Difference between generating function and Rodrigue's formula?

[sreerajt]

What is the difference between generating function and Rodrigue's formula? Some says that from generating function you can generate required polynomial (say for example from generating function of Legendre polynomial you can get Legendre polynomial.), but in that case,as far as i know, Rodrigues formula also does the same job. So?

 

[maajdl]

see:
http://en.wikipedia.org/wiki/Generating_function
http://en.wikipedia.org/wiki/Rodrigues'_formula

 

[sreerajt]

i saw this link before posting this question. but i don't think that it contain answer to my question...

 

[vanhees71]

The generating function and the Rodrigues formula are just two ways to define the Legendre polynomials. Both have their merits in deriving formulas involving this set of orthogonal polynomials on the Hilbert space $\mathrm{L}^2([-1,1])$. Of course both definitions give the same polynomials.

 

[blue_raver22]

Generating function and Rodrigue's formula are both mathematical tools used to generate polynomials, but they differ in their approach and application.

A generating function is a mathematical function that can be used to represent a sequence of numbers or coefficients in a polynomial. It allows for a compact representation of a polynomial and can be used to find specific coefficients or terms within the polynomial. For example, the generating function for the Legendre polynomial can be used to find the coefficients of the polynomial at different orders.

On the other hand, Rodrigue's formula is a specific method for generating polynomials, where the polynomial is expressed as a product of a power of a variable and a polynomial with lower degree. This formula is particularly useful for generating orthogonal polynomials, such as the Legendre polynomial.

While both generating function and Rodrigue's formula can be used to generate polynomials, they have different applications. Generating functions are more general and can be used for a wide range of polynomials, while Rodrigue's formula is specifically designed for orthogonal polynomials. Additionally, generating functions can also be used to find specific coefficients or terms within a polynomial, while Rodrigue's formula is mainly used for generating the entire polynomial.

In summary, the difference between generating function and Rodrigue's formula lies in their approach and application. Both are useful tools in generating polynomials, but they serve different purposes and are used in different scenarios.