Location of the zeros of polynomials satisfying three-term recurrence relations. I. General case with complex coefficients

Gilewicz, Jacek; Leopold, Elie

doi:10.1016/0021-9045(85)90142-x

Cited by 14 publications

(8 citation statements)

References 5 publications

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“…It can be noted that (1.1) is equivalent to the three-term recurrence relation of Frobenius-type [23] studied by Delsarte and Genin [16][17][18]. The weakened form of orthogonality, known as para-orthogonality after [29], that the polynomials {s n } n≥0 satisfy with respect to the orthogonality measure of the associated OPUC was also pointed out in [16].…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

Castillo

2015

Applied Mathematics and Computation

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Section: Introduction and Main Resultsmentioning

confidence: 91%

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

Castillo

2015

Applied Mathematics and Computation

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“…Location and properties of zeros of special families of polynomials are the subject of many works (see for instance [21,23,28,36,41]). More attention is devoted to orthogonal polynomials [11,38,42] and their extensions [17,32,33].…”

Section: Introductionmentioning

confidence: 99%

On the zeros of d-symmetric d-orthogonal polynomials

Romdhane¹

2008

Journal of Mathematical Analysis and Applications

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In this paper, we give some properties of the zeros of d-symmetric d-orthogonal polynomials and we localize these zeros on (d + 1) rays emanating from the origin. We apply the obtained results to some known polynomials. In particular, we partially solve the conjecture about the zeros of the Humbert polynomials stated by Milovanović and Dordević [G.V. Milovanović, G.B. Dordević, On some properties of Humbert's polynomials, II, Ser. Math. Inform. 6 (1991) 23-30]. A study of the eigenvalues of a particular banded Hessenberg matrix is done.

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“…Moreover, the right plot of Figure 4 suggests that for gcd(p, q) = 1, the distribution |Q t n ψ 0 | 2 has q peaks. This observation can be resolved by noting that letting τ = 4p/πyq leads to an extra factor of exp 2πik 2 p/q in equations (14) and (17). This causes the eigenvalues to align on a finite set of lines in the complex plane.…”

Section: Periodic Modal Mixingmentioning

confidence: 99%

Conditional probability distributions of finite absorbing quantum walks

Kuklinski

2020

Phys. Rev. A

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Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q n where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for t = O(n), periodic modal mixing for t = O(n 2 ), and stability for t = O(n 3 ). At the end of this paper, we demonstrate the existence of periodic modal mixing in other sufficiently regular quantum systems. arXiv:1909.12680v1 [quant-ph]

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Location of the zeros of polynomials satisfying three-term recurrence relations. I. General case with complex coefficients

Cited by 14 publications

References 5 publications

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

On the zeros of d-symmetric d-orthogonal polynomials

Conditional probability distributions of finite absorbing quantum walks

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