2023/01/07 by Mbouna, D.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.02776
We consider two sequences of orthogonal polynomials (Pn)n≥ 0 and (Qn)n≥ 0 with respect regular functionals \bf u and \bf v, respectively. We assume that ∑j=1 M aj,nDx k Pk+n-j (z)=∑j=1 N bj,nDx m Qm+n-j (z) , with k,m,M,N ∈ ℕ, aj,n and bj,n are sequences of complex numbers, 2Sxf(x(s))=(\triangle +2 I)f(z),~~ Dxf(x(s))=(\triangle)/(\triangle x(s-1/2))f(z), z=x(s-1/2), I is the identity operator, x defines a lattice, and \triangle f(s)=f(s+1)-f(s). We show that under some natural conditions, the functionals \bf u and \bf v are connected by a rational factor whenever m=k, and for k>m, \bf u and \bf Sx k-m\bf v are semiclassical functionals and in addition \bf Sx\bf u and \bf Sx k-m+1\bf v are connected by a rational factor. This leads to the notion of (M,N)-coherent pair of measures of order (m,k) extended to orthogonal polynomials on lattices.