Limit Discrete Meixner Series and Their Approximative Properties

In this article the problem of function approximation by discrete series by Meixner polynomials orthogonal on uniform net $\{0,1,\ldots\}$ is investigated. We constructed new series by these polynomials for which partial sums coincide with input function $f(x)$ in $x=0$. These new series were constructed by the passage to the limit of Fourier series $\sum\limits_{k=0}^\infty f^{\alpha}_km_k^{\alpha}(x)$ by Meixner polynomials when $\alpha\to-1$.