Kummer's Formulas
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 
, 
, 
, ....
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.
