 Normalization of the ordinary Laguerre polynomials

ID: normalization-of-the-ordinary-laguerre-polynomials

L_{n} (0) = 1

L_{n} (x) = \sum_{j = 0}^{n} (- 1)^{j} (j n) x^{j} / j!

. By the Leibniz rule, this is equivalent to Rodrigues' formula,

e^{- x} L_{n} (x) = D^{n} (e^{- x} x^{n}) / n!

. For any polynomial

q

of degree below

n

n

applications of integration by parts give

\int_{0}^{\infty} e^{- x} L_{n} q d x = 0

. All endpoint terms vanish: derivatives of

e^{- x} x^{n}

of order below

n

vanish at zero and decay at infinity. Since

L_{n}^{(n)} = (- 1)^{n}

, the same calculation gives

\int_{0}^{\infty} e^{- x} L_{n}^{2} d x = \frac{( - 1 ) ^{n}}{n !} \int_{0}^{\infty} e^{- x} x^{n} L_{n}^{(n)} d x = \frac{Γ ( n + 1 )}{n !} = 1.

e^{- x}

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