*(English)*Zbl 0639.33012

The author makes ingenious use of the Sturm comparison theorem to provide upper and lower bounds for the zeros of the Jacobi polynomial ${P}_{n}^{(\alpha ,\beta )}(cos\theta )$, 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 ${P}_{n}^{(\alpha ,\beta )}\left(x\right)={(-1)}^{n}{P}_{n}^{(\beta ,\alpha )}(-x))\xb7$ 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, ${\theta}_{n,k}\left(\alpha \right)$, $k=1,2,\xb7\xb7\xb7,[n/2]$, of the ultraspherical polynomial ${P}_{n}^{(\alpha ,\alpha )}(cos\theta )$, this leads to the inequalities

where $N=n+\alpha +$ and ${\varphi}_{n,k}\left(\alpha \right)=(k+\alpha /2-1/4)\pi /N$. Comparisons are made with known bounds and numerical examples are given to illustrate the sharpness of the new inequalities.

##### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type |

34C10 | Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory |

65D20 | Computation of special functions, construction of tables |