Clausen Formula

Clausen's 👁 _4F_3
identity

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

and

(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

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

See also

Generalized Hypergeometric Function, Hypergeometric Function

Explore with Wolfram|Alpha

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 Resource. https://mathworld.wolfram.com/ClausenFormula.html

Subject classifications