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A175838 Let rho(n) be the first positive root of Bessel function J_n(x). This sequence is decimal expansion of derivative rho'(0)=1.54288974...

%I A175838 #21 Jan 23 2025 22:11:21
%S A175838 1,5,4,2,8,8,9,7,4,2,5,9,9,3,1,3,6,8,8,0,7,0,3,2,1,4,2,1,4,7,1,4,3,5,
%T A175838 5,6,1,6,9,8,4,6,0,7,8,7,3,5,0,1,9,7,5,8,9,3,5,2,5,2,9,4,4,1,0,2,6,8,
%U A175838 2,5,6,4,6,9,7,2,9,1,1,2,6,0,5,0,2,3,8,2,7,4,6,7,3,8,1,0,4,7,5,6,6,1,5,4,6
%N A175838 Let rho(n) be the first positive root of Bessel function J_n(x). This sequence is decimal expansion of derivative rho'(0)=1.54288974...
%H A175838 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%p A175838 From _R. J. Mathar_, Sep 22 2010: (Start)
%p A175838 Digits := 120 : Jnudnu := proc(nu,z,kmax) -add( (-1)^k*Psi(nu+k+1)/GAMMA(nu+k+1)*(z/2)^(2*k+nu)/k! , k=0..kmax) ; evalf(%) ; end proc:
%p A175838 Jprime := diff(BesselJ(0,x),x) ; z := evalf(BesselJZeros(0,1)) ; denomin := subs(x=z,Jprime) ;
%p A175838 for kmax from 30 to 70 by 10 do numerat := Jnudnu(0,z,kmax) ; c := evalf(-numerat/denomin) ; print(c) ; end do: # Abramowitz-Stegun 9.1.64
%p A175838 (End)
%t A175838 N[(Pi BesselY[0,BesselJZero[0,1]])/(2 BesselJ[1,BesselJZero[0,1]]),200]
%o A175838 (PARI) besseljzero'(0) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A175838 Cf. A115368. - _R. J. Mathar_, Sep 22 2010
%K A175838 cons,nonn
%O A175838 1,2
%A A175838 _Vladimir Reshetnikov_, Sep 19 2010