# generating function of Legendre polynomials

For finding the generating function

 $F(t)\;=\;\sum_{n=0}^{\infty}P_{n}(z)t^{n}$

of the sequence of the Legendre polynomials
$P_{0}(z)\;=\;1$
$P_{1}(z)\;=\;z$
$P_{2}(z)\;=\;\frac{1}{2}(3z^{2}\!-\!1)$
$P_{3}(x)\;=\;\frac{1}{2}(5z^{3}\!-\!3z)$
$P_{4}(z)\;=\;\frac{1}{8}(35z^{4}\!-\!30z^{2}\!+\!3)$
$P_{5}(z)\;=\;\frac{1}{8}(63z^{5}\!-\!70z^{3}\!+\!15z)$
$\cdots\qquad\;\;\cdots$
we have to present $P_{n}(z)$ as the general coefficient of Taylor series in $t$, i.e. as the $n$th derivative of some $F(t)$ in the origin, divided by the factorial $n!$.  The Cauchy integral formula offers the chance to implement that.

Starting from the http://planetmath.org/node/11983Rodrigues formula of Legendre polynomials, we may write

 $P_{n}(z)\;=\;\frac{1}{2^{n}n!}\frac{d^{n}}{dz^{n}}(z^{2}\!-\!1)^{n}\;=\;\frac{% 1}{2^{n}n!}\frac{n!}{2i\pi}\oint_{c}\frac{(\zeta^{2}\!-\!1)^{n}}{(\zeta\!-\!z)% ^{n+1}}d\zeta\;=\;\frac{1}{2i\pi}\oint_{c}\left(\frac{1}{2}\frac{\zeta^{2}\!-% \!1}{\zeta\!-\!z}\right)^{n}\!\frac{d\zeta}{\zeta\!-\!z},$

where the contour $c$ runs anticlockwise once around the point $z$.  The change of variable

 $\frac{\zeta^{2}\!-\!1}{2(\zeta\!-\!z)}\;=\;\frac{1}{t},\qquad d\zeta\;=\;\frac% {zt\!-\!1\!-\!\sqrt{1\!-\!zt\!+\!t^{2}}}{t^{2}\sqrt{1\!-\!zt\!+\!t^{2}}}dt$

gives

 $P_{n}(z)\;=\;-\frac{1}{2i\pi}\oint_{c^{\prime}}\frac{dt}{t^{n}t\sqrt{1\!-\!zt% \!+\!t^{2}}}$

where $t$ must go round the origin clockwise, but in

 $P_{n}(z)\;=\;\frac{1}{n!}\cdot\frac{n!}{2i\pi}\oint_{c^{\prime}}\frac{dt}{% \sqrt{1\!-\!zt\!+\!t^{2}}\cdot(t\!-\!0)^{n+1}}$

anticlockwise.  This is, by Cauchy integral formula again,

 $P_{n}(z)\;=\;\frac{1}{n!}\left[\frac{d^{n}}{dt^{n}}\frac{1}{\sqrt{1\!-\!zt\!+% \!t^{2}}}\right]_{t=0}.$

This means that

 $F(t)\;:=\;\frac{1}{\sqrt{1\!-\!zt\!+\!t^{2}}}$

is the searched generating function of the Legendre polynomials:

 $\frac{1}{\sqrt{1\!-\!zt\!+\!t^{2}}}\;=\;P_{0}(z)+P_{1}(z)t+P_{2}(z)t^{2}+P_{3}% (z)t^{3}+\ldots$

Cf. the generating function of the Bessel functions.

Title generating function of Legendre polynomials GeneratingFunctionOfLegendrePolynomials 2015-03-08 20:29:06 2015-03-08 20:29:06 pahio (2872) pahio (2872) 17 pahio (2872) Result msc 33B99 msc 30D10 msc 30B10 GeneratingFunctionOfHermitePolynomials GeneratingFunctionOfLaguerrePolynomials VariantOfCauchyIntegralFormula