## Taiwanese Journal of Mathematics

### ASYMPTOTICS OF THE LANDAU CONSTANTS AND THEIR RELATIONSHIP WITH HYPERGEOMETRIC FUNCTIONS

#### Abstract

We examine the Landau constants defined by $$G_n:=\sum_{m\,=0}^{n}\frac{1}{2^{4 m}}\,\binom{2 m}{m}^2\qquad(n=0, 1, 2, \cdots)$$ by making use of the celebrated Ramanujan formula expressing $G_n$ in terms of the Clausenian ${}_3F_2$ hypergeometric series. It is shown that it could be used to deduce other, mostly new, Ramanujan type formulas for the Landau constants involving the terminating and non-terminating hypergeometric series. In addition, by this approach we derive once again, in a simple and unified manner, almost all of the known results and also establish several new results for $G_n$. These new results include (for example) the generating function and asymptotic expansions and estimates for $G_n$.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 3 (2009), 855-870.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405444

Digital Object Identifier
doi:10.11650/twjm/1500405444

Mathematical Reviews number (MathSciNet)
MR2526343

#### Citation

Cvijovi´c, Djurdje; Srivastava, H. M. ASYMPTOTICS OF THE LANDAU CONSTANTS AND THEIR RELATIONSHIP WITH HYPERGEOMETRIC FUNCTIONS. Taiwanese J. Math. 13 (2009), no. 3, 855--870. doi:10.11650/twjm/1500405444. https://projecteuclid.org/euclid.twjm/1500405444

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