本项目将着重研究拟共形特殊函数和 Heisenberg 群上的连分数理论中的若干前沿热点问题。具体研究内容和目标为：(1)给出完全椭圆积分和 Ramanujan R-函数的无穷级数公式和渐近精确上下界，将完全椭圆积分的经典性质推广到广义椭圆积分和零平衡超几何函数；(2)建立符号差为1/3，1/4和1/6的广义模方程的基本性质，寻找出带有参数的广义 Hübner上界函数的无穷乘积表达式，提供广义模方程解的精确估计，证明 Hersch-Pfluger 偏差函数的次可乘性并给出其渐近精确界，改进著名的拟共形 Schwarz 引理和 Mori 常数；(3)提出并解决 Heisenberg群上连分数的 Diophantine 逼近问题、Lagrange 定理和收敛准则，完善 Heisenberg群上的 Eisenstein 整数连分数理论。本项目具有很强的学科交叉与综合性和广阔的理论应用背景。

This project will focus mainly on several hot topics concerning special functions in quasiconformal mappings and the theory of continued fractions on Heisenberg group. The main contents and aims are as follows: (1) Giving the infinite series formulas and sharp upper and lower bounds for complete elliptic integrals and Ramanujan R-function, and extending some classical properties of complete elliptic integrals to the generalized elliptic integrals and zero-balanced hypergeometric functions; (2) Establishing the basic properties for the generalized Ramanujan modular equations with signatures 1/3, 1/4 and 1/6, finding the infinity-product representations for generalized Hübner upper bound functions with parameters, and providing the precise evaluations of the solutions of generalized Ramanujan modular equations, proving the submultiplicative property of Hersch-Pfluger distortion function and providing its asymptotic precise bounds, and improving the well-known Mori constant and Schwarz lemma in quasiconformal mappings; (3) Raising and solving the Diophantine approximation problem, Lagrange theorem and convergence criterions in the theory of continued fractions on Heisenberg group, and perfecting the theory of continued fractions on Heisenberg group using Eisenstein integrals. This project is interdisciplinary and comprehensive, and has broad theoretical application prospects.