Here are some aspects which might help to clarify the situation. At first I'd like to give a few definitions of Gegenbauer polynomials from relevant sources. This way we can get an impression what we typically might expect.
Higher Transcendental Functions, Vol I by A. Erdelyi and H. Bateman (author):
\begin{align*}
(1-2hz+h^2)^{-\nu}=\sum_{n=0}^\infty C_n^{\nu}(z)h^n\qquad\qquad|h|<|z\pm(z^2-1)^{1/2}|
\end{align*}
Handbook of Mathematical Functions by M. Abramowitz, I.A. Stegun:
- (22.9.3. Generating Functions)
\begin{align*}
(1-2xz+z^2)^{-\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x)z^n\qquad\qquad |z|<1,\alpha\ne 0
\end{align*}
NIST/DLMF
- (18.12.4. Ultraspherical)
\begin{align*}
(1-2xz+z^2)^{-\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x)z^n\qquad\qquad |z|<1
\end{align*}
Special Functions, Encyclopedia of mathematics and its applications 71 by G.E. Andrews, R. Askey and R. Roy:
\begin{align*}
C_n^{\lambda}(x):=\frac{(2\lambda)_n}{(\lambda+(1/2))_n}P_n^{(\lambda-(1/2),\lambda-(1/2))}(x)\tag{1}
\end{align*}
$\qquad$with the generating function
\begin{align*}
(1-2xr+r^2)^{-\lambda}=\sum_{n=0}^\infty C_n^{\lambda}(x)r^n.\tag{2}
\end{align*}
Note the exponent of $(1-2xr+r^2)^{\color{blue}{-\lambda}}$ and the upper index in $C_n^{\color{blue}{\lambda}}(x)$ in (2) are coupled by a multiplicative factor $-1$ and this notational convention is used in all these citations. The connection with the Jacobi polynomials is given in (1).
We take a look at OPs cited paper The Kissing Number in Four Dimensions by O.R. Musin, check a few statements and a cited reference we are interested in.
(3.B The Gegenbauer polynomials)
... Let us recall definitions of Gegenbauer polynomials $C_k^{(n)}(t)$, which are defined by the expansion
\begin{align*}
(1-2rt+r^2)^{(2-n)/2}=\sum_{k=0}^\infty r^kC_k^{(n)}(t)\tag{3}
\end{align*}
This definition looks somewhat peculiar, since the upper index $n$ of $C_k^{(\color{blue}{n})}$ is not coupled by a factor $-1$ with the exponent of $(1-2rt+r^2)^{\color{blue}{(2-n)/2}}$ of the generating function. A few lines above (3.B) the author states
... Schoenberg [29] extended this property to Gegenbauer polynomials $G_k^{(n)}$. He proved: The matrix $\left(G_k^{(n)}\left(\cos \phi_{i,j}\right)\right)$ is positive semidefinite for any finite $X\subseteq \mathbf{S}^{n-1}$.
The reference [29] addresses the paper Positive definite functions on spheres by I.J. Schoenberg. It is revealing to check the definition Schoenberg used for Gegenbauer (resp. ultraspherical) polynomials.
(Schoenberg [29], section 1)
... Let $P_n^{(\lambda)}(\cos t)$ be the ultraspherical polynomials defined by the expansion
\begin{align*}
(1-2r\cos t+r^2)^{-\lambda}=\sum_{n=0}^\infty r^nP_n^{(\lambda)}(\cos t),\qquad\qquad (\lambda>0).\tag{4}
\end{align*}
Note in definition (4) the parameter $\lambda$ is used in accordance with the citations above. A few lines later Schoenberg gives a series expansion of a function $g(t)$ in terms of Gegenbauer polynomials:
\begin{align*}
g(t)=\sum_{n=0}^\infty P_n^{(\lambda)}(\cos t),\qquad\qquad(\lambda=\frac{1}{2}(m-1)),\tag{5}
\end{align*}
Conclusion: Comparing Schoenberg's usage of the parameter $\lambda$ in (5) and (4) with Musin's parameter setting in (3) indicates he could have had some short-cut notation in mind. Regrettably this implies that recurrence relation, differential equations, etc. have a different form than usually expected. The differences in the recurrence relation do not occur, if Musin would have used (3') instead
\begin{align*}
(1-2rt+r^2)^{-\lambda}=\sum_{k=0}^\infty r^kC_k^{(\lambda)}(t)\qquad\qquad (\lambda=(n-2)/2)\tag{3'}
\end{align*}
