Riccati equations and convolution formulae for functions of Rayleigh type

Abstract

Kishore (1963 Proc. Am. Math. Soc. 14 527) considered the Rayleigh functions n ( ) = k = 1 j k -2n ,n = 1,2, ... , where ±j k are the (non-zero) zeros of the Bessel function J (z ) and provided a convolution-type sum formula for finding n in terms of 1 , ... , n -1 . His main tool was the recurrence relation for Bessel functions. Here we extend this result to a larger class of functions by using Riccati differential equations. We get new results for the zeros of certain combinations of Bessel functions and their first and second derivatives as well as recovering some results of Buchholz for zeros of confluent hypergeometric functions.