A2 - Math 239 Assignment 2 Due Friday May 21 1 Express the following coefficients as summations involving only ordinary binomial coefficients(a(4

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Unformatted text preview: Math 239 Assignment 2 Due: Friday, May 21 1. Express the following coefficients as summations involving only ordinary binomial coefficients. (a) (4 points) [xn ](1 − 2x)−m (1 − x2 )m . (b) (4 points) [xn ](1 + 2x − 3x2 )−4 . (c) (4 points) [xn ](1 + 2x)−5 (1 − x3 )m . 2. Let x1 , . . . , xk be variables. A monomial in these variables is a product xm1 · · · xmk 1 k where mi ’s are non-negative integers. (For example monomials in x1 and x2 include 1, x5 and x2 x2 .) The degree of a monomial is the sum of the exponents—in this case 1 2 m1 + · · · + mk . Let N denote the set of non-negative integers. (a) (2 points) Define a weight function on Nk such that the number of elements in Nk with weight m is equal to the number of monomials in x1 , . . . , xk with degree exactly m. (b) (3 points) Compute the generating series for Nk . (c) (3 points) Use the binomial theorem to compute the number of monomials in k variable with degree exactly n. 3. Let S be the set of all subsets of {0, . . . , n − 1} and define the weight of a subset to be the sum of its elements. Let T be the subset of S consisting of the subsets that do not contain n − 1. Let ΦS (x) and ΦT (x) be the generating series for S and T respectively. (a) (3 points) Compute ΦS (x) when n = 4. (b) (4 points) Prove that ΦS (x) = (1 + xn−1 )ΦT (x). (c) (2 points) Give an expression for ΦS (x) as a product of n polynomials. 4. Let A(x) and B (x) be series with constant terms equal to 1 such that B (x) = A(x)2 . Note that if C (x) is the series n≥0 cn xn , then its derivative C (X ) is n≥0 (n +1)cn+1 xn . (a) (2 points) Using the usual rules from Calculus, show that A(x)B (x) = 2A (x)B (x). (b) (2 points) Assuming that [xk ]A(x) = ak and [xk ]B (x) = bk , compute formulas for the coefficient of xn in A(x)B (x) and in A (x)B (x). (c) (5 points) Apply the previous material with B (x) = 1 − x to compute the first five coefficients of (1 − x)−1/2 . (d) (1 point) Which formal power series have a square root? ...
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This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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