Understanding the Legendre Recurrence Relation for Generating Functions

In summary, the conversation is about deriving the recurrence relation for Legendre polynomials using the generating function. The speaker is having trouble understanding the step where the powers of h are changed without shifting the series. They believe their method of expanding the series and then changing the powers is incorrect, but they are unsure of why. They are seeking clarification and understanding on this step.
  • #1
Taylor_1989
402
14

Homework Statement

I am having a slight issue with generating function of legendre polynomials and shifting the sum of the genertaing function.

So here is an example:

I need to derive the recurence relation ##lP_l(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}##

so I start with the following equation:

$$(1-2xh+h^2)\frac{\partial\phi}{\partial h}=(x-h)\phi $$

now taking the differential of the series of the Legendre generating function

$$\frac{\partial}{\partial h}(\sum_{l=0}^{\infty} h^l P_l(x))$$

I make it equal to $$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))$$

Some book I have read ignor the shift and keep the sum ##l=0## which I can see beacuse even at the ##l=0## the sums are the same.But when I expand the following is when I get in a bit of a mess.

So expanding both side I get the following:

$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)=x\sum_{l=0}^{\infty} h^l P_l(x)-\sum_{l=0}^{\infty} h^{l+1} P_l(x)[1]$$

So now if I make eq [1]=0 like so:

$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)-x\sum_{l=0}^{\infty} h^l P_l(x)+\sum_{l=0}^{\infty} h^{l+1} P_l(x)=0$$

So from here I know that I need to get all the powers of h to ##h^{l-1}##

So here what I do, which I believe is incorrect I just can't see why or to be honest understand why it wrong. My belief it it has some thing to do with how the generating function works, but every book I have read and youtube video I have watch completely ignores the step that I get confused with, and without any explanation just give the recurrence relation. Anyway here what I do.looking at the series I make all the powers of h equalt to ##l-1##

$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=2}^{\infty} (l-1)h^{l-1} P_{l-1}(x))+\sum_{l=3}^{\infty} (l-2)h^{l-1} P_{l-2}(x)-x\sum_{l=1}^{\infty} h^{l-1} P_{l-1}(x)+\sum_{l=2}^{\infty} h^{l-1} P_{l-2}(x)=0$$

Now youtube videos I have watched and book I have read just change the powers of h without make any shitf to the series like soo.

$$\sum_{l=0}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=0}^{\infty} (l-1)h^{l-1} P_{l-1}(x))+\sum_{l=0}^{\infty} (l-2)h^{l-1} P_{l-2}(x)-x\sum_{l=0}^{\infty} h^{l-1} P_{l-1}(x)+\sum_{l=0}^{\infty} h^{l-1} P_{l-2}(x)=0$$

Then just factor out the sum and the h and your left with the desired recurrence relation albeit with a little simplifying.

Where as when I do it my way I expand all the series unitl the are l=3 and then do the same thing. But as I said I believe this to be wrong, but I just can't see how that in the book I have read and video I have watched how they can just change the powers of the h without affecting the series itself.

I would much appreciate if someone could help me on understanding this.
 
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  • #2
The recurrence relation should state a range of l to which it applies. That range should exclude the terms which are different between your sum equation and the "book" one.
The book one, as you quote it, is clearly wrong since it includes h-1 and h-2 terms.
 

Related to Understanding the Legendre Recurrence Relation for Generating Functions

1. What is the Legendre recurrence relation?

The Legendre recurrence relation is a mathematical formula that relates the coefficients of Legendre polynomials, which are a set of orthogonal functions commonly used in physics and engineering. It is used to generate higher order Legendre polynomials from lower order ones.

2. What is the significance of the Legendre recurrence relation?

The Legendre recurrence relation is significant because it allows for the efficient calculation of higher order Legendre polynomials, which have many applications in physics and engineering, such as in solving differential equations and representing spherical harmonics.

3. How is the Legendre recurrence relation derived?

The Legendre recurrence relation is derived from the generating function of Legendre polynomials, which is a power series representation of the polynomials. By differentiating the generating function and equating coefficients, the recurrence relation can be obtained.

4. Can the Legendre recurrence relation be generalized to other orthogonal polynomials?

Yes, the Legendre recurrence relation can be generalized to other families of orthogonal polynomials, such as Chebyshev polynomials and Hermite polynomials. However, the specific form of the recurrence relation will differ for each family of polynomials.

5. What are the applications of the Legendre recurrence relation?

The Legendre recurrence relation has various applications in mathematics, physics, and engineering. It is commonly used in solving differential equations, approximating functions, and representing physical phenomena. It is also utilized in numerical methods, such as the Gauss-Legendre quadrature method for numerical integration.

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