Asymptotic of the Landau constants and their relationship with hypergeometric function (Q842010)

The Landau constants are defined by \[ G_n=\sum_{m=0}^n\frac{1}{2^{4m}}\binom{2m}{m}^2 \qquad (n=0,1,2,\dots). \] The following classical result is due to Ramanujan (1913): \[ \left(\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}\right)^2\cdot G_n=\frac{1}{n+1} \cdot{}_ 3F_ {2}(1/2, 1/2, n+1; 1, n+2; 1), \] where \({}_ 3F_ {2}\) is the generalized hypergeometric function. In the paper under review, the authors deduce other, mostly new, Ramanujan type formulas for the Landau constants involving the terminating and non-terminating hypergeometric series. They also derive the following nice formula for the generating function of the sequence \(G_n:\) \[ \sum_{n=0}^{\infty}G_nx^n=\frac{2}{\pi(1-x)}\int_0^{\pi/2} \frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}} \qquad (|x|<1). \] Finally, the authors establish several upper and lower bounds as well as asymptotic expansions for \(G_n\) in terms of the digamma and logarithm functions.