On the polynomials of Stieltjes

Vleck, Edward B. Van

doi:10.1090/s0002-9904-1898-00531-1

Cited by 29 publications

(24 citation statements)

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“…Similarly, for fixed k, the Van Vleck zeros are distinct and also lie within (α 1 , α 3 ). The proofs of these results can be found in [24,23,18,19].…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

On the distribution and interlacing of the zeros of Stieltjes polynomials

Bourget

McMillen

2010

Proc. Amer. Math. Soc.

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Abstract. Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of certain Stieltjes polynomials of successive degrees interlace.

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“…Similarly, for fixed k, the Van Vleck zeros are distinct and also lie within (α 1 , α 3 ). The proofs of these results can be found in [24,23,18,19].…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

On the distribution and interlacing of the zeros of Stieltjes polynomials

Bourget

McMillen

2010

Proc. Amer. Math. Soc.

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“…The polynomial C(x) is called a Van Vleck polynomial and the corresponding polynomial solution y(x) of (1.1) is called a Stieltjes polynomial. Van Vleck [18] was the first to prove that the zeros of C(x) lie in (a 0 , a p ). Klein [7], Bôcher [2] and Pólya [10] proved that the zeros of Stieltjes polynomials lie in the convex hull of a 0 , .…”

Section: Introductionmentioning

confidence: 99%

Lamé differential equations and electrostatics

Dimitrov

Assche

2000

Proc. Amer. Math. Soc.

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Abstract. The problem of existence and uniqueness of polynomial solutions of the Lamé differential equationwhere A(x), B(x) and C(x) are polynomials of degree p + 1, p and p − 1, is under discussion. We concentrate on the case when A(x) has only real zeros a j and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients r j in the partial fraction decomposition B(x)/A(x) = p j=0 r j /(x − a j ), we allow the presence of both positive and negative coefficients r j . The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges r j at a j is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.

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“…The above corollary expresses, for y = 0, the results stated in Theorems 1(a) and 1(b) in Marden [6], due, respectively, to Stieltjes [8] and Van Vleck [9].…”

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confidence: 70%

“…For the first time, Marden [6] gave the treatment of (1.1) subject to condition |arga | < y < tt/2 and obtained varied generalizations of the results (cf. Marden [6, Theorems l(a)-2(a)]) established earlier by Stieltjes [8], Van Vleck [9], Bôcher [1], Klein [3], and Pólya [7]. Our object in this paper is to study the zeros of the polynomials S(z) and V(z) in relation to the differential equation (1.2).…”

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confidence: 98%