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Slater's Formula

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Slater (1960, p. 31) terms the identity

_4F_3[a,1+1/2a,b,-n; 1/2a,1+a-b;1+a+n]=((1+a)_n(1/2+1/2a-b)_n)/((1/2+1/2a)_n(1+a-b)_n)

for n a nonnegative integer the "_4F_3[1] summation theorem." Here, _4F_3(a_1,...,a_4;b_1,b_2,b_3) is a generalized hypergeometric function with argument z=1 and (a)_z is a Pochhammer symbol.

This is a special case of the more general identity

_4F_3(a,b,c,1/2a+1;1/2a,a-b+1,a-c+1;1) =(Gamma((a+1)/2)Gamma(a-b+1)Gamma(a-c+1)Gamma((a+1)/2-b-c))/(Gamma(a+1)Gamma((a+1)/2-b)Gamma((a+1)/2-c)Gamma(a-b-c+1)),

which holds for R[a-2b-2b]>-1 (O. Marichev, pers. comm., May 16, 2008).

See also

Generalized Hypergeometric Function

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References

Slater, L. J. "An Elementary Proof of the _4F_3[1] Summation Theorem." §2.7.1 in Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, p. 31, 1960.

Referenced on Wolfram|Alpha

Slater's Formula

Cite this as:

Weisstein, Eric W. "Slater's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SlatersFormula.html

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