By applying the Euler–Rayleigh method to a specific representation of Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, derive upper and lower bound $$1-x_{nn}^2(\lambda )$$ , with $$x_{nn}(\lambda being zero n-th ultraspherical polynomial $$P_n^{(\lambda )}$$ . For every fixed $$\lambda >-1/2$$ limit ratio our does not exceed 1.6. This...

We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular a $k$-dimensional sphere for some $k < d$. For these we prove spectral multiplier theorem Mihlin-H\"ormander type, is optimal whenever $2k \leq d$, and corresponding Bochner-Riesz summability result. The proof hinges suitable weighted cluster bounds, in turn depend precise estimates ultrasph...

The paper deals with an approximate analysis of non-linear oscillation problems with slowly varying system parameters. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by using the generalized averaging method of Sinha and Srinivasan based on ultraspherical polynomial expansions. The Bogoliubov-Mitro...