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Monotonicity Properties of the Zeros of Ultraspherical Polynomials

✍ Scribed by Árpád Elbert; Panayiotis D. Siafarikas

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
111 KB
Volume
97
Category
Article
ISSN
0021-9045

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✦ Synopsis

Let x (*)

n, k , k=1, 2, ..., [nÂ2], denote the k th positive zero in increasing order of the ultraspherical polynomial P (*) n (x). We prove that the function [*+(2n 2 +1)Â (4n+2)] 1Â2 x (*) n, k increases as * increases for *> &1Â2. The proof is based on two integrals involved with the square of the ultraspherical polynomial P (*) n (x).

📜 SIMILAR VOLUMES

On a Conjecture Concerning Monotonicity
On a Conjecture Concerning Monotonicity of Zeros of Ultraspherical Polynomials
✍ Dimitar K. Dimitrov 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 235 KB

Let C \* n , n=0, 1, ..., \*>&1Â2 be the ultraspherical (Gegenbauer) polynomials, orthogonal on (&1, 1) with respect to the weight (1&x 2 ) \*&1Â2 . Denote by `n, k (\*), k=1, ..., [nÂ2] the positive zeros of C \* n enumerated in decreasing order. The problem of finding the ``extremal'' function f f

Zeros of Certain Sums of Two Polynomials
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✍ Seon-Hong Kim 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 103 KB

In this paper, we investigate the zero distribution of various sums of polynomials of the form A + B or A + tB 0 < t < ∞ or -∞ < t < ∞ , especially for A and B monic polynomials of the same degree. More precisely, we study generalizations and analogues of x -1 n + x + 1 n and their factorizations.