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

Abstract

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

Further on the Points of Inflection of Bessel Functions

“…The method in [2] includes the remaining cases of/^ as well. The method in [1] appeared to yield only partial results; the monotonicity of/'^, k = 2, 3,..., was established there only for 0 < v < 3838 [1, Theorem 10.1]. This was completed in [4] where the monotonicity of these remaining j" k was demonstrated for 10 < v < oo, using delicate asymptotic estimates.…”

Monotonicity of the zeros of the third derivative of Bessel functions

“…For j" k ,is > 0, monotonicity properties have been established only recently [6,7,8,11]. These state that j" k increases with is in 0 < is < 00, for each fixed k = 1, 2, In addition, it has been shown [5] that j" k increases in -1 < is < 00 when j" k > joi = 2.4048 ... and, further, that when 1/2 (= -0.199 ...) < is < 0, there exist precisely two zeros of J"(x) for which j^ < j" 2 < j v i.…”

The zeros of as functions of order

“…It is well-known that Jvk and 3vk (with k fixed in 1,2,-..) are increasing functions of v in v > 0 [11. Recently, in [2], it has been shown that Jl increases in v > 0 and that (with k fixed in 2,3,...) "' 3v/ increases in 0 < v < 3838. Also, in [3], Wong and Lang have extended these results to conclude that (with k fixed in 1,2,-..) Jk increases in v > 0.…”

“…In another recent paper [4] it has been proved that the k th positive zero of pJv(x)+ xJ'v(z) [2] other expression for w given nely, (j") so, comparing these, we conclude that j2(j,,) + Ju-l(J") Ju + l(J") f'Ju(J") J,'(J") Now it was also proved in [2] that if , > 0 (k 2,3,...) or if 0 < v _< 1 (k 1) then the right hand side here is negative. Then, so too, must be the left hand side in these cases.…”

“…Also, in [3], Wong and Lang have extended these results to conclude that (with k fixed in 1,2,-..) Jk increases in v > 0. Each of the papers [2] and [3] contains some very delicate analysis.…”