In mathematics, Clausen's formula, found by , expresses the square of a

Gaussian hypergeometric series Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...

as a

generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ...

. It states :

\;_F_1 \left begin
a & b  \\ 
a+b+1/2 \end 
; x \right 2   = \;_F_2 \left begin 
2a & 2b &a+b \\ 
a+b+1/2 &2a+2b \end 
; 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

* * * For a detailed proof of Clausen's formula: {{Citation , last1=Milla , first1=Lorenz , title= A detailed proof of the Chudnovsky formula with means of basic complex analysis , arxiv=1809.00533 , year=2018 Special functions