A differential equation for the zeros of bessel functions

Abstract

Let ρ(v) be a positive zero of the ordinary Bessel function Jv(z) of order v. It is shown that for every v in the interval -1>v>∞ the function ρ(v) satisfies the differential equation ρ(v)=ρ(vv)(L_vX(_v)) where L _v is the diagonal operator L_Ve_n=1/n+e_n on an abstract Hilbert space H with the oYtRonormal basis e_n=1,2,… and x(v) is a normalized element in H. A basic result which follows easily from this equation is that the differential inequality ρ′(V)>1 halds for every v in the interval -1<v<+∞. This inequality proves that the function ρ(v)-v is a strictly increasing function in the interval -1<v<∞ and unifies a number of simple lower and upper bounds for the positive zeros of J _v (z). Also from the above equation it follows easily that the functicn (l+v)^-1 .ρ(v) is a strictly increasing function in the interval --1<v<∞ for every positive zero of J _v (z). This generalizes a previous result.

AMS(OMS):

• 33A40
• 47B05
• 47A55