Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions
Abstract
We examine the Landau constants defined by G(n) := Sigma(n)(m=0) 1/2(4m) ((m) (2m))(2) (n = 0, 1, 2, . . .) by making use of the celebrated Ramanujan formula expressing Gn in terms of the Clausenian (3)F(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).
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 polynomialsSource:
Taiwanese Journal of Mathematics / TJM, 2009, 13, 3, 855-870Funding / projects:
- Ministry of Science and Environmental Protection of the Republic of Serbia [144004], Natural Sciences and Engineering Research Council of Canada [OGP0007353]
DOI: 10.11650/twjm/1500405444
ISSN: 1027-5487
WoS: 000266623400001
Scopus: 2-s2.0-74049142520
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Institution/Community
VinčaTY - JOUR AU - Cvijović, Đurđe AU - Srivastava, H. M. PY - 2009 UR - https://vinar.vin.bg.ac.rs/handle/123456789/3717 AB - We examine the Landau constants defined by G(n) := Sigma(n)(m=0) 1/2(4m) ((m) (2m))(2) (n = 0, 1, 2, . . .) by making use of the celebrated Ramanujan formula expressing Gn in terms of the Clausenian (3)F(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). T2 - Taiwanese Journal of Mathematics / TJM T1 - Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions VL - 13 IS - 3 SP - 855 EP - 870 DO - 10.11650/twjm/1500405444 ER -
@article{ author = "Cvijović, Đurđe and Srivastava, H. M.", year = "2009", abstract = "We examine the Landau constants defined by G(n) := Sigma(n)(m=0) 1/2(4m) ((m) (2m))(2) (n = 0, 1, 2, . . .) by making use of the celebrated Ramanujan formula expressing Gn in terms of the Clausenian (3)F(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).", journal = "Taiwanese Journal of Mathematics / TJM", title = "Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions", volume = "13", number = "3", pages = "855-870", doi = "10.11650/twjm/1500405444" }
Cvijović, Đ.,& Srivastava, H. M.. (2009). Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions. in Taiwanese Journal of Mathematics / TJM, 13(3), 855-870. https://doi.org/10.11650/twjm/1500405444
Cvijović Đ, Srivastava HM. Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions. in Taiwanese Journal of Mathematics / TJM. 2009;13(3):855-870. doi:10.11650/twjm/1500405444 .
Cvijović, Đurđe, Srivastava, H. M., "Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions" in Taiwanese Journal of Mathematics / TJM, 13, no. 3 (2009):855-870, https://doi.org/10.11650/twjm/1500405444 . .
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