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  "body": "\n## Summary\n\nThe calculus of finite differences studies functions and sequences through their [[finite differences|7JSQZ126BP]] and sums.\nIt does so in the way that ordinary calculus uses derivatives and integrals.\nIts main tools are the [[forward difference operator|8E0EBA8X07]] $\\Delta$, the [[shift operator|MBK6AE6734]] $E = I + \\Delta$, and the [[indefinite sum|SMAQN2N739]], which inverts $\\Delta$.\nMany rules of calculus have exact discrete counterparts, such as [[summation by parts|9G82Z17DKA]] and the [[fundamental theorem of the calculus of finite differences|MNSSV4QHD1]].\nIt is the basis of the theory of difference equations, of interpolation at equally spaced points, and of closed forms for finite sums.\n\n## Explanations\n\n### L1 human\n\nOrdinary calculus works with infinitely small changes; the calculus of finite differences works with steps of size one.\nDifferences replace derivatives and sums replace integrals.\nThe powers $x^k$ are replaced by the [[falling factorials|0SJNCJCCQG]] $(x)_k = x(x-1)\\cdots(x-k+1)$, because $\\Delta (x)_k = k\\,(x)_{k-1}$, just as the derivative of $x^k$ is $k x^{k-1}$.\nFor example, summing this rule gives $\\sum_{x=0}^{n-1} x = \\frac{(n)_2}{2} = \\frac{n(n-1)}{2}$, the discrete analogue of $\\int_0^n x\\,dx = \\frac{n^2}{2}$.\n",
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