International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 551-560
doi:10.1155/S0161171295000706
Transcendentality of zeros of higher dereivatives of functions involving Bessel functions
Department of Mathematics and Statistics, York University, Ontario, North York M3J 1P3, Canada
Received 3 February 1994; Revised 2 September 1995
Copyright © 1995 Lee Lorch and Martin E. Muldoon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that
(i) all zeros of Jv(x) and J′v(x) are transcendental when v is rational, x≠0, and (ii)
J′v(x)/Jv(x) is transcendental when v is rational and x algebraic. As usual, Jv(x) is the
Bessel function of first kind and order v. Here it is shown that simple arguments permit one to
infer from Siegel's results analogous but not identical properties of the zeros of higher derivatives
of x−uJv(x) when μ is algebraic and v rational. In particular, J‴1(±3)=0 while all
other zeros of J‴1(x) and all zeros of J‴v(x), v2≠1, x≠0, are transcendental. Further,
J0(4)(±3)=0 while all other zeros of J0(4)(x), x≠0, and of Jv(4)(x), v≠0, x≠0, are
transcendental. All zeros of Jv(n)(x), x≠0, are transcendental, n=5,…,18, when v
is rational. For most values of n, the proofs used the symbolic computation package Maple V
(Release 1).