Transcendentality of zeros of higher dereivatives of functions involving Bessel functions

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).

Keywords

• Bessel functions
• zeros
• transcendentality
• differential equations.