Consider the Bessel DEQ: $$x^2y''+xy'+(x^2-\alpha^2)y=0$$
Find a function $F$ such that $$F(x,t)=\sum_{\alpha=-\infty}^\infty J_{\alpha}(x)t^{\alpha}$$
I received a few instructions:
a) first compute $\frac{\partial^n}{\partial x^n}F(x,t)$ and $\frac{\partial}{\partial t}(t\frac{\partial}{\partial t}F(x,t))$
b) Then exploiting the results of the first hint, write down the PDE that $F(x,t)$ has to satisfy.
c) Then given the ansatz $F(x,t)=\exp(xT(t))$ find the two ODEs that $T(t)$ has to satisfy ($T(t)=t^r$)
d) Last, show that $T(t)=-T(1/t)$ which will fix all constants in the generating function which gives the generating function of the Bessel functions.
I'm already running into problems with the first hint, I'm not sure how to take the $n$th derivative of the $F(x,t)$ function and I would really appreciate some assistance!
EDITs: Calculated the partial derivatives according to the helpful comments, but now stuck on the second part!
$$\frac{\partial}{\partial x}J_{\alpha}(x)=\frac{1}{2}(J_{\alpha-1}(x)-J_{\alpha+1}(x))$$
$$\frac{\partial^2}{\partial x^2}J_{\alpha}(x)=\frac{1}{4}(J_{\alpha-2}(x)-2J_{\alpha}(x)+J_{\alpha+2}(x))$$
$$\frac{\partial}{\partial t}F(x,t)=\Sigma_{-\infty}^{\infty}J_{\alpha}(x)\alpha^2t^{\alpha-1}$$
Now for the second part, as @Gary mentioned in the comments I need to have $\sum\limits_{n = - \infty }^\infty {(x^2 J''_n (x) + xJ'_n (x) + (x^2 - n^2 )J_n (x))t^n } \;\; (=0)$. But as the second hint suggests, i need to have a PDE but not sure where the time partial dertivative should be...Thanks
2nd Edit
With the help of Gary i have :$$x^2\Sigma_{\alpha=-\infty}^{\infty}J_{\alpha}^2(x)t^{\alpha}+x\Sigma_{\alpha=-\infty}^{\infty}J_{\alpha}(x)t^{\alpha}+x^2\Sigma_{\alpha=-\infty}^{\infty}J_{\alpha}(x)t^{\alpha}-t^2\Sigma_{\alpha=-\infty}^{\infty}J_{\alpha}(x)t^{\alpha}$$
and the PDE follows:
$$x^2F_{xx}+xF_x+x^2F-tF_t-t^2F_{tt}=0$$
Can someone give me a hint on how to find the two ordinary differential equations asked in c)?
The exercise asked you to compute $\frac{{\partial ^n F(x,t)}}{{\partial x^n }}$ and not $\frac{{\partial ^n J_\alpha (x)}}{{\partial x^n }}$. Likewise, it was suggested to compute $\frac{\partial }{{\partial t}}\left( {t\frac{{\partial F(x,t)}}{{\partial t}}} \right)$ and not just ${\frac{{\partial F(x,t)}}{{\partial t}}}$ (which is probably just a typo). We have $$ \frac{{\partial ^n F(x,t)}}{{\partial x^n }} = \sum\limits_{\alpha = - \infty }^\infty {J_\alpha ^{(n)} (x)t^\alpha } $$ and $$ \frac{\partial }{{\partial t}}\left( {t\frac{{\partial F(x,t)}}{{\partial t}}} \right) = \sum\limits_{\alpha = - \infty }^\infty {\alpha ^2 J_\alpha (x)t^{\alpha-1} } , $$ respectively. Now what is the series form of $$ x^2 \frac{{\partial ^2 F(x,t)}}{{\partial x^2 }} + x\frac{{\partial F(x,t)}}{{\partial x}} + x^2 F(x,t) - t\frac{\partial }{{\partial t}}\left( {t\frac{{\partial F(x,t)}}{{\partial t}}} \right)\,? $$