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ALGEBROGEOMETRIC ASPECTS OF HEINESTIELTJES THEORY
, 2008
"... The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the co ..."
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Cited by 9 (2 self)
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The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild nondegeneracy assumptions there exist exactly n+r ´ such polynomials Vn,i(z) whose corresponding eigenpolynomials Sn,i(z)
ON HIGHER HEINESTIELTJES POLYNOMIALS
"... Abstract. Take a linear ordinary differential operator d(z) = P k i=1 Qi(z) di dz i with polynomial coefficients and set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical e ..."
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Cited by 3 (2 self)
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Abstract. Take a linear ordinary differential operator d(z) = P k i=1 Qi(z) di dz i with polynomial coefficients and set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [6] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has a polynomial solution S(z) of degree n. We have shown that under some mild nondegeneracy assumptions on T there exist exactly `n+r ´ spectral polyn nomials Vn,i(z) of degree r and their corresponding eigenpolynomials Sn,i(z) of degree n. Localization results of [6] provide the existence of abundance of converging as n → ∞ sequences of normalized spectral polynomials { e Vn,in (z)} where e Vn,in (z) is the monic polynomial proportional to Vn,in (z). Below we calculate for any such converging sequence { e Vn,in (z)} the asymptotic rootcounting measure of the corresponding family {Sn,in(z)} of eigenpolynomials. We also conjecture that the sequence of sets of all normalized spectral polynomials { e Vn,i(z)} having eigenpolynomials S(z) of degree n converges as n → ∞ to the standard measure in the space of monic polynomials of degree r which depends only on the leading coefficient Qk(z). 1.
ON SPECTRAL POLYNOMIALS OF THE HEUN EQUATION. II.
"... Abstract. The wellknown Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stie ..."
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Abstract. The wellknown Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in the late 19th century is for a given positive integer n to find all possible polynomials V (z) such that the above equation has a polynomial solution S(z) of degree n. Below we prove a conjecture of the second author, see [17] claiming that the union of the roots of such V (z)’s for a given n tends when n → ∞ to a certain compact connecting the three roots of Q(z) which is given by a condition that a certain
Polynomial Solutions of the Heun Equation
, 2011
"... We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present. ..."
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We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present.
AATLDE, Boris Shapiro, B2
"... Aspects of the asymptotic theory of linear ordinary differential equations ..."
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Aspects of the asymptotic theory of linear ordinary differential equations