“…It is well known that R(a) is essential in some fields of mathematics such as the zero-balanced Gaussian hypergeometric functions 2 F 1 (a, 1−a; 1; z), the theories of Ramanujan's modular equations and quasiconformal mappings, and the properties of R(a) are indispensable for us to show the properties of 2 are often simultaneously appear in the study of the properties and applications of R(a), and we often need to compare R(a) with B(a). In [11,Section 1], such kind of importance and applications of R(a), and the relation between R(a) and B(a) were described in details. (See also [2, 4-10, 12, 15-17].)…”