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Monotonicity of zeros of Jacobi polynomials

✍ Scribed by Dimitar K. Dimitrov; Fernando R. Rafaeli


Book ID
111713133
Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
205 KB
Volume
149
Category
Article
ISSN
0021-9045

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Denote by x nk (Ξ±), k = 1, . . . , n, the zeros of the Laguerre polynomial L (Ξ±) n (x). We establish monotonicity with respect to the parameter Ξ± of certain functions involving x nk (Ξ±). As a consequence we obtain sharp upper bounds for the largest zero of L (Ξ±) n (x).

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