Taiwanese Journal of Mathematics

ASYMPTOTICS OF THE LANDAU CONSTANTS AND THEIR RELATIONSHIP WITH HYPERGEOMETRIC FUNCTIONS

Djurdje Cvijovi´c and H. M. Srivastava

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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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405444

Digital Object Identifier
doi:10.11650/twjm/1500405444

Mathematical Reviews number (MathSciNet)
MR2526343

Subjects
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, $_2F_1$

Keywords
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

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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References

  • 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.