[math-fun] Re: Ratio of two Bessel function values
Many thanks! R. On Tue, 4 Oct 2005, Eric W. Weisstein wrote:
On Tue, 4 Oct 2005, Richard Guy wrote:
I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated?
Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions?
There are a few examples which are rational functions of e. R.
Yes; see http://mathworld.wolfram.com/ContinuedFractionConstant.html
Cheers, -Eric
Regarding the following equality shown in http://mathworld.wolfram.com/ContinuedFractionConstant.html [A + D, A + 2D, A + 3D,...] = I_A/D(2/D) ---------- I_1+A/D(2/D) for real A and D not = 0 Has this been proven for complex D not = 0? Based on a few calculations, it seems likely to be true. In PARI/GP, for h(A,D)=besseli(A/D,2/D)/besseli(1+A/D,2/D) some values are h(0,1+I) = 1.267934387137125272910 + 0.7730503905741197348306*I h(0,-2-2*I) = -2.127520168632410874081 - 1.877682745590767507315*I h(1,1+I) = 2.233277826497280740363 + 0.8608766633022917990395*I h(1,-1-I) = -0.2588902240312738217853 - 0.6069437181652639961115*I h(2,-2+3*I) = -0.05227475025018962554575 + 2.848780558976335541278*I h(3,-Pi+exp(1)*I) = -0.2253429231897860436837 + 2.583985060897120442589*I h(3,tan(1)-44.44*I) = 4.558178321956313652916 - 44.42880095256962018668*I The following PARI/GP code (varying the A and D) suggests that the continued fraction values are the same; the computed differences are extremely small: A = 3; D = tan(1)-44.44*I; n = 100; L = listcreate(n); listput(L,A + D,1); for(i=1,n,x = A + i*D;listput(L,x,i));vcf = 1; forstep(i=n,1,-1,vcf=L[i] + 1/vcf); 1.0*vcf-h(A,D) (any suggestions on improvements to the code are welcome) ----------------- This somewhat similar looking equality appears to be true: BesselJ(2,2/z)/BesselJ(1,2/z) = [0, 2*z-1, 1, 3*z-2, 1, 4*z-2, 1, 5*z-2, 1, ...] for z not = 0 (this can likely be generalized in some similar manner). Has this been proven? Some partial PARI/GP code for this: g(z) = besselj(2,2/z)/besselj(1,2/z); cg(z) = contfrac(g(z)) cg(4) = [0, 7, 1, 10, 1, 14, 1, 18, 1, 22, 1, 26, 1, 30, 1, 34, 1, 38, 1, 43] I only checked this possible equality for I in a spreadsheet; it looks like it's true for I. Regards, Gerald At 06:19 PM 10/4/2005, Richard Guy wrote:
Many thanks! R.
On Tue, 4 Oct 2005, Eric W. Weisstein wrote:
On Tue, 4 Oct 2005, Richard Guy wrote:
I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated? Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions? There are a few examples which are rational functions of e. R.
Yes; see http://mathworld.wolfram.com/ContinuedFractionConstant.html
Cheers, -Eric
--- Gerald McGarvey <Gerald.McGarvey@comcast.net> wrote:
Regarding the following equality shown in http://mathworld.wolfram.com/ContinuedFractionConstant.html
[A + D, A + 2D, A + 3D,...] =
I_A/D(2/D) ---------- I_1+A/D(2/D)
for real A and D not = 0
Has this been proven for complex D not = 0? Based on a few calculations, it seems likely to be true. ...
The continued fraction [A, A, A, ...] converges for all complex A except the open interval of the imaginary axis {iy | -2 < y < 2}. On that basis, I would expect the continued fraction formula to hold, and be a meromorphic function of A and D, when D is nonzero or A is outside the divergence interval. Gene ______________________________________________________ Yahoo! for Good Donate to the Hurricane Katrina relief effort. http://store.yahoo.com/redcross-donate3/
participants (3)
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Eugene Salamin -
Gerald McGarvey -
Richard Guy