Why are there two different recurrences for Gegenbauer polynomials? As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method.
To make a long story short, the method involves expressing a particular function as a non-negative linear sum of Gegenbauer polynomials.  In various publications on this topic (see, for example, Musin, "The Kissing Number in Four Dimensions" in the July 2008 Annals of Mathematics), these polynomials have the following recurrence:
$$
G^{(n)}_0(t) = 1
$$
$$
G^{(n)}_1(t) = t
$$
$$
G^{(n)}_k(t) = \frac{(2k+n-4)tG^{(n)}_{k-1}(t)-(k-1)G^{(n)}_{k-2}(t)}{k+n-3}
$$
However, in the Wikipedia and Wolfram MathWorld plot summaries for Gegenbauer polynomials, the recurrences are different (and not just up to a constant factor).  In both, converting to consistent symbols, we have
$$
G^{(n)}_0(t) = 1
$$
$$
G^{(n)}_1(t) = 2nt
$$
$$
G^{(n)}_k(t) = \frac{2t(k+n-1)G^{(n)}_{k-1}(t)-(k+2n-2)G^{(n)}_{k-2}(t)}{k}
$$
Both definitions are normalized to $G^{(n)}_0(t) = 1$.  What accounts for the difference between the two?

I'll use $G$ for the second definition and $\widetilde G$ for the first definition. $G$ has the generating function $(1 - 2 x t + t^2)^{-n}$:
$$(1 - 2 x t + t^2)^{-n} = \sum_{k \geq 0} G_k^{(n)}(x) \,t^k.$$
$\widetilde G$ is constructed by first taking $C$ with the gf $(1 - 2 x t + t^2)^{1 -n/2}$ and then normalizing by $C(1)$:
$$(1 - 2 x t + t^2)^{1 - n/2} = \sum_{k \geq 0} C_k^{(n)}(x) \,t^k, \\
\widetilde G_k^{(n)}(x) = \frac {C_k^{(n)}(x)} {C_k^{(n)}(1)}.$$
Therefore
$$\widetilde G_k^{(n)}(x) =
\frac {(-1)^k} {\binom {2 - n} k } G_k^{(n/2 - 1)}(x).$$
