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

$\;_{2}F_1 \left[\begin{matrix} a & b \\ a+b+1/2 \end{matrix} ; x \right]^2 = \;_{3}F_2 \left[\begin{matrix} 2a & 2b &a+b \\ a+b+1/2 &2a+2b \end{matrix} ; x \right]$

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.