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

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

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, vol. 71, Cambridge University Press, ISBN 978-0-521-62321-6, MR 1688958
• 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
• For a detailed proof of Clausen's formula: Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533