A Note on Type 2 Degenerate <i>q</i>-Euler Polynomials

• Main Authors: Taekyun Kim, Dae San Kim, Han Young Kim, Sung-Soo Pyo
• Format: Article
• Language: English
• Published: MDPI AG 2019-07-01
• Series: Mathematics
• Subjects: type 2 degenerate <i>q</i>-Euler polynomials; fermionic <i>p</i>-adic <i>q</i>-integral
• Online Access: https://www.mdpi.com/2227-7390/7/8/681

Recently, type 2 degenerate Euler polynomials and type 2 <i>q</i>-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a <i>q</i>-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate <i>q</i>-Euler polynomials, which are derived from the fermionic <i>p</i>-adic <i>q</i>-integrals on <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>, and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 <i>q</i>-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate <i>q</i>-Euler polynomials arise naturally by means of the fermionic <i>p</i>-adic <i>q</i>-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously.