Inflection Points of Bessel Functions of Negative Order

Abstract

We consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

The zeros of the third derivative of Bessel functions of order less than one

“…Theorems 4.1 and 4.2 verify the case n = 3. For n = 2, the conjecture is verified in [6]. It holds also for n = 1 [5].…”

The real zeros of the derivatives of cylinder functions of negative order

“…Results analogous to what we have found for the special case c νk = j ν1 , were found for j (2) νk in Theorems 1 and 2 of [13], and for j (3) νk in [12]. Together, these three sets of properties suggest the following conjecture: Conjecture 6.1.…”

On the Points of Inflection of Bessel Functions of Positive Order, I

“…This result has been achieved by R. Wong and T. Lang [12] who applied to (3.1) delicate asymptotic estimates with numerical estimates of the remainder terms to establish monotonicity fori/ > 10 when/: > 2. For -1 < v < 0 (and hence in combination with the present paper and [12] for -1 < v < oo) monotonicity for f vk ,k = 3,4..., have also been demonstrated, but by different methods [6]. These cover analogous properties for k = 1,2, as well.…”