Kummer's Formulas
Kummer's first formula is
| _2F_1(1/2+m-k,-n;2m+1;1)=(Gamma(2m+1)Gamma(m+1/2+k+n))/(Gamma(m+1/2+k)Gamma(2m+1+n)), |
(1)
|
where _2F_1(a,b;c;z) is the hypergeometric function with m!=-1/2, -1, -3/2, ..., and Gamma(z) is the gamma function. The identity can be written in the more symmetrical form as
where a-b+c=1 and b is a positive integer (Bailey 1935, p. 35; Petkovšek et al. 1996; Koepf 1998, p. 32; Hardy 1999, p. 106). If b is a negative integer, the identity takes the form
(Petkovšek et al. 1996).
Kummer's second formula is
![画像:z^(m+1/2)[1+sum_(k=1)^(infty)(z^(2k))/(2^(4k)k!(m+1)_k)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KummersFormulas/Inline14.svg){kind=link}
where M_(0,m)(z) is a Whittaker function, _1F_1(a;b;z) is the confluent hypergeometric function of the first kind, (a)_n is a Pochhammer symbol, I_n(z) is a modified Bessel function of the first kind, and m!=-1/2, -1, -3/2, ....
See also
Confluent Hypergeometric Function of the First Kind, Hypergeometric FunctionExplore with Wolfram|Alpha
References
Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 42-43 and 126, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Referenced on Wolfram|Alpha
Kummer's FormulasCite this as:
Weisstein, Eric W. "Kummer's Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KummersFormulas.html