Glaisher-kinkelin constant
WebGlaisher provided an asymptotic formula for the hyperfactorials, ... where is the Glaisher–Kinkelin constant. Other properties. According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when is an odd prime number ) () / ()!! (), where !! is the notation for the double factorial. ... WebThe constants of Landau and Lebesgue are defined, for all integers n⩾0, in order, byGn=∑k=0n116k2kk2andLn=12π∫-ππsinn+12tsin12tdt,which play important…
Glaisher-kinkelin constant
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WebCatalan (or Glaisher) combinatorial constant. glaisher A. 1.28242 Decimal expansion of Glaisher-Kinkelin constant. khinchin k. 2.685452 Decimal expansion of Khinchin constant. extreme_value_skewness 12√6 ζ(3)/ π 3. 1.139547 Extreme value distribution ... WebFeb 21, 2024 · In this paper, we provide some new sequences to approximate the Glaisher–Kinkelin constant and Bendersky–Adamchik constant, which are faster than the approximations in literature (Dawei and Mortici in J Number Theory 144:340–352, 2014; Mortici in J Number Theory 133:2465–2469, 2013 ). Download to read the full article text.
WebAbstract. (i) The Glaisher–Kinkelin constant A =1.28242712… is defined as the limit of the sequence . We establish the asymptotic representation of the sequence (ln A n ) n∈ℕ and obtain the upper and lower bounds for ln A n −ln A. (ii) Also, two constants analogous to the Glaisher–Kinkelin constant are considered and the results ... WebM. Bresse (1867) computed 24 decimals of using a technique from E. Kummer's work. J. Glaisher (1877) evaluated 20 digits of the Catalan constant, which he extended to 32 digits in 1913. The Catalan constant is applied in number theory, combinatorics, and different areas of mathematical analysis.
WebAug 1, 2013 · The results. Regarding the problem of approximation of the Glaisher–Kinkelin constant, we give the following. Theorem 1. For every n ⩾ 1, we have w n − 1 720 n 2 + 1 5040 n 4 − 1 10 080 n 6 < ln A < w n − 1 720 n 2 + 1 5040 n 4, where w n = ∑ k = 1 n k ln k − ( n 2 2 + n 2 + 1 12) ln n + n 2 4. Web(OEIS A074962) is called the Glaisher-Kinkelin constant and is the derivative of the Riemann zeta function (Kinkelin 1860; Jeffrey 1862; Glaisher 1877, 1878, 1893, 1894; Voros 1987). The constant is implemented as Glaisher, and appears in a number of …
WebFeb 9, 2016 · In this paper, some new continued fraction approximations, inequalities and rates of convergence of Glaisher–Kinkelin’s and Bendersky–Adamchik’s constants are provided. To demonstrate the superiority of our new convergent sequences over the classical sequences and Mortici’s sequences, some numerical computations are also …
Webwhere is the Glaisher-Kinkelin constant. Using equation ( ) gives the derivative (38) which can be derived directly from the Wallis formula (Sondow 1994). can also be derived directly from the Euler-Maclaurin summation formula (Edwards 2001, pp. 134-135). trading standards east lothianWebThe decimal expansion of the Glaisher-Kinkelin constant is given by A=1.28242712... (OEIS A074962). A was computed to 5×10^5 decimal digits by E. Weisstein (Dec. 3, 2015). The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 7, 14, 2264, 1179, 411556, ... (OEIS A225763). trading standards dundee contact numberWebThe Glaisher-Kinkelin constant \(A = \exp(\frac{1}{12}-\zeta'(-1))\). EXAMPLES: sage: float ( glaisher ) 1.2824271291006226 sage: glaisher . n ( digits = 60 ) 1.28242712910062263687534256886979172776768892732500119206374 sage: a = glaisher + 2 sage: a glaisher + 2 sage: parent ( a ) Symbolic Ring trading standards electrical goodsWebOct 15, 2012 · (i) The Glaisher–Kinkelin constant A=1.28242712… is defined as the limit of the sequence . We establish the asymptotic representation of the sequence (ln A n ) n∈ℕ and obtain the upper and lower bounds for ln A n −ln A. (ii) Also, two constants analogous to the Glaisher–Kinkelin constant are considered and the results corresponding to (i) are … the salt rock grill indian shores flWebThe constant in Moron's answer is C = log A, where A is the Glaisher-Kinkelin constant. Thus C = 1 12 − ζ ′ ( − 1). The expression H ( n) = ∏ k = 1 n k k is called the hyperfactorial, and it has the known asymptotic expansion. H ( n) = A e − n 2 / 4 n n ( n + 1) / 2 + 1 / 12 ( 1 + 1 720 n 2 − 1433 7257600 n 4 + ⋯). the same as ... the salt room appletonWebMar 19, 2024 · The Glaisher–Kinkelin constant, usually denoted by the symbol \(A\), is a mathematical constant which is approximately equal to \[ 1.2824271291006226368753425688697917277676889273250011920637400217. trading standards dundee city councilWebMay 8, 2024 · In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function.The constant appears in a number of sums and integrals, especially those involving gamma functions and zeta functions.It is named after mathematicians James … trading standards great yarmouth