A monotonicity property of Bessel functions

We extend and unify the proof of a result of L. Lorch (Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), 209–216) by showing that the ratio sgn(ν)Jν+1(|ν|)/Jν(|ν|) increases from −∞ to 1 as ν increases from ν0 (= −0.8375 . . .) to ∞. Here ν0 is the largest zero of Jν(|ν|). One approach is based on an expansion of this ratio involving Rayleigh sums. Another is based on a continued fraction representation. In fact, this approach shows also that the graph of sgn(ν)Jν+1(|ν|)/Jν(|ν|) consists of increasing branches between the zeros of Jν(|ν|). AMS (1991) Classification: 33A40