Some special series by general Laguerre polynomials and Fourier series by Laguerre polynomials, orthogonal in Sobolev sense

Some special series on Laguerre polynomials are considered and their approximative properties are investigated. In particular, the upper estimate for the Lebesgue function of introduced special series by Laguerre polynomials is obtained. The polynomials $ l_{r,k}^\alpha(x)$ $(k=0,1,\ldots)$ , generated by classical orthogonal Laguerre polynomials $L_k (x) (k = 0; 1; \ldots)$ and orthonormal with respect to the Sobolev-type inner product
\begin{equation*} <f,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_0^\infty f^{(r)}(t)g^{(r)}(t)t^\alpha e^{-t}dt, \end{equation*}
are introduced and investigated. The representations of these polynomials in the form of certain expressions containing Laguerre polynomials $L_n^{\alpha-r}(x)$ are\linebreak obtained. An explicit form of the polynomials $ l^\alpha_{r,k+r}(x)$ which is an expansion in powers of $x^{r+l}$ with $l=0,\ldots,k$ is established. These results can be used in the study of asymptotic properties of polynomials $l^\alpha_{r,k}(x)$ when $k\to\infty$ and in the study of approximative properties of partial sums of Fourier series by these polynomials. It is shown that Fourier series by polynomials $l^\alpha_{r,k}(x)$ coincides with the mixed series by Laguerre polynomials introduced and studied earlier by the author. Besides it is shown if $\alpha=0$, then mixed series on Laguerre polynomials and, as a corollary, the Fourier series by polynomials $l^0_{r,k}(x)$ represents the particular cases of special series, introduced in present paper.