Clausen's formula

In mathematics, Clausen's formula, found by Thomas Clausen (1828), expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states

${}_2{F}_1{[\begin{array}{cc}a& b\\ a+b+1/2\end{array};x]}^2={}_3{F}_2[\begin{array}{ccc}2a& 2b& a+b\\ a+b+1/2& 2a+2b\end{array};x]$

In particular, it gives conditions for a hypergeometric series to be positive. This can be used to prove several inequalities, such as the Askey–Gasper inequality used in the proof of de Branges's theorem.

References

• For a detailed proof of Clausen's formula: