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  "body": "\n## Definition\n\nLet $F(n, k)$ be a hypergeometric term in both variables, as in the definition of a [[WZ pair|WC0QNABPZ4]].\n*Zeilberger's algorithm* searches, for $J = 0, 1, 2, \\ldots$ in turn, for polynomials $a_0(n), \\ldots, a_J(n)$, not all zero, and a hypergeometric term $G(n, k)$ with $G/F$ rational, such that\n$$\\sum_{j=0}^{J} a_j(n)\\,F(n + j, k) = G(n, k + 1) - G(n, k).$$\nFor each $J$ it applies [[Gosper's algorithm|NHMJQBS04M]] to the term $t_k = \\sum_{j} a_j(n) F(n + j, k)$ in $k$, with unknown $a_j$.\nThe polynomial equation of Gosper's method is then linear in the $a_j$ and in the unknown coefficients of $x$.\nIt stops at the first $J$ for which a solution exists and returns the $a_j$ and $G$.\n\n## Remarks\n\nIf $F(n, k)$ vanishes for $k$ outside a finite range depending on $n$, summing the relation over all $k$ makes the right side a [[telescoping sum|0GXKM1R7E4]] equal to $0$.\nSo $f(n) = \\sum_k F(n, k)$ satisfies the [[recurrence|0FYX44GP92]] $\\sum_{j=0}^{J} a_j(n) f(n + j) = 0$.\nThe search terminates when $F$ is a proper hypergeometric term.\nSuch a term is a polynomial, times a product of factorials of integer-linear forms in $n$ and $k$, times $x^k$.\nThis termination result is part of the fundamental theorem of Wilf and Zeilberger.\nExample: for $F(n, k) = \\binom{n}{k}$ the algorithm stops at $J = 1$ with $a_0 = -2$, $a_1 = 1$ and $G(n, k) = -\\binom{n}{k - 1}$.\nCheck by Pascal's rule: $\\binom{n + 1}{k} - 2\\binom{n}{k} = \\binom{n}{k - 1} - \\binom{n}{k} = G(n, k + 1) - G(n, k)$, which gives $f(n + 1) = 2f(n)$.\nNon-example: for $J = 0$ no solution exists here, since that would need $\\sum_{j\u003ck}\\binom{n}{j}$ to be a hypergeometric term in $k$, and Gosper's algorithm shows it is not.\n",
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