I am currently working with Gegenbauer/ultraspherical polynomials, which can be defined by their relation to Jacobi polynomials:
$$ C_n^\lambda(x)=\frac{(2\lambda)_n}{\left(\lambda+\frac{1}{2}\right)_n}P_n^{\lambda-\frac{1}{2},\lambda-\frac{1}{2}}(x) $$
Using this definition it is obvious (assuming we know Rodrigues's formula for Jacobi polynomials), that these Gegenbauer polynomials satisfy their own Rodrigues's formula. At this point I realised that there are several distinct ways to define these orthogonal polynomials, all of which have to be equivalent. So I went on a chase to prove equivalency for every definition I could find. Currently I am working from the definition in terms of their generating function:
$$ (1-2xr+r^2)^{-\lambda}=\sum_{k=0}^\infty C_k^\lambda(x)r^k. $$
From here I want to get to their definition in terms of Rodrigues's formula, which, for Gegenbauer polynomials, is given by
$$ (1-z^2)^{\lambda-\frac{1}{2}}C_n^\lambda(z)=\frac{(-1)^n}{2^n n!}\frac{(2\lambda)_n}{\left(\lambda+\frac{1}{2}\right)_n}\omega_n^{(n)}(z), $$ where $\omega_n(z)=(1-z^2)^{n+\lambda-\frac{1}{2}}$. I am well aware that this might not be the best path from one definition to the next, but I wanted to be thorough. I went through every proof of Rodrigues's formula for Jacobi polynomials and found one very intriguing proof using Lagrange Inversion, without any orthogonality or anything similar. That proof simply used Lagrange inversion to show that the LHS of the formula is given by the generating function of Jacobi polynomials, while the right is given by a power series whose coefficients were exactly the Rodrigues's formula. That would in fact be perfect. I tried to adapt the proof to Gegenbauer Polynomials, but in the end ran into the following problem:
The two generating functions are very different, in particular because there is the exta factor for Gegenbauer Polynomials, which depends on $n$, thus changing the entire series. In fact, I'd need to find some other function to apply Lagrange Inversion to, in order for the proof to still work out.
My question now is: Do you know any way to prove Rodrigues's formula for Gegenbauer Polynomials only using the generating function, but not their relation to Jacobi polynomials nor their orthogonality, preferably by using Lagrange Inversion (just because I really like the proof for Jacobi)?
The question is already long enough but if more details about the Jacobi Proof are required, I can supply them.