Difference between generating function and Rodrigue's formula?

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Discussion Overview

The discussion centers on the differences between generating functions and Rodrigues' formula, particularly in the context of Legendre polynomials. Participants explore how each method can be used to derive or define these polynomials, considering their respective merits and applications.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the distinction between generating functions and Rodrigues' formula, suggesting both can generate Legendre polynomials.
  • Another participant notes that both methods define Legendre polynomials and have merits in deriving related formulas in the context of orthogonal polynomials.
  • A participant expresses dissatisfaction with existing resources, indicating they do not adequately address the question posed.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the differences between generating functions and Rodrigues' formula, and the discussion reflects multiple viewpoints on their equivalence and utility.

Contextual Notes

Some assumptions about the definitions and applications of generating functions and Rodrigues' formula may not be fully explored, and the discussion does not resolve the nuances between the two methods.

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

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