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 an id="mmlsi1" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Galse">(aq,b,λqalse">)alse">/Galse">(a,b,λalse">)ath>an>an>an> may be derived from each other via Bauer&ndash;Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer&ndash;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 an id="mmlsi2" class="mathmlsrc"><a 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">ass="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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">ϕ12ath>an>an>an> functions, or other similar continued fraction expansions of ratios of an id="mmlsi2" class="mathmlsrc"><a 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">ass="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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">ϕ12ath>an>an>an> functions. Further, by employing essentially the same methods, a new continued fraction for an id="mmlsi1" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Galse">(aq,b,λqalse">)alse">/Galse">(a,b,λalse">)ath>an>an>an> 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: