An inverse problem associated with (1,1) symmetric coherent linear functionals

In this paper we study an inverse problem associated with the non-coherent algebraic relation Pn+1[i](x)+an[1]Pn[i](x)+an[2]Pn−1[i](x)+bnQn+1(x)+cnQn(x)=1+bnRn+1(x)+dnRn(x),n≥0," role="presentation">P[i]n+1(x)+a[1]nP[i]n(x)+a[2]nP[i]n−1(x)+bn(Qn+1(x)+cnQn(x))=(1+bn)Rn+1(x)+dnRn(x),n≥0, where the sequences of monic polynomials Pn(x)n≥0," role="presentation">{Pn(x)}n≥0, Qn(x)n≥0" role="presentation">{Qn(x)}n≥0 and Rn(x)n≥0" role="presentation">{Rn(x)}n≥0 are orthogonal with respect to quasi-definite linear functionals u, v and w, respectively, and Pk[i](x)=Pk+i(i)(x)(k+1)i" role="presentation">P[i]k(x)=P(i)k+i(x)(k+1)i for i=0,1." role="presentation">i=0,1. We assume that v is a monic polynomial perturbation of u. Moreover, in the case i=1," role="presentation">i=1, we assume that u is a semiclassical linear functional of class s. In this way, we discuss the relation between the formal Stieltjes series associated with u and w. This inverse problem is motivated by the analysis of sequences of polynomials orthogonal with respect to (1,1)" role="presentation">(1,1) symmetric coherent pairs of linear functionals when a symmetrization process is implemented.

An inverse problem associated with (1,1) symmetric coherent linear functionals Articles