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Bailey's Theorem

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Let Gamma(z) be the gamma function and n!! denote a double factorial, then

[(Gamma(m+1/2))/(Gamma(m))]^2[1/m+(1/2)^21/(m+1)+((1·3)/(2·4))^21/(m+2)+...]_()_(n) =[(Gamma(n+1/2))/(Gamma(n))]^2[1/n+(1/2)^21/(n+1)+((1·3)/(2·4))^21/(n+2)+...]_()_(m).

Writing the sums explicitly, Bailey's theorem states

[(Gamma(m+1/2))/(Gamma(m))]^2sum_(k=0)^(n-1)1/(m+k)[((2k-1)!!)/((2k)!!)]^2 =[(Gamma(n+1/2))/(Gamma(n))]^2sum_(k=0)^(m-1)1/(n+k)[((2k-1)!!)/((2k)!!)]^2.

See also

Gamma Function

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References

Bailey, W. N. "The Partial Sum of the Coefficients of the Hypergeometric Series." J. London Math. Soc. 6, 40-41, 1931.Bailey, W. N. "On One of Ramanujan's Theorems." J. London Math. Soc. 7, 34-36, 1932.Darling, H. B. C. "On a Proof of One of Ramanujan's Theorems." J. London Math. Soc. 5, 8-9, 1930.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 106-107 and 112, 1999.Hodgkinson, J. "Note on One of Ramanujan's Theorems." J. London Math. Soc. 6, 42-43, 1931.Watson, G. N. "Theorems Stated by Ramanujan (VIII): Theorems on Divergent Series." J. London Math. Soc. 4, 82-86, 1929.Watson, G. N. "The Constants of Landau and Lebesgue." Quart. J. Math. (Oxford) 1, 310-318, 1930.Whipple, F. J. W. "The Sum of the Coefficients of a Hypergeometric Series." J. London Math. Soc. 5, 192, 1930.

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Bailey's Theorem

Cite this as:

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

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