I would like to find the relation between the solutions of a differential equation obtained by two different authors. The first solution is given in terms of the hypergeometric function $_2F_1$:
$$\left(1+\beta p^2\right)^{\frac{1}{4} (-1-2 n-2 \xi)} \, _2F_1\left(-n,-n-2 \xi ;-n-\xi +\frac{1}{2};\frac{1}{2}\left(1 + i p \sqrt{\beta}\right)\right),$$
where $p$ is a real variable, $\beta>0$, $n$ is a positive integer and $\xi= \sqrt{4 + \beta^2}/(2\beta)$.
The second one involves the Gegenbauer polynomials $C_n^\lambda(z)$
$$ \left(1 + \beta p^2\right)^{-\frac{1}{2} \left(\xi +\frac 12\right)} C_n^{\left(\xi +\frac{1}{2}\right)}\left(\frac{\sqrt{\beta } p}{\sqrt{\beta p^2+1}}\right).$$
I have omitted some normalization constants. Perhaps it is easy, but I do not see it. I know the relations between the hypergeometric function and the Gegenbauer polynomials but I can't apply them to this problem. In particular, I'm not sure how to transform the argument of $_2F_1$ to match the argument of $C_n^\lambda(z)$.
I will consider the case when $n$ is even. The fact that these two solutions are equivalent when $n$ is odd can be proved in an analogous manner.
Let $n=2\nu$ be even nonnegative integer, $\lambda=\xi+\frac{1}{2}$ and let's absorb $\sqrt{\beta}$ into the definition of $p$. Then the ratio of the two expressions above can be written as $$ \displaystyle\frac{ \displaystyle\phantom{}_2F_1\left(-2\nu,-2\nu-2\lambda+1,-2\nu-\lambda+1;\frac{1+pi}{2}\right)}{ \displaystyle(1+p^2)^\nu C_{2\nu}^\lambda\left(\frac{p}{\sqrt{p^2+1}}\right)}. $$ By eq. (20): $$ C_{2\nu}^\lambda\left(\frac{p}{\sqrt{p^2+1}}\right)=\left({\textstyle 2\nu+2\lambda-1\atop \textstyle 2\nu}\right)\phantom{}_2F_1\left(-\nu,\nu+\lambda,\lambda+\frac{1}{2};\frac{1}{p^2+1}\right). $$ By eq. (15.8.2) this is equal, up to a constant coefficient: $$ \frac{1}{(p^2+1)^\nu}\cdot \phantom{}_2F_1\left(-\nu,-\nu-\lambda+\frac{1}{2},-2\nu-\lambda+1;p^2+1\right). $$ By Kummer's quadratic transformation, eq. 15.8.18: $$ \phantom{}_2F_1\left(-\nu,-\nu-\lambda+\frac{1}{2},-2\nu-\lambda+1;p^2+1\right)=\phantom{}_2F_1\left(-2\nu,-2\nu-2\lambda+1,-2\nu-\lambda+1;\frac{1\pm pi}{2}\right). $$ So we proved that the expression $$ \displaystyle\frac{ \displaystyle\phantom{}_2F_1\left(-2\nu,-2\nu-2\lambda+1,-2\nu-\lambda+1;\frac{1+pi}{2}\right)}{ \displaystyle(1+p^2)^\nu C_{2\nu}^\lambda\left(\frac{p}{\sqrt{p^2+1}}\right)}. $$ is constant, i.e. doesn't depend on $p$, as required.