generating function of Laguerre polynomials

We start from the definition of Laguerre polynomials via their http://planetmath.org/node/11983Rodrigues formula

$L_n(z):=e^z{\displaystyle \frac{d^n}{d{z}^n}}{e}^{-z}{z}^n\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}0},\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }).$ (1)

The consequence

$f^{(n)}(z)={\displaystyle \frac{n!}{2\pi i}}{\displaystyle {\oint }_C}{\displaystyle \frac{f(\zeta )}{{(\zeta -z)}^{n+1}}}𝑑\zeta$ (2)

of http://planetmath.org/node/1150Cauchy integral formula allows to write (1) as the complex integral

$$L_n(z)=\frac{n!}{2i\pi }{\oint }_C\frac{e^z{e}^{-\zeta }}{{(\zeta -z)}^{n+1}}𝑑\zeta =\frac{n!}{2i\pi }{\oint }_C\frac{e^{z-\zeta }d\zeta }{{(1-\frac{z}{\zeta })}^n(\zeta -z)},$$

where $C$ is any contour around the point $z$ and the direction is anticlockwise.  The http://planetmath.org/node/11373substitution

$$\zeta -z:=\frac{zt}{1-t},\zeta =\frac{z}{1-t},t=\mathrm{\hspace{0.33em}1}-\frac{z}{\zeta }\mathit{\hspace{1em}}d\zeta =\frac{zdt}{{(1-t)}^2}$$

here yields

$$L_n(z)=\frac{n!}{2i\pi }{\oint }_{C^{\prime }}\frac{e^{-\frac{zt}{1-t}}zdt}{{(1-t)}^2{t}^n\cdot \frac{zt}{1-t}}=\frac{n!}{2i\pi }{\oint }_{C^{\prime }}\frac{e^{-\frac{zt}{1-t}}dt}{(1-t)t^{n+1}}$$

where the contour $C^{\prime }$ goes round the origin.  Accordingly, by (2) we can infer that

$$L_n(z)={\left[\frac{d^n}{d{t}^n}\frac{e^{-\frac{zt}{1-t}}}{1-t}\right]}_{t=0},$$

whence we have found the generating function

$$\frac{e^{-\frac{zt}{1-t}}}{1-t}=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n(z)}{n!}{t}^n$$

of the Laguerre polynomials.

Title	generating function of Laguerre polynomials
Canonical name	GeneratingFunctionOfLaguerrePolynomials
Date of creation	2013-03-22 19:06:51
Last modified on	2013-03-22 19:06:51
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Derivation
Classification	msc 33B99
Classification	msc 30B10
Classification	msc 26C05
Classification	msc 26A09
Classification	msc 33E30
Related topic	ExampleOfFindingTheGeneratingFunction
Related topic	GeneratingFunctionOfHermitePolynomials
Related topic	VariantOfCauchyIntegralFormula