Clausen Formula

Clausen's ${}_4F_3$ identity

$${}_4F_3\left({\matrix{a, b, c, d\cr e, f, g\cr}; 1}\right)={... ...}\over(2a+2b)_{\vert d\vert}a_{\vert d\vert}b_{\vert d\vert}},$$

holds for $a+b+c-d=1/2$, $e=a+b+1/2$, $a+f=d+1=b+g$, $d$ a nonpositive integer, and $(a)_n$ is the Pochhammer Symbol (Petkovsek et al. 1996).

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

$$\left[{{}_2F_1\left({\matrix{a, b\cr a+b+{\textstyle{1\over ... ..., a+b, 2b\cr a+b+{\textstyle{1\over 2}}, 2a+2b\cr}; x}\right).$$

See also Generalized Hypergeometric Function, Hypergeometric Function

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43 and 127, 1996.