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

JOURNAL OF NUMBER THEORY

卷:168

Article

Lee, Yoonjin¹  Park, Yoon Kyung² 

[1] Ewha Womans Univ, Dept Math, Seoul 03760, South Korea

[2] Ewha Womans Univ, Inst Math Sci, Seoul 03760, South Korea

关键词: Rogers-Ramanujan continued fraction;    Modular function;    Modular unit;   

DOI  :  10.1016/j.jnt.2016.04.009

来源: Elsevier

【 摘 要 】

We prove the modularity of the level 13 analogue r(13)(tau) of the Rogers-Ramanujan continued fraction. We establish some properties of r13(tau) using the modular function theory. We first prove that r13(tau) is a generator of the function field on Gamma(0)(13). We then find modular equations of 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 to every level n. We further show that the value r(13)(tau) is an algebraic unit for any tau is an element of K - Q, where K is an imaginary quadratic field. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free