On Bochner-Krall orthogonal polynomial systems

Abstract

In this paper we address the classical question going back to S. Bochner and H. L. Krall to describe all systems {p n (x)} ∞ n=0 of orthogonal polynomials (OPS) which are the eigenfunctions of some finite order differential operator. SummaryConsider a sequence of polynomials {p n } ∞ n=0 in a variable x, where deg p n = n. This sequence is orthogonal with respect to a measure µ if p n (x)p m (x) dµ(x) is nonzero precisely when n = m. We are here concerned with polynomials orthogonal with respect to a measure of the so-called Nevai class, see [11] and below. Furthermore we say that {p n } is a sequence of eigenpolynomials if there exists a differential operator ᒁ = N k=1 a k (x) (d/dx) k where a k are polynomials in x. Finally, {p n } is called a Bochner-Krall system if it is both orthogonal and a system of eigenpolynomials. If it is orthogonal with respect to a measure of Nevai class, we say that it is a Bochner-Krall system of Nevai type.It is an open problem to classify all Bochner-Krall systems. In [4] it is conjectured that the leading coefficient a N for any Bochner-Krall system is a power of a polynomial of degree at most 2. Our main result is an affirmative answer to this conjecture for Bochner-Krall systems of Nevai type.

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

“…Applying Theorem 1, we see that the last sum on the right-hand side of inequality (4) in Lemma 3 tends to zero when n → ∞. Now consider the double sum on the right-hand side of (4).…”

“…In the pictures below large dots represent the zeros of Q 5 and small dots represent the zeros of the eigenpolynomials p 50 , p 75 and p 100 , respectively. As a consequence of the above results we were able to prove a special case of a general conjecture describing the leading terms of all Bochner-Krall operators; see [4].…”

“…But for any such we can form = +b 2 for which < b and < , and thus lim n→∞ r n n = ∞ for all < b. 4 On the left-hand side in the expression for b above we take the minimum over i ∈ [0, deg Q j ], so we can put…”

“…j 0 ,j 0 = 1. Table 1 Numerical evidence of the growth of r n = max{|z|:p n (z) = 0} Operator n r n Experimental r n Conjectured 50 2.7 · 50 0.967 595 c 1 · 50 1 T 1 = zD + zD 2 + zD 3 + zD 4 such that (j − j 0 )/(j − deg Q j ) < d is (conjecturally) irrelevant for the zero distribution when n → ∞.…”

“…and where the notation min + means that we only take the minimum over positive terms 4 Then every term in the double sum tends to zero when n → ∞, and we obtain a contradiction to (4) when n → ∞. Thus, for all sufficiently large integers n we must have r n > c 0 (n − k + 1) for all < b, and hence lim inf n→∞ r n n > c 0 for any < b.…”