be a prime number with p 1 (mod 2). 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 jÁjp be the normalized p-adic absolute value with j pjp 1⁄4. From (1), we have qIÀqðf1Þ þ IÀqð f Þ 1⁄4 1⁄22qf ð0Þ; ð2Þ where f1ðxÞ 1⁄4 f ðx þ 1Þ. QnIÀqðfnÞ þ IÀqð f Þ 1⁄4 1⁄22q ðÀ1Þlqlf ðlÞ; ð4Þ l1⁄40 for n 0 (mod 2), qnIÀqðfnÞ À IÀqð f Þ 1⁄4 À1⁄22q ðÀ1Þlqlf ðlÞ: ð5Þ. The ordinary Euler polynomials are defined by the generating function to be et 2 þ ext. When x 1⁄4 0, En 1⁄4 Enð0Þ are called the Euler numbers
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