# Alternative definition of Legendre polynomials

I'm studying Panofsky and Phillips' Classical Electricity and Magnetism. In writing the potential of a linear $$2^n$$-pole lying along the $$x$$-axis, they make use of the following definition for the Legendre polynomials. $$\frac{P_n(\cos\theta)}{r^{n+1}} = \frac{(-1)^n}{n!}\frac{\partial^n}{\partial x^n} \left(\frac{1}{r}\right).$$

Here, $$r:=|\mathbf{r - r_0}|$$ and I guess that $$\theta$$ is the angle that $$(\mathbf{r-r_0})$$ makes with the $$x$$-axis (the authors don't mention abot $$\theta$$). Further I'm not even sure if the RHS is evaluated at $$\mathbf r = 0$$.

I'm only barely familiar with the Legendre functions and I am not able to show that the above matches with the standard definitions of the Legendre functions. I'll be happy if you can help me see the equivalence with the Rodrigue's formula.

• Can you give us the exact equation in the book and perhaps all other details. What is x? Commented Feb 13, 2021 at 15:29
• @PraharMitra $x$ is the first coordinate, corresponding to the $x$-axis. Writing more explicitly, $r=\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2}$.
– Atom
Commented Feb 13, 2021 at 15:31
• Would Mathematics be a better home for this question? Commented Feb 13, 2021 at 15:38
• @Qmechanic I agree. Can you move it? Or should I ask separately there?
– Atom
Commented Feb 13, 2021 at 19:36

I think that this is the standard generating-function formula $$\frac{1}{\sqrt{1+x^2-2x\cos\theta}}= \sum_{n=0}^\infty x^n P_n(\cos\theta), \quad |x|<1$$ in disguise as the formula for its Taylor series coefficients $$P_n(\cos \theta) =\left. \frac 1 {n!} \frac{\partial^n}{\partial x^n} \frac{1}{\sqrt{1+x^2-2x\cos\theta}}\right|_{x=0}$$