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  "body": "\n## Statement\n\nLet $n$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers.\nAssume that none of $\\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n$ and $d + e - a - n$ 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]] satisfy\n$${}_7F_6\\left( \\begin{matrix} a,\\ 1 + \\frac{a}{2},\\ b,\\ c,\\ d,\\ e,\\ -n \\\\ \\frac{a}{2},\\ 1 + a - b,\\ 1 + a - c,\\ 1 + a - d,\\ 1 + a - e,\\ 1 + a + n \\end{matrix} ; 1 \\right)$$\n$$= \\frac{(1 + a)_n\\, (1 + a - d - e)_n}{(1 + a - d)_n\\, (1 + a - e)_n}\\; {}_4F_3\\left( \\begin{matrix} -n,\\ d,\\ e,\\ 1 + a - b - c \\\\ 1 + a - b,\\ 1 + a - c,\\ d + e - a - n \\end{matrix} ; 1 \\right).$$\n\n## Remarks\n\nThe left side is a very well-poised ${}_7F_6$: each upper parameter plus the matching lower parameter gives $1 + a$.\nThe ${}_4F_3$ on the right is balanced: its lower parameters sum to $1$ more than its upper parameters.\nSo the transformation trades well-poised structure for balanced structure, and the number of free parameters stays the same.\nA standard proof uses the Pfaff–Saalschütz summation together with an exchange of the order of summation.\nFor $n = 0$ both sides equal $1$.\nIf moreover $1 + 2a = b + c + d + e - n$, then $d + e - a - n = 1 + a - b - c$, the ${}_4F_3$ reduces to a balanced ${}_3F_2$, and the Pfaff–Saalschütz theorem sums it; this proves Dougall's formula.\n\n## Explanations\n\n### L1 human\n\nWhipple's transformation rewrites one long finite hypergeometric sum as a shorter one with a simple factor in front.\nThe two sums look very different, but they have the same value.\nIdentities of this kind let one move a hard sum into a form where a known summation formula applies.\n\n### L2 human\n\nThe theorem connects the two main families of hypergeometric series at $1$: the well-poised ones and the balanced ones.\nCombined with the Pfaff–Saalschütz theorem it gives Dougall's ${}_7F_6$ summation.\nIts $q$-analogue is Watson's transformation between a very well-poised ${}_8\\phi_7$ and a balanced ${}_4\\phi_3$.\nThe name Whipple is attached to several other transformations of ${}_3F_2$ and ${}_7F_6$ series as well.\n",
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