Quadratic symmetry of modified q-Euler polynomials

We use the p-adic q-integral and group action to count the number of the generating functions of modified q-Euler polynomials in a prescribed set. Some generating function yields modified q-Euler polynomials with the isotropy group D4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{4}$\end{document} and some gives Euler polynomials with the isotropy group V4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{4}$\end{document}.