Clausen Formula

Clausen's  identity

(1)

holds for , , , where  a nonpositive integer and  is the Pochhammer symbol (Petkovšek et al. 1996). Closely related identities include

(2)

and

(3)

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

Another identity ascribed to Clausen which involves the hypergeometric function  and the generalized hypergeometric function  is given by

(4)

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

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  ein quadrat von der Form  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.