In this paper we consider the system of discrete functions {ϕr,k(x)} ∞ k=0 , which is orthonormal with respect to the Sobolev-type inner product hf, gi = Xr−1 ν=0 ∆ ν f(−r)∆ν g(−r) + X t∈Ωr ∆ r f(t)∆r g(t)µ(t), where µ(t) = q t (1−q), 0 < q < 1. It is shown that the shifted classical Meixner polynomials © M−r k (x + r) ª∞ k=r together with functions n (x+r) [k] k! or−1 k=0 form a complete orthogonal system in the space l2,µ(Ωr) with respect to the Sobolev-type inner product. It is shown that the Fourier series on Meixner polynomials © akM−r k (x + r) ª∞ k=r (ak— 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 Mα k (x) with α > −1 are considered. In the case when α = r these special series coincide with mixed series on Meixner polynomials M0 k (x) and Fourier series on the system © akM−r k (x + r) ª∞ k=r orthonormal with respect to the Sobolev-type inner product.