# differentiating the generating function of the Legendre equation

I need to differentiate the generating function

$$G(x,t)=\sum a_n(x)t^n$$ w.r.t. x in order to solve

$$\tfrac{d}{dx}[(1-x^2)\tfrac{dG}{dx}]+\tfrac{d}{dt}[t^2\tfrac{dG}{dt}]$$.

But I don't understand how this can be done ..... any suggestions ?

context:I know that not giving an attempt is frowned upon in this community. I almost always do. In this particular instance this is part of a much larger question which I have in fact given a big attempt at but it is too long to type out. I do'nt want the question done for me I just want to know how to differentiate G wrt x

Note: these were supposed to be partial derivatives ( i didn't know the symbol)

Note : $$G(x,t)=(1-2xt+t^2)^{-1/2}$$

• @DevashishKaushik I know that not giving an attempt is frowned upon in this community. I almost always do. In this particular instance this is part of a much larger question which I have in fact given a big attempt at but it is too long to type out. I do'nt want the question done for me I just want to know how to differentiate G wrt x – excalibirr Oct 30 '18 at 14:27
• Well, maybe you should add that to the question (as context) :). – Devashish Kaushik Oct 30 '18 at 14:32
• Also, I think you need to clarify whether you are looking for partial or total derivatives. In case you explicitly need $\frac{d}{dx}$, then you are looking for the latter. You can easily Google how to do that. – Devashish Kaushik Oct 30 '18 at 14:33
• In case you are not sure which one to use, you should provide more context so others can figure out which one is applicable, imho. – Devashish Kaushik Oct 30 '18 at 14:33

While taking partial derivatives, you treat the other variable as if it were a constant :

$$\frac{\partial{G}}{\partial{x}} = \frac{\partial{{(1 - 2xt + t^2)}^{-1/2}}}{\partial{x}}$$

$$= {-1 \over 2}\cdot{(1-2xt + t^2)}^{-3/2}\cdot \frac{\partial{(1-2xt+t^2)}}{\partial{x}}$$ (Using the chain rule)

$$= {-1 \over 2}\cdot{(1-2xt + t^2)}^{-3/2}\cdot(-2)$$

$$= {(1-2xt + t^2)}^{-3/2}$$

# UPDATE -

For the series version, simply use the same procedure (all variables except $$x$$ are to be treated as constants , mentally replace variable $$t$$ by constant $$a$$; $$x \mapsto t$$ ). Here you will get something along the lines of :

$$\frac{\partial}{\partial{x}} { \{ \Sigma (a_n(x) \cdot (t)^n) \} } = \Sigma \{ \frac{\partial{ (a_n(x) \cdot (t)^n) }}{\partial{x}} \}$$

$$= \Sigma \{ (t)^n \cdot \frac{\partial{a_n(x)}}{\partial{x}} \}$$

However, I do not think that is what you should be looking for.

What you need is :

$$\frac{\partial{G}}{\partial{x}} = \frac{1}{{(1-2xt + t^2)}^{3/2}}$$

and

$$\frac{\partial{G}}{\partial{t}} = \frac{(x-t)}{{(1-2xt + t^2)}^{3/2}}$$

Be sure to accept if that solves your problem ;) (@exodius)

• I think I have to use the series version of G though. cos the next part of the question say show that a_n satisfies the legendre eq and that indeed a_n=p_n – excalibirr Oct 30 '18 at 15:04
• I think what I need to do is find an equation for A_n then put this equation into the legendre eq to show that it is satisfied – excalibirr Oct 30 '18 at 15:32