广义Grtzsch环函数的几个性质(英文)

Some Properties of Generalized Grtzsch Ring Function

设0<a,r<1,r′=1-r2,Ka(r)为第一类广义椭圆积分,定义广义Gr¨otzsch环函数为μa(r)={π/[2sin(πa)]}Ka(r′)/Ka(r),μa(r)出现于广义Ramanujan模方程,而μ1/2(r)即为拟共形理论中平面Gr¨otzsch环B2\[0,r]的共形模。本文揭示了μa(r)和广义椭圆积分的若干性质,这些结果将被用于研究广义Ramanujan模方程及其解的性态。

For a∈(0,1/2],r∈(0,1) and r ′=1-r 2, let K a(r) be the generalized elliptic integral of the first kind,and μ a(r)=πK a(r ′)/[2K a(r) sin (πa)] the so-called generalized Grtzsch ring function which appears in Ramanujan's generalized modular equations and whose special case μ(r)=μ 1/2 (r) is the modulus of Grtzsch ring B 2\[0,r]R 2. In this paper,some properties of μ a(r) and generalized elliptic integrals are obtained.These results can be used to study Ramanujan's generalized modular equations.