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 Explore with Wolfram|Alpha 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)]](http://mathworld.wolfram.com/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 
, 
, 
, .... 