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Clausen Formula

Clausen's _4F_3 identity

_4F_3(a,b,c,d; e,f,g;1)=((2a)_(|d|)(a+b)_(|d|)(2b)_(|d|))/((2a+2b)_(|d|)a_(|d|)b_(|d|)),
(1)

holds for a+b+c-d=1/2, e=a+b+1/2, a+f=d+1=b+g, where d a nonpositive integer and (a)_n is the Pochhammer symbol (Petkovšek et al. 1996). Closely related identities include

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

and

_4F_3[1/2a,1/2(a+1),b+n,-n; 1/2(b+1),1/2(b+2),a;1]=((b-a+1)_n)/((b+1)_(n-1)(b+2n))
(3)

(Bailey 1935; Slater 1966, p. 245; Andrews and Burge 1993).

Another identity ascribed to Clausen which involves the hypergeometric function _2F_1(a,b;c;z) and the generalized hypergeometric function _3F_2(a,b,c;d,e;z) is given by

(_2F_1[a,b; a+b+1/2;x])^2=_3F_2[2a,a+b,2b; a+b+1/2,2a+2b;x]
(4)

(Clausen 1828; Bailey 1935, p. 86; Hardy 1999, p. 106).

See also

Generalized Hypergeometric Function, Hypergeometric Function

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References

Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158, 1-14, 1993.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Clausen, T. "Ueber die Falle wenn die Reihe y=1+(alpha·beta)/(1·gamma)x+... ein quadrat von der Form x=1+(alpha^'beta^'gamma^')/(1·delta^'epsilon^')x+... hat." J. für Math. 3, 89-95, 1828.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 43 and 127, 1996.Slater, L. J. Generalized Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1966.

Referenced on Wolfram|Alpha

Clausen Formula

Cite this as:

Weisstein, Eric W. "Clausen Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClausenFormula.html

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