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

[math]\displaystyle{ \;_{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] }[/math]

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 
• 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, http://www.digizeitschriften.de/main/dms/toc/?PPN=PPN243919689_0003 
• For a detailed proof of Clausen's formula: Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis