I have different formulas for Legendre polynomials

Question

I have 3 different formulas for the Legendre polynomials and I want to know what is the difference. During lecture we had:

$$L_m(x) = \frac{1}{m!\cdot 2^m}\cdot \frac{\mathrm{d}^m}{\mathrm{d}x^m} \underbrace{\left[ (x-a)^m(x-b)^m \right]}_{different}$$

In the script I have:

$$M_m(x) = \underbrace{\sqrt{m+\frac{1}{2}}}_{different}\frac{1}{m!\cdot 2^m}\cdot \frac{\mathrm{d}^m}{\mathrm{d}x^m} \left[ (x^2-1)^m \right]$$

On Wikipedia: $$Q_m(x) = \frac{1}{m!\cdot 2^m}\cdot \frac{\mathrm{d}^m}{\mathrm{d}x^m} \left[ (x^2-1)^m \right]$$

Answer 1

These are Rodrigues' Formulas for the Legendre polynomial.

The first form defining $L_m(x)$, provided $a$ and $b$ take certain values, can be related to a commonly stated form which is your final form $Q_m(x)$. It's curious your $L_m(x)$ isn't labelled by $a$ and $b$ though and it's a form I've never seen before in the context of Legendre polynomials.

The factor $\sqrt{m+1/2} = \sqrt{(2m+1)/2}$ is a different normalisation. These polynomials are orthogonal on $[-1,1]$. The value of the orthogonality integral depends on the normalisation. The result with standard normalisation is $2/(2m+1)$. With the normalisation of $M_m(x)$ it would be $1$.

Look at e.g. Chapter 22, Orthogonal Polynomials in Handbook of Mathematical Functions (Abramowitz & Stegun)

An additional note, $Q_m(x)$ is the notation usually reserved for the Legendre function of the 2nd kind and $P_m(x)$ is usually used for the Legendre polynomial of degree m.

Answer 2

The factor of $\sqrt{m + 1/2}$ is a normalization constant so that $\displaystyle \int_{-1}^1 M_m(x) M_m(x)dx = 1$. This is a convention for where the Legendre polynomials pop up in physics: the quantum mechanical treatment of angular momentum.

If you integrate by parts $m$ times you can show that $\displaystyle \int_{-1}^1 Q_m(x) Q_m(x)dx = \dfrac{2}{2m+1} = \dfrac{1}{m + 1/2}$.