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]](../I/m/296d1e79605afdac79ac51693a5a4156.png)
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, 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