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

			(4)
			(5)
			(6)

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.