2022/04/29 by Mbouna, D., Mañas-Mañas, Juan F., Moreno-Balcázar, Juan J.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2204.14098
We consider two sequences of orthogonal polynomials (Pn)n≥ 0 and (Qn)n≥ 0 such that ∑j=1 M aj,nSxDx 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, both involved orthogonal polynomials sequences (Pn)n≥ 0 and (Qn)n≥ 0 are semiclassical whenever k=m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.