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Least integer nu such that the first zero of the Bessel j-function of index nu is at least nu + n.
1

%I #8 Feb 01 2023 23:04:45

%S 0,2,7,16,29,48,73,106,148,199,260,333,417,515,627,754,897,1057,1234,

%T 1431,1647,1884,2142,2423,2727,3056,3410,3791,4198,4634,5099,5594,

%U 6120,6678,7268,7893,8552,9247,9979,10748,11555,12402,13290

%N Least integer nu such that the first zero of the Bessel j-function of index nu is at least nu + n.

%C Tricomi proved that the first zero of j_nu occurs at nu + a*nu^(1/3) + b*nu^(-1/3) + O(1/nu). The PARI program below uses an estimate with a = 1.85575708087 and b = 1.

%D Francesco Tricomi, Sulle funzioni di Bellel di ordine e argomento pressochè uguali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 83:3-20 (1949).

%H Charles R Greathouse IV, <a href="/A360284/b360284.txt">Table of n, a(n) for n = 2..100</a>

%H Árpád Elbert and Andrea Laforgia, <a href="https://www.sciencedirect.com/science/article/pii/0022247X84902658">An asymptotic relation for the zeros of Bessel functions</a>, Journal of Mathematical Analysis and Applications, Volume 98, Issue 2 (February 1984), pp. 502-511.

%H Roger C. McCann, <a href="https://www.ams.org/journals/proc/1977-064-01/S0002-9939-1977-0442316-6/">Lower bounds for the zeros of Bessel functions</a>, Proc. Amer. Math. Soc. 64 (1977), pp. 101-103.

%F Tricomi (cited in Elbert & Laforgia and McCann) proved that a(n) ~ kn^3. It seems that k is approximately 0.15647199543.

%o (PARI) esta(n)=my(a=1.85575708087); ((n+sqrt(n^2-4*a))/2/a)^3

%o a(n)=if(n==2, return(0)); my(k=esta(n)\1,t=besseljzero(k)-k); if(t<n, while(besseljzero(k++)-k<n, ); k, while(besseljzero(k--)-k>=n, ); k+1)

%K nonn

%O 2,2

%A _Charles R Greathouse IV_, Feb 01 2023