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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

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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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