  The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted
by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and
${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$
and real $\nu\ge 0,$ are special functions (usually non-polynomials), which are
defined upon the hypergeometric representation of the classical Gegenbauer
polynomial by allowing integer degree to be real fractional degree. Remarkably,
the GGF-Fs become indispensable for optimal error estimates of polynomial
approximation to singular functions, and have intimate relations with several
families of nonstandard basis functions recently introduced for solving
fractional differential equations. However, some properties of GGF-Fs, which
are important pieces for the analysis and applications, are unknown or under
explored. The purposes of this paper are twofold. The first is to show that for
$\lambda,\nu>0$ and $x=\cos\theta$ with $\theta\in (0,\pi),$
  \begin{equation*}\label{IntRep-0N}
  (\sin \varphi)^{\lambda}\,{}^{r\!}G_\nu^{(\lambda)}(\cos \varphi)=
  \frac{2^\lambda\Gamma(\lambda+1/2)}{\sqrt{\pi} {(\nu+\lambda)^{\lambda}}} \,
{\cos ((\nu+\lambda)\varphi- \lambda\pi/2)}
  +{\mathcal R}_\nu^{(\lambda)} (\varphi),
  \end{equation*} and derive the precise expression of the "residual" term
${\mathcal R}_\nu^{(\lambda)} (\varphi).$ With this at our disposal, we obtain
the bounds of GGF-Fs uniform in $\nu.$ Under an appropriate weight function,
the bounds are uniform for $\theta\in [0,\pi]$ as well. Moreover, we can study
the asymptotics of GGF-Fs with large fractional degree $\nu.$ The second is to
present miscellaneous properties of GGF-Fs for better understanding of this
family of useful special functions.
