Some properties of the difference between the Ramanujan constant and beta function

Sharp Approximations for the Ramanujan Constant

“…Let γ = 0.5772156649 · · · be the Euler-Mascheroni constant, and for a ∈ (0, 1), let R(a) be defined by R(a) = −2γ − ψ(a) − ψ(1 − a) (1.2) which is called the Ramanujan constant in literature although it is actually a function of a and probably better to call R(a) the Ramanujan R-function (cf. [11]). By the symmetry, we may assume that a ∈ (0, 1/2] in (1.2).…”

“…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].)…”

“…Some authors have obtained some properties, including lower and upper bounds, for R(a). In [11], for instance, power series expansion, integral representation and bounds were obtained for the difference R(a) − B(a). Some related studies showed that B(a) is one of good approximation functions for R(a), and we often require the properties of certain combinations defined in terms of R(a), B(a), a(1 − a) and other polynomials.…”

“…Some related studies showed that B(a) is one of good approximation functions for R(a), and we often require the properties of certain combinations defined in terms of R(a), B(a), a(1 − a) and other polynomials. (See [3,8,10,11,[15][16][17]. )…”

“…which is called the Ramanujan constant in literature although it is actually a function of a and probably better to call R(a) the Ramanujan R-function (cf. [11]). By the symmetry, we may assume that a ∈ (0, 1/2] in (1.2).…”