The Fourier series of the Meixner polynomials orthogonal with respect to the Sobolev-type inner product

In this paper we consider the system of discrete functions $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty,$ which is orthonormal with respect to the Sobolev-type inner product
\begin{equation*} \langle f,g \rangle=\sum_{\nu=0}^{r-1}\Delta^{\nu} f(-r)\Delta^{\nu} g(-r) + \sum_{t\in\Omega_r}\Delta^r f(t) \Delta^r g(t)\mu(t), \end{equation*}
where $\mu(t)=q^t(1-q)$, $0<q<1.$ It is shown that the shifted classical Meixner polynomials $\left\{M_k^{-r}(x+r)\right\}_{k=r}^\infty$ together with functions $\left\{{(x+r)^{[k]}\over k!}\right\}_{k=0}^{r-1}$ form a complete orthogonal system in the space $l_{2,\mu}(\Omega_r)$ with respect to the Sobolev-type inner product. It is shown that the Fourier series on Meixner polynomials $\left\{a_kM_k^{-r}(x+r)\right\}_{k=r}^\infty$ ($a_k$ — normalizing factors), orthonormal in terms of Sobolev, is a special case of mixed series on Meixner polynomials. Some new special series on Meixner orthogonal polynomials $M_k^\alpha(x)$ with $\alpha>-1$ are considered. In the case when $\alpha=r$ these special series coincide with mixed series on Meixner polynomials $M_k^0(x)$ and Fourier series on the system $\left\{a_kM_k^{-r}(x+r)\right\}_{k=r}^\infty$ orthonormal with respect to the Sobolev-type inner product.