Taiwanese Journal of Mathematics


Djurdje Cvijovi´c and H. M. Srivastava

Full-text: Open access


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

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

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11Y60: Evaluation of constants 26D15: Inequalities for sums, series and integrals 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
Secondary: 30B10: Power series (including lacunary series) 33C05: Classical hypergeometric functions, $_2F_1$

Landau constants inequalities psi function Ramanujan formula generalized Gauss hypergeometric functions generating functions asymptotic expansions and estimates Clausenian hypergeometric function central binomial coefficients and central factorials Bernoulli polynomials


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

Export citation


  • H. Alzer, Inequalities for the constants of Landau and Lebesgue, J. Comput. Appl. Math., 139 (2002), 215-230.
  • H. Alzer, D. Karayannakis and H. M. Srivastava, Series representations for some mathematical constants, J. Math. Anal. Appl., 320 (2006), 145-162.
  • L. Brutman, A sharp estimate of the Landau constants, J. Approx. Theory, 34 (1982), 217-220.
  • D. Cvijović and J. Klinowski, Inequalities for the Landau constants, Math. Slovaca, 50 (2000), 159-164.
  • J. Dutka, Two results of Ramanujan SIAM J. Math. Anal. 12 (1981), 471-476.
  • A. Eisinberg, G. Franzè and N. Salerno, Asymptotic expansion and estimate of the Landau constant, Approx. Theory Appl. (New Ser.), 17 (2001), 58-64.
  • L. P. Falaleev, Inequalities for the Landau constants, Siberian Math. J., 32 (1991), 896-897.
  • S. Finch, Mathematical Constants, Cambridge University Press, London, 2003.
  • J. Gurland, On Wallis' formula, Amer. Math. Monthly, 63 (1956), 643-645.
  • Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969.
  • T. M. Mills and S. J. Smith, On the Lebesgue function for Lagrange interpolation with equidistant nodes, J. Austral. Math. Soc. $($Ser. A$)$, 52 (1992), 111-118.
  • E. Montaldi and G. Zucchelli, Some formulas of Ramanujan, revisited, SIAM J. Math. Anal., 23 (1992), 562-569.
  • A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions, Gordon and Breach Science Publishers, New York, London and Tokyo, 1989.
  • S. Ramanujan, Collected Papers of Srinivasa Ramanujan, (G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, Editors), American Mathematical Society and Chelsea Publications, New York, 2000.
  • A. Schönhage, Fehlerfortpflanzung bei Interpolation, Numer. Math., 3 (1961), 62-71.
  • H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  • H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  • G. N. Watson, Theorems stated by Ramanujan (VIII): Theorems on divergent series, J. London Math. Soc., 4 (1929), 82-86.
  • G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford $($Ser. $1)$, 2 (1930), 310-318.
  • J. E. Wilkins, The Landau constants, in Progress in Approximation Theory (P. Nevai and A. Pinkus, Editors), pp. 829-842, Academic Press, Boston, 1991.