Abstract :
We discuss an inverse problem in the theory of (standard) orthogonal polynomials
involving two orthogonal polynomial families (Pn)n and (Qn)n whose derivatives of higher
orders m and k (resp.) are connected by a linear algebraic structure relation such as
N
i=0
ri,n P(m)
n−i+m(x) =
M
i=0
si,n Q (k)
n−i+k(x)
for all n = 0, 1, 2, . . ., where M and N are fixed nonnegative integer numbers, and ri,n and
si,n are given complex parameters satisfying some natural conditions. Let u and v be the
moment regular functionals associated with (Pn)n and (Qn)n (resp.). Assuming 0 m k,
we prove the existence of four polynomials ΦM+m+i and ΨN+k+i , of degrees M +m+i and
N +k + i (resp.), such that
Dk−m(ΦM+m+iu) = ΨN+k+iv (i = 0, 1),
the (k −m)th-derivative, as well as the left-product of a functional by a polynomial, being
defined in the usual sense of the theory of distributions. If k = m, then u and v are
connected by a rational modification. If k = m + 1, then both u and v are semiclassical
linear functionals, which are also connected by a rational modification. When k > m,
the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary
differential equation of order k −m with polynomial coefficients.
