## 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. If is a negative integer, the identity takes the form
 (3)

(Petkovsek et al. 1996).

Kummer's second formula is

 (4)

where is the Confluent Hypergeometric Function and , , , ....

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 42-43 and 126, 1996.