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

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• 作者：Jongsil Lee<sup>asup><sup> ; sup><sup>jsglocke@yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding authorsup> ; James Mc Laughlin<sup>bsup><sup> ; sup><sup>jmclaughlin@wcupa.edu" class="auth_mail" title="E-mail the corresponding authorsup> ; Jaebum Sohn<sup>asup><sup> ; sup><sup>jsohn@yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding authorsup>
• 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 <span id="mmlsi1" class="mathmlsrc"><span 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)span><span class="mathContainer hidden"><span class="mathCode">scroll">Gstretchy="false">(aq,b,λqstretchy="false">)stretchy="false">/Gstretchy="false">(a,b,λstretchy="false">)span>span>span> 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 <span id="mmlsi2" class="mathmlsrc">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">ss="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">script>style="vertical-align:bottom" width="25" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306485-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">scripts>ϕ1scripts>scripts>2scripts>span>span>span> functions, or other similar continued fraction expansions of ratios of <span id="mmlsi2" class="mathmlsrc">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">ss="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">script>style="vertical-align:bottom" width="25" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306485-si2.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">scripts>ϕ1scripts>scripts>2scripts>span>span>span> functions. Further, by employing essentially the same methods, a new continued fraction for <span id="mmlsi1" class="mathmlsrc"><span 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)span><span class="mathContainer hidden"><span class="mathCode">scroll">Gstretchy="false">(aq,b,λqstretchy="false">)stretchy="false">/Gstretchy="false">(a,b,λstretchy="false">)span>span>span> 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: