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  "body": "\n## Definition\n\nLet $a, b, c$ be distinct points of the Riemann sphere, and let $\\alpha, \\alpha', \\beta, \\beta', \\gamma, \\gamma' \\in \\C$ with $\\alpha + \\alpha' + \\beta + \\beta' + \\gamma + \\gamma' = 1$.\nThe *Riemann P-symbol* is the table\n$$P\\begin{Bmatrix} a \u0026 b \u0026 c \u0026 \\\\ \\alpha \u0026 \\beta \u0026 \\gamma \u0026 z \\\\ \\alpha' \u0026 \\beta' \u0026 \\gamma' \u0026 \\end{Bmatrix}.$$\nIt denotes the set of solutions $w(z)$ of [[Riemann's differential equation|RS8Z48E784]] with singular points $a, b, c$.\nThe exponents are $\\alpha, \\alpha'$ at $a$, $\\beta, \\beta'$ at $b$ and $\\gamma, \\gamma'$ at $c$.\nWhen one of the points is $\\infty$, the equation is the limiting form in which the factors involving that point are dropped.\n\n## Remarks\n\nThe order of the columns, and the order of the two exponents within a column, do not matter.\nTwo basic rules hold.\n\n1. If $T$ is a [[Möbius transformation|0SG74WBFA2]], then the P-symbol with points $a, b, c$ and variable $z$ equals the P-symbol with points $T(a), T(b), T(c)$, the same exponents, and variable $T(z)$.\n2. If $a, b, c$ are finite, then for all $k, l \\in \\C$,\n$$\\left(\\frac{z - a}{z - b}\\right)^{k} \\left(\\frac{z - c}{z - b}\\right)^{l} P\\begin{Bmatrix} a \u0026 b \u0026 c \u0026 \\\\ \\alpha \u0026 \\beta \u0026 \\gamma \u0026 z \\\\ \\alpha' \u0026 \\beta' \u0026 \\gamma' \u0026 \\end{Bmatrix} = P\\begin{Bmatrix} a \u0026 b \u0026 c \u0026 \\\\ \\alpha + k \u0026 \\beta - k - l \u0026 \\gamma + l \u0026 z \\\\ \\alpha' + k \u0026 \\beta' - k - l \u0026 \\gamma' + l \u0026 \\end{Bmatrix}.$$\n\nExample: the [[hypergeometric function|463X5V31D9]] satisfies\n$${}_2F_1(a, b; c; z) \\in P\\begin{Bmatrix} 0 \u0026 \\infty \u0026 1 \u0026 \\\\ 0 \u0026 a \u0026 0 \u0026 z \\\\ 1 - c \u0026 b \u0026 c - a - b \u0026 \\end{Bmatrix},$$\nand the exponents add up to $0 + (1 - c) + a + b + 0 + (c - a - b) = 1$, as required.\n\nNon-example: a table whose six exponents do not add up to $1$ is not a valid P-symbol.\nNo second-order Fuchsian equation has exactly those three singular points with those exponents.\n\nRule 2 is consistent with the sum condition: the shifts $k$, $-k-l$, $l$ appear twice each and add up to $0$.\n",
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