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 eponymo ...

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, which ...

. 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 In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \fr ...

used in the proof of

de Branges's theorem In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...

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