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  "body": "\n## Statement\n\nLet $m$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers with\n$$1 + 2a = b + c + d + e - m.$$\nAssume that none of the lower parameters $\\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + m$ lies in $\\set{0, -1, \\dots, 1 - m}$, and that $(1 + a - b - c - d)_m \\ne 0$.\nWrite $(x)_k = x (x + 1) \\cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThen the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_7F_6$ at $1$ is\n$${}_7F_6\\left( \\begin{matrix} a,\\ 1 + \\frac{a}{2},\\ b,\\ c,\\ d,\\ e,\\ -m \\\\ \\frac{a}{2},\\ 1 + a - b,\\ 1 + a - c,\\ 1 + a - d,\\ 1 + a - e,\\ 1 + a + m \\end{matrix} ; 1 \\right) = \\frac{(1 + a)_m\\, (1 + a - b - c)_m\\, (1 + a - b - d)_m\\, (1 + a - c - d)_m}{(1 + a - b)_m\\, (1 + a - c)_m\\, (1 + a - d)_m\\, (1 + a - b - c - d)_m}.$$\n\n## Remarks\n\nThe series is *very well-poised*: each upper parameter plus the matching lower parameter gives $1 + a$, and the pair $1 + \\frac{a}{2}, \\frac{a}{2}$ is present.\nThe condition $1 + 2a = b + c + d + e - m$ says that the series is $2$-balanced: the lower parameters sum to $2$ more than the upper ones.\nOne proof applies [[Whipple's transformation|QNGGCTXK9S]], which turns the ${}_7F_6$ into a multiple of ${}_4F_3(-m, d, e, 1 + a - b - c; 1 + a - b, 1 + a - c, d + e - a - m; 1)$.\nUnder the condition $1 + 2a = b + c + d + e - m$ one has $d + e - a - m = 1 + a - b - c$, so this ${}_4F_3$ reduces to a balanced ${}_3F_2$.\nThe [[Pfaff–Saalschütz theorem|9X6KG770GV]] then sums it, and the product on the right comes out.\nFor $m = 0$ both sides equal $1$.\nLetting $m \\to \\infty$ term by term gives the summation of the well-poised ${}_5F_4$ with upper parameters $a, 1 + \\frac{a}{2}, b, c, d$, which in turn contains part 3 of [[Dixon's identity|QC4TP5D6HE]].\n\n## Explanations\n\n### L1 human\n\nDougall's formula evaluates a finite sum of ratios of rising factorials with seven parameters on top and six on the bottom.\nThe parameters must follow a rigid pattern, and one linear relation must hold among them.\nWhen they do, the whole sum is a single product, which is rare for sums of this size.\n\n### L2 human\n\nDougall's formula sits near the top of the classical summation formulas for hypergeometric series at $1$.\nSpecialising parameters gives many other summations, which is why it is a hub in tables of identities.\nIts $q$-analogue is Jackson's summation for a very well-poised ${}_8\\phi_7$.\n",
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