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  "body": "\n## Statement\n\nLet $n$ be a nonnegative integer and let $a, b, c$ be complex numbers.\nAssume that neither $c$ nor $1 + a + b - c - n$ lies in $\\set{0, -1, \\dots, 1 - n}$.\nWrite $(x)_k = x (x + 1) \\cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThen the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_3F_2$ at $1$ is\n$${}_3F_2\\left( \\begin{matrix} -n,\\ a,\\ b \\\\ c,\\ 1 + a + b - c - n \\end{matrix} ; 1 \\right) = \\frac{(c - a)_n\\, (c - b)_n}{(c)_n\\, (c - a - b)_n}.$$\n\n## Remarks\n\nThe series is *balanced* (Saalschützian): the sum of the lower parameters exceeds the sum of the upper parameters by $1$.\nA standard proof starts from Euler's transformation $(1 - x)^{a + b - c}\\, {}_2F_1(a, b; c; x) = {}_2F_1(c - a, c - b; c; x)$.\nComparing the coefficients of $x^n$ on both sides gives the identity after a rearrangement of Pochhammer symbols.\nFor $n = 1$ the left side is $1 - \\frac{ab}{c(a + b - c)}$.\nThis equals $\\frac{c(a + b - c) - ab}{c(a + b - c)} = \\frac{(c - a)(c - b)}{c(c - a - b)}$, which is the right side.\nThe hypothesis on $c$ and $1 + a + b - c - n$ keeps the denominators of the terms of the series nonzero; it also gives $(c - a - b)_n \\ne 0$, since $(c - a - b)_n = (-1)^n (1 + a + b - c - n)_n$.\n\n## Explanations\n\n### L1 human\n\nA hypergeometric series with a negative integer $-n$ on top stops after $n + 1$ terms, so it is a finite sum.\nIn general such a sum at $1$ has no simple closed form.\nThe theorem says that when the parameters are balanced, the sum collapses to a single product of rising factorials.\n\n### L2 human\n\nThe theorem is the ${}_3F_2$ analogue of the Chu–Vandermonde identity for terminating ${}_2F_1$ sums at $1$.\nIt is one of the basic summation formulas used to evaluate sums of products of binomial coefficients.\nIt has a $q$-analogue in the theory of basic hypergeometric series.\n",
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