4.3 Laguerre'sche Funktionen

Partikuläre Lösungen der Laguerre'schen Differentialgleichung, einer speziellen hypergeometrischen Differentialgleichung. ^4.2

$$\leq \leq \infty$$ (4.12)

$${{n!} \over {\Gamma(n+k+1)}} \infty$$ (4.13)

k rell, für ganze k ergeben sich die Laguerre-Polynome

L₀(x) = 1        L₁^(k)(x) = k + 1 - x (4.14)

$${1 \over 2} \left[\vphantom{(k+1)(k+2)-2(k+2)x+x^2 }\right. \left.\vphantom{(k+1)(k+2)-2(k+2)x+x^2 }\right]$$ (4.15)

$${1 \over n!} {{d^n} \over {dx^n}} \sum_{\nu=0}^{n} \left(\vphantom{ \begin{array}{cc} {n+k} \\ {n-\nu} \end{array}}\right. \begin{array}{cc} {n+k} \\ {n-\nu} \end{array} \left.\vphantom{ \begin{array}{cc} {n+k} \\ {n-\nu} \end{array}}\right) {{(-x)^{\nu}} \over {\nu!}}$$ (4.16)

$${{x^k} \over {k!}} \left(\vphantom{ \begin{array}{cc} {n+k}\\ {k} \end{array}}\right. \begin{array}{cc} {n+k}\\ {k} \end{array} \left.\vphantom{ \begin{array}{cc} {n+k}\\ {k} \end{array}}\right)$$ (4.17)

$$\left(\vphantom{x{{d^2} \over {dx^2}} +(k+1-x){d \over dx}+n }\right. {{d^2} \over {dx^2}} {d \over dx} \left.\vphantom{x{{d^2} \over {dx^2}} +(k+1-x){d \over dx}+n }\right)$$ (4.18)

$$\prime$$

$${{1 \over x}(n L_n^{(k)}(x)-(n+k) L_{n-1}^{(k)}(x))}$$

Fortsetzung für k nicht ganz:

$$\left(\vphantom{ \begin{array}{cc} {n+k}\\ {n} \end{array}}\right. \begin{array}{cc} {n+k}\\ {n} \end{array} \left.\vphantom{ \begin{array}{cc} {n+k}\\ {n} \end{array}}\right)$$ (4.20)

(₁F₁ ist die konfluente hypergeometrische Funktion)