n
Let L(x) = SUM [a_i*x^i]. Then
i=0
(x - 243)*L(3x) = 243*(x - 1)*L(x) becomes
n n
3^5*(x - 1)*SUM [a_i*x^i] = (x - 3^5)*SUM [3^i*a_i*x^i]
i=0 i=0
or
n n
SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_i*(3^i - 3^5)*x^(i+1)]
i=0 i=0
or
n n+1
SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_(i-1)*(3^(i-1) - 3^5)*x^i]
i=1 i=1
or
n n
SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_(i-1)*(3^(i-1) - 3^5)*x^i]
i=1 i=1
+ a_n*(3^n - 3^5)*x^(n+1)
Let n = 5 and we get the difference equation for i = 1 to 5:
3^(i-1) - 3^5
a_i = ---------------
3^5*(3^i - 1)
Using a computer for the dirty work we get the following monic polynomial:
L(x) = x^5 - 363x^4 + 32,670x^3 - 882,090x^2 + 7,144,929x - 14,348,907
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Posted by Bractals
on 2005-12-23 18:35:52 |
Solution
