On Stieltjes polynomials

Marden, Morris

doi:10.1090/s0002-9947-1931-1501624-1

Cited by 18 publications

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“…, a p if a j are complex provided the coefficients r j are positive. Marden [8] proved the same result under a weaker condition. He allowed the coefficients r j to be complex numbers with positive real parts.…”

Section: Introductionmentioning

confidence: 64%

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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Section: Introductionmentioning

confidence: 64%

Lamé differential equations and electrostatics

Dimitrov

Assche

2000

Proc. Amer. Math. Soc.

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“…Precisely, we have: If P and Q are one-variable polynomials (with the identification between ¿ZoaJzJ and ^la;Z7Vw->), the polynomial adQ/dz + a'dQ/dz' is the homogeneous version of the polynomial mQ(z) + (a-z)Q'(z), which is called the polar derivative of Q at the point a (cf. Marden [9]) and denoted by Q\ (a, z). So, for one-variable polynomials, the above formula becomes…”

Section: Differential Identitiesmentioning

confidence: 99%

“…But this fact will become obvious once we establish Proposition 7. The form L coincides with the multilinear form generated by the hypercube; that is, (9) L,(Z<«,. :vrzW)-E ciu,.,imzlil)-z™.…”

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confidence: 99%

Differential identities

Beauzamy¹,

Dégot²

1995

Trans. Amer. Math. Soc.

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Abstract.We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements.The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.

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“…it is the intersection of all confocal ellipses having common foci at cx and c2), we obtain from Corollary (2.3) the following result due to Zaheer [12 We note that the above corollary for q = 1 is essentially a result due to Marden [6, Theorem 6b]. We also note that the part of our Theorem (2.1) that concerns Stieltjes polynomials may be regarded as a generalization of Lucas' theorem [5,Theorems (6,1), (6, 1)', (6, 2)] or [4] to systems of partial fraction sums.…”

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confidence: 56%

Zeros of Stieltjes and Van Vleck polynomials

Alam¹

1979

Trans. Amer. Math. Soc.

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Abstract. The study of the polynomial solutions of the generalized Lamé differential equation gives rise to Stieltjes and Van Vleck polynomials. Marden has, under quite general conditions, established varied generalizations of the results proved earlier by Stieltjes, Van Vleck, Bôcher, Klein, and, Pólya, concerning the location of the zeros of such polynomials. We study the corresponding problem for yet another form of the generalized Lamé differential equation and generalize some recent results due to Zaheer and to Alam. Furthermore, applications of our results to the standard form of this differential equation immediately furnish the corresponding theorems of Marden. Consequently, our main theorem of this paper may be considered as the most general result obtained thus far in this direction.

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On Stieltjes polynomials

Abstract: t A generalization of the Lamé equation (a, = 1/2, ally) and of the hypergeometric equation (p = 3).

Cited by 18 publications

References 0 publications

Lamé differential equations and electrostatics

Lamé differential equations and electrostatics

Differential identities

Zeros of Stieltjes and Van Vleck polynomials

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