Question

Derive the Rodrigues formula. ​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

Consider the function

fn(x)=(x2−1)n

..........(20)

Differentiating this equation we get the second order differential equation,

(1−x2)f′′n+2(n−1)xf′n+2nfn=0

..................(22)

We wish to differentiate this n times by use of Leibniz's formula,

dndxnA(x)B(x)=∑k=0nn!k!(n−k)!dkAdxkdn−kBdxn−k

......................(23)

Applying this to (22) we easily get

(1−x2)f(n+2)n−2xf(n+1)n+n(n+1)f(n)n=0

......................................(24)

which is exactly Lergendre's differential equation (1-49). This equation is therefore satisfied by the polynomials

y=dndxn(x2−1)n

.....................(25)

The Legendre polynomials Pn(x) are normalized by the requirement Pn(1)=1. Using

y=2nn!

...............(26) for x=1,

We get

Pn(x)=12nn!dndxn(x2−1)n