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  "body": "\n## Statement\n\nLet $a, b, c \\in \\C$ be such that none of $c$, $c - a - b$, $a - b$ is an integer, and consider the [[hypergeometric differential equation|W8VE25R5A7]] with these parameters.\nLet ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]].\n\n1. The following six functions are solutions, each on a suitable simply connected domain:\n$$\\begin{aligned} w_1 \u0026= {}_2F_1(a, b; c; z), \u0026 w_2 \u0026= z^{1-c} \\, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z), \\\\ w_3 \u0026= {}_2F_1(a, b; a + b - c + 1; 1 - z), \u0026 w_4 \u0026= (1 - z)^{c-a-b} \\, {}_2F_1(c - a, c - b; c - a - b + 1; 1 - z), \\\\ w_5 \u0026= z^{-a} \\, {}_2F_1(a, a - c + 1; a - b + 1; 1/z), \u0026 w_6 \u0026= z^{-b} \\, {}_2F_1(b, b - c + 1; b - a + 1; 1/z). \\end{aligned}$$\n2. Applying the [[Pfaff and Euler transformations|KAAJ0JJ2GF]] to each $w_i$ gives four expressions $z^{\\lambda} (1 - z)^{\\mu} \\, {}_2F_1(a', b'; c'; \\xi)$.\nIn each, $\\xi$ is one of $z$, $1 - z$, $1/z$, $1/(1 - z)$, $z/(z - 1)$, $(z - 1)/z$.\nThis gives $24$ expressions of solutions, falling into six groups of four equal functions.\n\n## Remarks\n\nThe six variables are the images of $z$ under the six [[Möbius transformations|0SG74WBFA2]] that permute $\\set{0, 1, \\infty}$.\nEach such map carries the equation to a hypergeometric equation with permuted local exponents.\nThe solutions $w_1, \\ldots, w_6$ are the pairs of local solutions at $0$, $1$ and $\\infty$, each with a pure power behaviour there.\nThe hypothesis that $c$, $c - a - b$ and $a - b$ are not integers makes the two exponents at each singular point differ by a non-integer, so each pair is a basis.\nWithout it, some of the six formulas are undefined or coincide, and logarithmic solutions appear.\n\nExample: the group of $w_1$ consists of\n$${}_2F_1(a, b; c; z) = (1 - z)^{-a} \\, {}_2F_1\\left(a, c - b; c; \\tfrac{z}{z - 1}\\right) = (1 - z)^{-b} \\, {}_2F_1\\left(c - a, b; c; \\tfrac{z}{z - 1}\\right) = (1 - z)^{c - a - b} \\, {}_2F_1(c - a, c - b; c; z).$$\n\nSince the solution space is two-dimensional, any three of the $w_i$ are linearly dependent.\nFor instance, on $\\C \\setminus \\big((-\\infty, 0] \\cup [1, \\infty)\\big)$,\n$$w_1 = \\frac{\\Gamma(c) \\Gamma(c - a - b)}{\\Gamma(c - a) \\Gamma(c - b)} \\, w_3 + \\frac{\\Gamma(c) \\Gamma(a + b - c)}{\\Gamma(a) \\Gamma(b)} \\, w_4,$$\nand the first coefficient is the value given by [[Gauss's hypergeometric theorem|1FC0T0GCCX]].\n\n## Explanations\n\n### L1 human\n\nThe hypergeometric equation has three special points, $0$, $1$ and $\\infty$, and near each of them there are two natural solutions that behave like powers.\nEach of these six solutions can be written as a hypergeometric function in four different ways.\nThat gives Kummer's list of $24$ formulas.\n\n### L2 human\n\nThe list is the tool for continuing ${}_2F_1$ from one singular point to another.\nLinear relations between solutions at different points, the connection formulas, follow from Gauss's theorem and give the monodromy explicitly.\n",
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