A level 16 analogue of Ramanujan series for 1/π

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:456

Article

Lee, Yoonjin¹  Park, Yoon Kyung² 

[1] Ewha Womans Univ, Dept Math, 52 Seodaemun Gu, Seoul 03760, South Korea

[2] Ewha Womans Univ, Inst Math Sci, 52 Seodaemun Gu, Seoul 03760, South Korea

关键词: Ramanujan's series for 1/pi;    Modular function;    Modular equation;    Ray class field;   

DOI  :  10.1016/j.jmaa.2017.06.082

来源: Elsevier

【 摘 要 】

The modular function h(tau) = q(n=1)Pi(infinity) (1-q(16n))(2)(1-q(2n))/(1-q(n))(2) (1-q(8n)) is called a level 16 analogue of Ramanujan's series for 1/pi. We prove that h(tau) generates the field of modular functions on Gamma(0)(16) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K(h(tau)) modulo 4 over an imaginary quadratic field K for tau is an element of K boolean AND S such that Z[4 tau] is the integral closure of Z in K, where Sj is the complex upper half plane. For any tau is an element of K boolean AND f., it turns out that the value 1/h(tau) is integral, and we can also explicitly evaluate the values of h(tau) if the discriminant of K is divisible by 4. (C) 2017 Elsevier Inc. All rights reserved.

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