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.
References

Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, ISBN 978-0-521-62321-6; 978-0-521-78988-2, MR1688958
Clausen, Thomas (1828), "Ueber die Fälle, wenn die Reihe von der Form y = 1 + ... etc. ein Quadrat von der Form z = 1 ... etc.hat", Journal für die reine und angewandte Mathematik 3

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Mathematics Encyclopedia