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Clausen's formula
In mathematics, Clausen's formula, found by , 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] 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:
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