Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0639.33012
Gatteschi, Luigi
New inequalities for the zeros of Jacobi polynomials. (English)
[J] SIAM J. Math. Anal. 18, 1549-1562 (1987). ISSN 0036-1410; ISSN 1095-7154/e

The author makes ingenious use of the Sturm comparison theorem to provide upper and lower bounds for the zeros of the Jacobi polynomial $P\sb n\sp{(\alpha,\beta)}(\cos \theta)$, is case -$\le \alpha,\beta \le$. He shows that an asymptotic formula, involving zeros of Bessel functions, due to Frenzen and Wong, in fact provides a lower bound for these zeros (and also an upper bound, using $P\sb n\sp{(\alpha,\beta)}(x)=(-1)\sp n P\sb n\sp{(\beta,\alpha)}(-x)).$ He also shows that between any pair of zeros there occurs at least one root of a certain transcendental equation involving elementary functions. In the case of the kth zero, $\theta\sb{n,k}(\alpha)$, $k=1,2,...,[n/2]$, of the ultraspherical polynomial $P\sb n\sp{(\alpha,\alpha)}(\cos \theta)$, this leads to the inequalities $$ \phi\sb{n,k}(\alpha)\le \theta\sb{n,k}(\alpha)\le \phi\sb{n,k}(\alpha)+N\sp{-2}((1/8)-\alpha\sp 2/2)\cot \phi\sb{nk}(\alpha), $$ where $N=n+\alpha +$ and $\phi\sb{n,k}(\alpha)=(k+\alpha /2-1/4)\pi /N$. Comparisons are made with known bounds and numerical examples are given to illustrate the sharpness of the new inequalities.
[M.E.Muldoon]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
65D20 Computation of special functions

Keywords: approximations; Sturm comparison theorem; Jacobi polynomial; zeros; ultraspherical polynomial; inequalities

Cited in: Zbl 0805.33013 Zbl 0722.33001

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]