Gegenbauer functional equations

Question

Let $C_n^\lambda$ be the Gegenbauer Polynomial with the parameter $\lambda$. We have the defining recurrence relation $$nC_n^\lambda(x)=2(n+\lambda-1)x\,C_{n-1}^\lambda(x)-(n+2\lambda-2)C_{n-2}^\lambda(x),$$ and $C_0^\lambda=1$, $C_1^\lambda(x)=2\lambda x$. Using some machinery it is fairly easy to show that $$C_n^\lambda(x)=\frac{\lambda}{n+\lambda}\left(C_n^{\lambda+1}(x)-C_{n-2}^{\lambda+1}(x)\right).$$ This equation relates a Gegenbauer Polynomial with parameter $\lambda$ to Gegenbauer Polynomials with parameter $\lambda+1$. I wanted to see whether there are more such functional equations, in particular whether there is an equation that relates a Gegenbauer Polynomials with parameter $\lambda$ to a (potentially infinite) sum of Gegenbauer Polynomials with parameter $\lambda\pm 1/2$. Even more precisely, I am interested in rewriting $C_n^1$ in terms of any Gegenbauer Polynomials, but preferably in terms of a sum of $C_j^{1/2}$. Going from $C_n^1$ to Chebyshev polynomials (which are, in a way, the correct replacement for Gegenbauer Polynomials for $\lambda=0$) yields a sum of all Chebyshev polynomials down to the first, which can be written as $$C_n^1(s)=\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}\alpha_{-1,j,n}T_{n-2j}(s),$$ where $\alpha_{-1,n/2,n}=1$ for even $n$ and $\alpha_{-1,j,n}=2$ for $j<n/2$. Going up by one can be done easily via the above equation. Is it possible to do something simily with half steps? In particular, are there coefficients $c_j$ such that $$C_n^1(s)=\sum_{j=0}^\infty c_jC_j^{\pm 1/2}(s)?$$

Answer 1

There is, in fact, a statement of the following type: $$C_n^\lambda(x)=\sum_{j=0}^{\lfloor \frac{n}2 \rfloor}\frac{(\lambda)_{n-j}(\lambda-\mu)_{j}(n+\mu-2j)}{(\mu+1)_{n-j}\cdot j!\cdot \mu}C_{n-2j}^\mu(x)\quad \text{for all }x\in\mathbb{C}, \lambda,\mu>0.$$ A proof can be found in e.g. George E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. The Press Syndicate of the University of Cambridge, 1999, Theorem 7.1.4'.