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A of degree has the form r N rational function () () (), where and are polynomials whose degrees sum to p x r x q x p x q x N = The City College of New York CSc – Scientific Programming Spring – Erik K Grimmelmann, PhD Rational function approximation 66 from Wikipedia. H^r standirig with lhe BBUalcal publle of NewYork Hor \oice though ample ln volume dld not aeem last nrht acreeable in qualiiy, ard ws BBOreorer, agheted wltb an nncaaatng tranaolo, so pronounced aa to envelop her intentions as to ihe pt1 ' continual uneenainiy lha has gueney, l>ut h*r. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomialAccording to the theorem, it is possible to expand the polynomial (x y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b c = n, and the coefficient a of each term is a specific positive.
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$\begin{aligned} \bigl( D^\pm & J B D^\pm J B D^\pm J B \bigr)_{xw} \\ &= (-1)^x \sum_y \sum_z (-1)^{y+z} \binom{n-1-x}{y}\binom{n-1-y}{z} \\ & \hspace*{3in} \binom{n-1-z}{w} \\ &= (-1)^{x+n-1} \sum_z \Bigl( \sum_y \binom{n-1-x}{y} \binom{-z-1}{n-1-y-z} \Bigr) \\& \hspace*{3in} \binom{n-1-z}{w} \\ &= (-1)^{x+n-1} \sum_z \binom{n-2-x-z}{n-1-z}\binom{n-1-z}{w} \\ &= (-1)^{x} \sum_z (-1)^{z}\binom{x}{n-1-z}\binom{n-1-z}{w} \\ &= (-1)^{x+n-1} \sum_r (-1)^r \binom{x}{r} \binom{r}{w} \\&= (-1)^{x+n-1} (-1)^x [x=w] = (-1)^{n-1}. \end{aligned}$

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