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Orr's Theorem

If

(1-z)^(alpha+beta-gamma-1/2)_2F_1(2alpha,2beta;2gamma;z)=sum_(n=0)^inftya_nz^n,
(1)

where _2F_1(a,b;c;z) is a hypergeometric function, then

_2F_1(alpha,beta;gamma;z)_2F_1(gamma-alpha+1/2,gamma-beta+1/2;gamma+1;z)=sum_(n=0)^infty((gamma+1/2)_n)/((gamma+1)_n)a_nz^n,
(2)

where (a)_n is a Pochhammer symbol.

Furthermore, if

(1-z)^(alpha+beta-gamma-1/2)_2F_1(2alpha-1,2beta;2gamma-1;z)=sum_(n=0)^inftya_nz^n,
(3)

then

_2F_1(alpha,beta;gamma;z)Gamma(gamma-alpha+1/2,gamma-beta-1/2;gamma;z)=sum_(n=0)^infty((gamma-1/2)_n)/((gamma)_n)a_nz^n,
(4)

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

See also

Cayley's Hypergeometric Function Theorem, Generalized Hypergeometric Function

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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.

Referenced on Wolfram|Alpha

Orr's Theorem

Cite this as:

Weisstein, Eric W. "Orr's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrrsTheorem.html

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