Kummer's first formula is
(1)
where
is the hypergeometric function with , , , ..., and is the gamma function .
The identity can be written in the more symmetrical form as
(2)
where
and
is a positive integer (Bailey 1935, p. 35; Petkovšek et al. 1996;
Koepf 1998, p. 32; Hardy 1999, p. 106). If is a negative integer, the identity takes the form
(3)
(Petkovšek et al. 1996).
Kummer's second formula is
where
is a Whittaker function , is the confluent
hypergeometric function of the first kind , is a Pochhammer symbol ,
is a modified
Bessel function of the first kind , and , , , ....
See also Confluent Hypergeometric Function of the First Kind ,
Hypergeometric
Function
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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 Formulas
Cite this as:
Weisstein, Eric W. "Kummer's Formulas."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/KummersFormulas.html
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