BOUNDS AND MONOTONICITIES FOR THE ZEROS OF DERIVATIVES OF ULTRASPHERICAL BESSEL-FUNCTIONS

The positive zeros p(nuk)(l) of [x(-nu+1) J(nu+l-1)(x)]', nu + l > 0, where J(nu)(x) denotes the standard Bessel function, arise in the study of the eigenvalues of Neumann Laplacians in N dimensions [M. S. Ashbaugh and R. D. Benguria, SIAM J. Math. Anal., 24 (1993), pp. 557-570]. The case l = 1 is particularly relevant. To pave the way for these applications, the authors present here inter alia (i) lower and upper bounds for p(nu1)(l) and (ii) an explicit representation for dp(nuk)(l)/dnu. The latter implies that p(nuk)(l) is increasing in nu for fixed k,l, provided nu + l > 1.