Inner product and Gegenbauer polynomials in Sobolev space

In this paper we consider the system of functions $G_{r,n}^{\alpha }(x)$ ($r\in\mathbb{N},$ $n=0,1,...$) which is orthogonal with respect to the Sobolev-type inner product on $(-1,1)$ and generated by orthogonal Gegenbauer polynomials. The main goal of this work is to study some properties related to the system $\{\varphi _{k,r}(x)\}_{k\geq 0}$ of the functions generated by the orthogonal system\linebreak $\{G_{r,n}^{\alpha }(x)\}$ of Gegenbauer functions. We study the conditions on a function $f(x)$ given in a generalized Gegenbauer orthogonal system for it to be expandable into a generalized mixed Fourier series of the form
$$ f(x)\sim \sum_{k=0}^{r-1}f^{(k)}(-1)\frac{(x+1)^{k}}{k!}+\sum_{k=r}^{\infty }C_{r,k}^{\alpha }(f)\varphi _{r,k}^{\alpha }(x),$$
as well as the convergence of this Fourier series. The second result of this paper is the proof of a recurrence formula for the system $\{\varphi _{k,r}(x)\}_{k\geq 0}.$ We also discuss the asymptotic properties of these functions, and this represents the latter result of our contribution.