Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions

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• 作者：Jongsil Lee^a ; ^{jsglocke@yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; James Mc Laughlin^b ; ^{jmclaughlin@wcupa.edu" class="auth_mail" title="E-mail the corresponding author} ; Jaebum Sohn^a ; ^{jsohn@yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Heine's continued fraction ; Rogers&ndash ; Ramanujan continued fraction ; The Bauer&ndash ; Muir transformation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：1126-1141
• 全文大小：408 K

文摘

In this paper we show that various continued fractions for the quotient of general Ramanujan functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306485&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306485&_rdoc=1&_issn=0022247X&md5=b88ba68cb3274317b37d98e9081e0393" title="Click to view the MathML source">G(aq,b,λq)/G(a,b,λ)class="mathContainer hidden">class="mathCode">$G(aq,b,\lambda q)/G(a,b,\lambda )$ may be derived from each other via Bauer–Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer–Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine's continued fraction for a ratio of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306485&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306485&_rdoc=1&_issn=0022247X&md5=972701359f6b7d8d3ff947b0b550ecd4">class="imgLazyJSB inlineImage" height="15" width="25" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306485-si2.gif">class="mathContainer hidden">class="mathCode">${}_2\varphi _1$ functions, or other similar continued fraction expansions of ratios of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306485&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306485&_rdoc=1&_issn=0022247X&md5=972701359f6b7d8d3ff947b0b550ecd4">class="imgLazyJSB inlineImage" height="15" width="25" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306485-si2.gif">class="mathContainer hidden">class="mathCode">${}_2\varphi _1$ functions. Further, by employing essentially the same methods, a new continued fraction for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306485&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306485&_rdoc=1&_issn=0022247X&md5=b88ba68cb3274317b37d98e9081e0393" title="Click to view the MathML source">G(aq,b,λq)/G(a,b,λ)class="mathContainer hidden">class="mathCode">$G(aq,b,\lambda q)/G(a,b,\lambda )$ is derived. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: