There is [this paper][1] which treats the special case of "half-range Chebyshev polynomials" (both kinds, corresponding to the weights $\dfrac1{\sqrt{1-x^2}}$ and $\sqrt{1-x^2}$ over $[0,1]$) to deal with Fourier expansions of nonperiodic functions. I have a feeling that half-range Gegenbauer polynomials have been treated before, and I'll try to see what I can dig up.

In the meantime, one can use the Stieltjes procedure to build up the recursion relations for these half range Gegenbauers. Letting

$$\langle f(x),g(x) \rangle^{(\alpha)}=\int_0^1 (1-t^2)^{\alpha-1/2} f(t)g(t)\mathrm dt$$

be the associated inner product, the Stieltjes procedure for generating monic orthogonal polynomials $\phi_k(x)$ uses the formulae

$$\begin{align*}b_k&=\frac{\langle x\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}{\langle\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}\\c_k&=\frac{\langle\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}{\langle\phi_{k-1}(x),\phi_{k-1}(x)\rangle^{(\alpha)}}\end{align*}$$

to give the coefficients $b_k,c_k$ for the recursion relation

$$\phi_{k+1}(x)=(x-b_k)\phi_k(x)-c_k\phi_{k-1}(x)$$

Here, the result

$$\int_0^1 (1-t^2)^{\alpha-1/2}t^k \mathrm dt=\frac{\Gamma\left(\frac{1+k}{2}\right)\Gamma\left(\alpha+\frac12\right)}{2\Gamma\left(\alpha+\frac{k}{2}+1\right)}$$

is useful.

  [1]: http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW534.pdf