## 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 s
Some monotonicity properties of the zeros of ultraspherical polynomials
✍ Scribed by Á. Elbert; A. Laforgia
- Publisher
- Akadmiai Kiad
- Year
- 1986
- Tongue
- English
- Weight
- 199 KB
- Volume
- 48
- Category
- Article
- ISSN
- 1588-2632
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📜 SIMILAR VOLUMES
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
Denote by x nk (α), k = 1, . . . , n, the zeros of the Laguerre polynomial L (α) n (x). We establish monotonicity with respect to the parameter α of certain functions involving x nk (α). As a consequence we obtain sharp upper bounds for the largest zero of L (α) n (x).