Ratio of two Bessel function values: true for complex D?; a BesselJ equality

[Gerald McGarvey]

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