What is the Proof for the Relation Between Legendre Polynomials and Sums?

[JiriV]

Following relation seems to hold:

\int^{1}_{-1}\left(\sum \frac{b_{j}}{\sqrt{1-μ^{2}}} \frac{∂P_{j}(μ)}{∂μ}\right)^{2} dμ = 2\sum \frac{j(j+1)}{2j+1} b^{2}_{j}

the sums are for j=0 to N and P_{j}(μ) is a Legendre polynomial. I have tested this empirically and it seems correct.

Anyway, I would like to have i) either a proof, or ii) a reference in a book, by which it is obtained easily. Do you have some suggestions?

Thank you.

 

[chiro]

Hey JiriV and welcome to the forums.

I'm not too familiar with the legendre polynomial myself but I think I might be able to offer a few suggestions.

One suggestion is to expand the results of the integral and simplify. Another suggestion is to use the properties of the projection via an integral transform and a correct basis. For more of the specifics on this check out the following link and scroll down to the Gram-Schmidt process for getting the bases:

http://mathworld.wolfram.com/LegendrePolynomial.html

You might actually be better off doing the projection or using properties of the derivative in conjunction with the projection, but you would have to do a bit of investigation on your part.

 

[JiriV]

I will answer myself:

The relation above can be found in the book

W. E. Byerly, An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics. (Ginn & company, Boston, 1893).

(article 106). The book is available online at www.gutenberg.org.

Jiri