The Real Zeros of the Derivatives of Cylinder Functions of Negative Order

We study the real x–zeros of the derivative C ν (x) of the cylinder function Cν(x) = Cν(x, α) = cosαJν(x) − sinαYν(x) for ν < 0. For fixed α, 0 ≤ α < π and n = 1, 2, . . ., we show the existence of a number νn in the interval −n+ α/π − 1 < ν < −n+ α/π such that the first positive zero of C ν (x) occurs after the first positive zero of Cν(x) when −n− 1 + α/π < ν < νn and before the first positive zero of Cν(x) when νn ≤ ν < −n + α/π. In case νn < ν < −n − α/π, C ν(x) has two zeros which precede the first positive zero of Cν(x, α). For ν = νn, C ν(−ν) is a double zero. The first is decreasing and the second is increasing as ν increases. Moreover, in the case α = 0, for ν1 = −1.117 . . . ≤ ν < −1, J ′ ν(x) has exclusively real zeros. The present results and earlier ones on the zeros of J ′′ ν (x), J ′′′ ν (x) lead to some conjectures on the zeros of J (n)