 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/20/a/solution

Apply the Leibniz rule to the

n

j

e^{- x}

and

n - j

x^{n}

, the corresponding term is

(j n) (- 1)^{j} e^{- x} (n! / j!) x^{j}

. Cancelling the prefactor gives

L_{n} (x) = j = 0 \sum n (- 1)^{j} (j n) \frac{x ^{j}}{j !} .

(- 1)^{n} / n!

is nonzero, so this is a polynomial of degree exactly

n

. In particular, the requested coefficients in

L_{10}

are

[x^{10}] L_{10} = \frac{1}{10 !}, [x^{9}] L_{10} = - \frac{10 !}{( 9 ! ) ^{2}} .

These are the ordinary Laguerre polynomials, with normalization

L_{n} (0) = 1

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