### Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team. ### Learn More → # A Comprehensive Treatment of q-CalculusThe first q-functions A Comprehensive Treatment of q-Calculus: The first q-functions [We introduce the first q-functions, the tilde operator, two other tilde operators and the △-operator. The q-integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{0}^{a} f(t,q) \,d_q(t)\equiv{a}(1-q) \sum_{n=0}^{\infty}f(aq^{n},q)q^{n}$$\end{document} can be written in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int f d\mu=\sum_{n=0}^{\infty}b_n\mu(E_n),$$\end{document} where bn means the function value f(aqn,q) times a and μ(En)=(1−q)qn denotes the measure in the point x=aqn. We use a σ-algebra , where the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{E_{n}\}_{0}^{\infty}$\end{document} are disjoint.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png # A Comprehensive Treatment of q-CalculusThe first q-functions 46 pages Loading next page... /lp/springer-journals/a-comprehensive-treatment-of-q-calculus-the-first-q-functions-IS6rONvuYr Publisher Springer Basel Copyright © Springer Basel 2012 ISBN 978-3-0348-0430-1 Pages 195 –240 DOI 10.1007/978-3-0348-0431-8_6 Publisher site See Chapter on Publisher Site ### Abstract [We introduce the first q-functions, the tilde operator, two other tilde operators and the △-operator. The q-integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{0}^{a} f(t,q) \,d_q(t)\equiv{a}(1-q) \sum_{n=0}^{\infty}f(aq^{n},q)q^{n}$$\end{document} can be written in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int f d\mu=\sum_{n=0}^{\infty}b_n\mu(E_n),$$\end{document} where bn means the function value f(aqn,q) times a and μ(En)=(1−q)qn denotes the measure in the point x=aqn. We use a σ-algebra , where the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{E_{n}\}_{0}^{\infty}\$\end{document} are disjoint.]

Published: Jun 18, 2012

Keywords: Elliptic Function; Theta Function; Fundamental Domain; Jacobi Elliptic Function; Addition Theorem

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