Modular identities for some special cases of Ramanujan's general continued fraction

被引：0
作者：
Bhat, Shruthi C. [1 ]
Srivastava, H. M. [2 ,3 ,4 ,5 ,6 ,7 ]
Kumar, B. R. Srivatsa [1 ]
机构：
[1] Manipal Acad Higher Educ, Manipal Inst Technol, Dept Math, Manipal 576104, Karnataka, India
[2] Univ Victoria, Dept Math & Stat, Victoria, BC V8W 3R4, Canada
[3] China Med Univ, China Med Univ Hosp, Dept Med Res, Taichung 40402, Taiwan
[4] Kyung Hee Univ, Ctr Converging Humanities, 26 Kyungheedae Ro, Seoul 02447, South Korea
[5] Chung Yuan Christian Univ, Dept Appl Math, Taoyuan City 320314, Taiwan
[6] Azerbaijan Univ, Dept Math & Informat, 71 Jeyhun Hajibeyli St, AZ-1007 Baku, Azerbaijan
[7] Int Telemat Univ Uninettuno, Sect Math, I-00186 Rome, Italy
来源：
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS | 2025年 / 119卷 / 02期
关键词：
Continued fractions; Modular equations; Theta functions; Jacobi's triple-product identity; Dedekind's eta-function; Rogers-Ramanujan continued fraction; Ramanujan-Selberg continued fraction; Ramanujan-G & ouml; llnitz-Gordon continued fraction; THETA-FUNCTION IDENTITIES; EQUATIONS;
D O I：
10.1007/s13398-025-01706-3
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this article, we first introduce the continued fractions K(q), L(q) and M(q), which are based on the Ramanujan-Selberg continued fraction C(q). We then obtain several modular identities which connect K(q) with K(-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(-q)$$\end{document} and K(qi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(q<^>i)$$\end{document}(i=2,3,5,7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i=2,3,5,7)$$\end{document}, L(q) with L(-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(-q)$$\end{document} and L(qj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(q<^>j)$$\end{document}(j=2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\;(j=2,3)$$\end{document}, M(q) with M(-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(-q)$$\end{document} and M(qj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(q<^>j)$$\end{document}(j=2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(j=2,3)$$\end{document}, and C(q) with C(-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(-q)$$\end{document}. We also provide some algebraic results associated with K(q) and L(q). In addition, as an application of our results, we deduce a few fascinating colored partitions which also validate the results.
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页数：13
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