The level 13 analogue of the Rogers-Ramanujan continued fraction and its modularity

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• 作者：Yoonjin Lee^a ; ^{yoonjinl@ewha.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Yoon Kyung Park^b ; ^{ykp@ewha.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11J70 ; 11F11 ; 11R04 ; 14H55
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：168
• 期：Complete
• 页码：306-333
• 全文大小：446 K

文摘

We prove the modularity of the level 13 analogue f463cfea9707be" title="Click to view the MathML source">r₁₃(τ)$r_{13}(\tau )$ of the Rogers–Ramanujan continued fraction. We establish some properties of f463cfea9707be" title="Click to view the MathML source">r₁₃(τ)$r_{13}(\tau )$ using the modular function theory. We first prove that f463cfea9707be" title="Click to view the MathML source">r₁₃(τ)$r_{13}(\tau )$ is a generator of the function field on Γ₀(13)${\mathrm{\Gamma }}_0(13)$. We then find modular equations of f463cfea9707be" title="Click to view the MathML source">r₁₃(τ)$r_{13}(\tau )$ of level n for every positive integer n   by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7$n=2,3\text{and}7$ to every level n  . We further show that the value f463cfea9707be" title="Click to view the MathML source">r₁₃(τ)$r_{13}(\tau )$ is an algebraic unit for any τ∈K−Q$\tau \in K-\mathbb{Q}$, where K is an imaginary quadratic field.