| 释义 |
G-FunctionDefined in Whittaker and Watson (1990, p. 264) and also called the Barnes G-Function.
 | (1) |
 is the Euler-Mascheroni Constant. This is an Analytic Continuation of the  function defined in the construction of the Glaisher-Kinkelin Constant
 | (2) |
 | (3) |
 . This function is what Sloane and Plouffe (1995) call the Superfactorial, and the first fewvalues for  , 2, ... are 1, 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ...(Sloane's A000178).
The -function is the reciprocal of the Double Gamma Function. It satisfies
 | (4) |
 | (5) |
 | (6) |
 | (7) |
where
 | (10) |
An unrelated pair of functions are denoted  and  and are known as G-Functions">Ramanujan g- and G-Functions. See also Euler-Mascheroni Constant, Glaisher-Kinkelin Constant, K-Function,Meijer's G-Function, Ramanujan g- andG-Functions, Superfactorial
References
Barnes, E. W. ``The Theory of the -Function.'' Quart. J. Pure Appl. Math. 31, 264-314, 1900.Glaisher, J. W. L. ``On a Numerical Continued Product.'' Messenger Math. 6, 71-76, 1877. Glaisher, J. W. L. ``On the Product  .'' Messenger Math. 7, 43-47, 1878. Glaisher, J. W. L. ``On Certain Numerical Products.'' Messenger Math. 23, 145-175, 1893. Glaisher, J. W. L. ``On the Constant which Occurs in the Formula for  .'' Messenger Math. 24, 1-16, 1894. Kinkelin. ``Über eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung.'' J. Reine Angew. Math. 57, 122-158, 1860. Sloane, N. J. A. SequenceA000178/M2049in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Voros, A. ``Spectral Functions, Special Functions and the Selberg Zeta Function.'' Commun. Math. Phys. 110, 439-465, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990. |
