On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Abstract

Let D_v the difference operator and q-difference operators defined by D_ωp(x) = \frac{p(x+ω)-p(x)}ω and D_q p(x) = \frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let (P_n)_n and Q_n)_n be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the above two OPS assuming that their difference derivatives $D_{\nu }$ of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as $$\sum _{i=0}^M{a}_{i,n}{D}_{\nu }^m{P}_{n+m-i}(x)=\sum _{i=0}^N{b}_{i,n}{D}_{\nu }^k{Q}_{n+k-i}(x),n\ge 0,$$ where $M,N,m,k=0,1,2,... Under certain conditions, we prove that U and V are related by a rational factor ç (in the distributional sense). Moreover, when m\neq k then both U and V are D_v-semiclassical functionals. This leads us to the concept of (M,N)-D_v-coherent pair of order (m,k) extending to the discrete case several previous works. As an application we consider the OPS with respect to a certain following Sobolev-type discrete inner product.

20 pages, Journal of Computational and Applied (2014) Journal of Computational and Applied Mathematics

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