Hermite functions and inner product in Sobolev space

Let us consider the orthogonal Hermite system $\left\{ \varphi_{2n}(x)\right\} _{n\geq 0}$ of even index defined on $\left( -\infty,\infty \right),$ where
\begin{equation*} \varphi _{2n}(x)=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{\left( 2n\right) !}\pi ^{\frac{1}{4}}2^{n}}H_{2n}(x), \end{equation*}
by $H_{2n}(x)$ we denote a Hermite polynomial of degree $2n.$ In this paper, we consider a generalized system $\left\{ \psi_{r,2n}(x)\right\} $ with $r>0,$ $n\geq 0$ which is orthogonal with respect to the Sobolev type inner product on $\left(-\infty ,\infty \right),$ i.e.
\begin{equation*} \langle f,g \rangle =\lim_{t\rightarrow -\infty }\sum_{k=0}^{r-1}f^{\left(k\right) }(t)g^{\left( k\right) }(t)+\int_{-\infty }^{\infty }f^{\left(r\right) }(x)g^{\left( r\right) }(x)\rho (x)dx \end{equation*}
with $\rho (x)=e^{-x^{2}},$ and generated by $\left\{\varphi_{2n}(x)\right\}_{n\geq 0}.$ The main goal of this work is to study some properties related to the system $\left\{ \psi_{r,2n}(x)\right\}_{n\geq 0},$
\begin{gather*} \psi _{r,n}(x)=\frac{(x-a)^{n}}{n!},\quad n=0,1,2,\ldots,r-1, \\[2pt] \psi _{r,r+n}(x)=\frac{1}{(r-1)!}\int_{a}^{b}(x-t)^{r-1}\varphi _{n}(t)dt, \quad n=0,1,2,\ldots\, . \end{gather*}
We study the conditions on a function $f(x),$ given in a generalized Hermite orthogonal system, for it to be expandable into a generalized mixed Fourier series as well as the convergence of this Fourier series. The second result of the paper is the proof of a recurrent formula for the system $\left\{ \psi _{r,2n}(x)\right\} _{n\geq 0}.$ We also discuss the asymptotic properties of these functions, and this concludes our contribution.