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  "body": "\n## Statement\n\nLet $n \\ge 0$ be an integer, let $q \\in \\C \\setminus \\set{0}$ with $q^k \\ne 1$ for $1 \\le k \\le n$, and let $a, c \\in \\C$ with $a \\ne 0$ and $(c; q)_n \\ne 0$.\nThen the terminating [[basic hypergeometric series|EB4N9RQR51]] satisfy\n$${}_2\\phi_1 \\left( \\begin{matrix} a, q^{-n} \\\\ c \\end{matrix} ; q, q \\right) = \\frac{(c/a; q)_n}{(c; q)_n}\\, a^n$$\nand\n$${}_2\\phi_1 \\left( \\begin{matrix} a, q^{-n} \\\\ c \\end{matrix} ; q, \\frac{c q^n}{a} \\right) = \\frac{(c/a; q)_n}{(c; q)_n},$$\nwhere $(\\cdot\\,; q)_n$ is the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\n\n## Remarks\n\nBoth sides are rational functions, and the series stop at index $n$ since $(q^{-n}; q)_k = 0$ for $k \u003e n$.\nProof idea: the second form is the case $b = q^{-n}$ of the $q$-Gauss sum, which follows from [[Heine's transformation|A93A5VATSD]].\nThe first form is the second one with the order of summation reversed.\nExample: $n = 1$.\nThe first sum is $1 + \\frac{(1 - a)(1 - q^{-1})}{(1 - c)(1 - q)}\\, q = 1 - \\frac{1 - a}{1 - c} = \\frac{a - c}{1 - c}$, and the right side is $\\frac{(1 - c/a)\\, a}{1 - c} = \\frac{a - c}{1 - c}$.\nIn terms of [[Gaussian binomial coefficients|YS09Z1YNEK]] it reads $\\binom{m+n}{k}_q = \\sum_{j} \\binom{m}{j}_q \\binom{n}{k-j}_q\\, q^{(m-j)(k-j)}$.\nAt $q = 1$ this becomes [[Vandermonde's identity|01JADB9VW5]].\n\n## Explanations\n\n### L1 human\n\nVandermonde's identity counts the ways to choose $n$ people from a group of men and women by splitting according to the number of men.\nThe $q$-Vandermonde identity is the same sum with each binomial coefficient replaced by its $q$-analogue and suitable powers of $q$.\nIt is the simplest closed-form sum of a terminating basic hypergeometric series.\n\n### L2 human\n\nIts Gaussian binomial form can also be proved, for $q$ a prime power, by counting subspaces of $\\F_q^{m+n}$.\nIt is the $q$-analogue of the [[Chu–Vandermonde identity|4YK46R1QDR]] for ${}_2F_1$, and it is a terminating case of the $q$-Gauss sum.\nIt is used to evaluate connection coefficients between $q$-orthogonal polynomials.\n",
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