Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions
| Autor: | Lee, Jongsil, Laughlin, James Mc, Sohn, Jaebum |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | The Journal of Mathematical Analysis and Applications Volume 447, Issue 2, 15 March 2017, Pages 1126-1141 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jmaa.2016.10.052 |
| Popis: | In this paper we show that various continued fractions for the quotient of general Ramanujan functions $G(aq,b,\l q)/G(a,b,\l)$ 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 Heine's continued fraction for a ratio of $_2\phi_1$ functions and other continued fractions of a similar type, and by this method derive a new continued fraction for $G(aq,b,\l q)/G(a,b,\l)$. 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: \begin{multline*} \frac{(-a,b;q)_{\infty} - (a,-b;q)_{\infty}}{(-a,b;q)_{\infty}+ (a,-b;q)_{\infty}} = \frac{(a-b)}{1-a b} \- \frac{(1-a^2)(1-b^2)q}{1-a b q^2}\\ \- \frac{(a-bq^2)(b-aq^2)q}{1-a b q^4} %\phantom{sdsadadsaasdda}\\ \- \frac{(1-a^2q^2)(1-b^2q^2)q^3}{1-a b q^6} \- \frac{(a-bq^4)(b-aq^4)q^3}{1-a b q^8} \- \cds . \end{multline*} Comment: 18 pages |
| Databáze: | arXiv |
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