Orr's Theorem

If

(1)

where  is a hypergeometric function, then

(2)

where  is a Pochhammer symbol.

Furthermore, if

(3)

then

(4)

where  is the gamma function (Bailey 1935, p. 84).

REFERENCES:

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.

Edwards, D. "An Expansion in Factorials Similar to Vandermonde's Theorem, and Applications." Messenger Math. 52, 129-136, 1923.

Orr, W. M. "Theorems Relating to the Product of Two Hypergeometric Series." Trans. Cambridge Philos. Soc. 17, 1-15, 1899.

Watson, G. N. "The Theorems of Clausen and Cayley on Products of Hypergeometric Functions." Proc. London Math. Soc. 22, 163-170, 1924.

Whipple, F. J. W. "Algebraic Proofs of the Theorems of Cayley and Orr Concerning the Products of Certain Hypergeometric Series." J. London Math. Soc. 2, 85-90, 1927.

Whipple, F. J. W. "On a Formula Implied in Orr's Theorems Concerning the Products of Hypergeometric Series." J. London Math. Soc. 4, 48-50, 1929.