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  "body": "\n## Statement\n\nLet ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]], with principal powers, and let $a, b \\in \\C$.\n\n1. If $a + b + \\frac12 \\notin \\set{0, -1, -2, \\ldots}$, then for all $z \\in \\C$ with $\\operatorname{Re} z \u003c \\frac12$,\n$${}_2F_1\\left(2a, 2b; a + b + \\tfrac12; z\\right) = {}_2F_1\\left(a, b; a + b + \\tfrac12; 4z(1 - z)\\right).$$\n2. If $2b \\notin \\set{0, -1, -2, \\ldots}$, then for all $z \\in \\C$ with $\\abs{z} \u003c 1$,\n$${}_2F_1(a, b; 2b; z) = (1 - z)^{-a/2} \\, {}_2F_1\\left(\\tfrac{a}{2}, b - \\tfrac{a}{2}; b + \\tfrac12; \\frac{z^2}{4z - 4}\\right).$$\n3. If $1 + a - b \\notin \\set{0, -1, -2, \\ldots}$, then for all $z \\in \\C$ with $\\abs{z} \u003c 1$,\n$${}_2F_1(a, b; 1 + a - b; z) = (1 - z)^{-a} \\, {}_2F_1\\left(\\tfrac{a}{2}, \\tfrac{1 + a}{2} - b; 1 + a - b; \\frac{-4z}{(1 - z)^2}\\right).$$\n\n## Remarks\n\nIn each formula both sides solve the same second-order hypergeometric equation in $z$.\nIndeed, the substitution of a quadratic rational function of $z$ into a hypergeometric equation with special parameters gives back a hypergeometric equation.\nBoth sides are holomorphic near $z = 0$ with value $1$ there and the exponent $0$ there, which singles out one solution; analytic continuation then extends the identity to the stated domains.\nFor instance in part 1, $4z(1 - z) = 1 - (1 - 2z)^2$ maps the half-plane $\\operatorname{Re} z \u003c \\frac12$ into $\\C \\setminus [1, \\infty)$.\n\nSuch transformations exist only for special parameters.\nConsider the exponent differences $1 - c$, $c - a - b$ and $a - b$ of the equation.\nA quadratic transformation exists if two of them are equal up to sign or one of them is $\\pm\\frac12$.\nAll three parts fit this pattern: in part 1, the right side has $c - a - b = \\frac12$; in part 2, $c = 2b$ gives $c - a - b = -(a - b)$; in part 3, $c = 1 + a - b$ gives $1 - c = -(a - b)$.\n\nExample: with $a = b = \\frac12$ in part 3, $1 + a - b = 1$ and the formula expresses ${}_2F_1(\\frac12, \\frac12; 1; z)$, which is $\\frac{2}{\\pi}$ times the complete elliptic integral of the first kind with parameter $z$, through $(1 - z)^{-1/2} \\, {}_2F_1(\\frac14, \\frac14; 1; \\frac{-4z}{(1-z)^2})$.\n\n## Explanations\n\n### L1 human\n\nSome hypergeometric functions can be rewritten in terms of another hypergeometric function of a squared or quadratic expression in $z$, such as $4z(1 - z)$.\nThis only works when the three parameters satisfy a special relation.\nThese formulas connect hypergeometric functions with elliptic integrals and the arithmetic–geometric mean.\n\n### L2 human\n\nLinear transformations such as Pfaff and Euler exist for all parameters and come from the Möbius symmetries of $\\set{0, 1, \\infty}$.\nQuadratic transformations come instead from rational maps of degree $2$ of the sphere that send singular points to singular points, which is possible only for special exponents.\nKummer and Goursat studied them systematically; for more special parameters there are also transformations of higher degree.\n",
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