Kummer's first formula is
(1)
where hypergeometric function with gamma function .
The identity can be written in the more symmetrical form as
(2)
where et al. 1996;
Koepf 1998, p. 32; Hardy 1999, p. 106). If
(3)
(Petkovšek et al. 1996).
Kummer's second formula is
where Whittaker function , confluent
hypergeometric function of the first kind , Pochhammer symbol ,
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. Hardy,
G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. Koepf, W. Hypergeometric
Summation: An Algorithmic Approach to Summation and Special Function Identities. Petkovšek, M.; Wilf, H. S.;
and Zeilberger, D. A=B. 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 https://mathworld.wolfram.com/KummersFormulas.html
Subject classifications
is the hypergeometric function with 
, 
, 
, ..., and 
is the gamma function.
The identity can be written in the more symmetrical form as
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
![z^(m+1/2)[1+sum_(k=1)^(infty)(z^(2k))/(2^(4k)k!(m+1)_k)]](/images/equations/KummersFormulas/Inline14.svg)
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 
, 
, 
, ....
