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  "body": "\n## Statement\n\nLet $a$ and $b$ be complex numbers such that neither $a + b + \\frac12$ nor $2a + 2b$ lies in $\\set{0, -1, -2, \\dots}$.\nThen for every complex $x$ with $\\abs{x} \u003c 1$,\n$$\\left[ {}_2F_1\\left( a, b; a + b + \\tfrac12; x \\right) \\right]^2 = {}_3F_2\\left( \\begin{matrix} 2a,\\ 2b,\\ a + b \\\\ a + b + \\frac12,\\ 2a + 2b \\end{matrix} ; x \\right),$$\nwhere ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and ${}_3F_2$ is the [[generalized hypergeometric series|VDBAW9PN88]].\n\n## Remarks\n\nOne proof compares differential equations.\nIf $y$ solves a second order linear equation, then $y^2$ solves a third order linear equation, the symmetric square.\nFor the hypergeometric equation with $c = a + b + \\frac12$, the symmetric square turns out to be the generalized hypergeometric equation of the ${}_3F_2$ on the right.\nBoth sides are analytic at $0$ with value $1$, and that equation has only one such solution, so they agree.\nCheck of the first coefficient: on the left, the coefficient of $x$ is $\\frac{2ab}{a + b + \\frac12}$.\nOn the right it is $\\frac{2a \\cdot 2b \\cdot (a + b)}{(a + b + \\frac12)(2a + 2b)} = \\frac{2ab}{a + b + \\frac12}$, as it should be.\nFor $a = b = \\frac14$ the formula reads ${}_2F_1(\\frac14, \\frac14; 1; x)^2 = {}_3F_2(\\frac12, \\frac12, \\frac12; 1, 1; x)$.\nThe relation $c = a + b + \\frac12$ between the parameters is what makes the symmetric square equation hypergeometric, and the proof uses it.\n\n## Explanations\n\n### L1 human\n\nClausen's formula says that the square of a certain hypergeometric function is again a hypergeometric function, with one more parameter.\nSquaring a power series normally mixes the coefficients in a complicated way.\nHere the coefficients of the square come out as a simple ratio of rising factorials.\n\n### L2 human\n\nThe formula links two levels of the hypergeometric hierarchy, ${}_2F_1$ and ${}_3F_2$.\nIt gives positivity results: when the ${}_2F_1$ is real, the ${}_3F_2$ on the right is a square, hence nonnegative.\nIt is also used to turn identities for squares of elliptic integrals into series for $1/\\pi$.\n",
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