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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 nonnegative 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 nonnegative 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.
 Spring '09
 M.PEI
 Math, Binomial

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