From: "G. A. Edgar" <edgar@math.ohio-state.edu>
Subject: Re: e and pi, and continued fractions
Date: Tue, 23 Jan 2001 11:31:59 -0500
Newsgroups: sci.math
To: WSelf@msubillings.edu
Summary: Closed forms for continued fractions with quotients in arithmetic progressions

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I wrote:
> 
> When the denominators form an arithmetic sequence, there is
> a closed form for the continued fraction.  Or even certain
> cases of two interleaved arithmetic sequences.  Try this one:
> 
> [1;2,3,4,5,6,7,...]

OK, you can stop asking.  I searched this newsgroup at deja.com
and found my previous post with the information...

  
> 
>  Yes, denominators in arithmetic progression is covered
>  by Lehmer [Scripta Math. 29 (1973) 17--24].  For any b and any
>  nonzero a, the continued fraction [b; b+a, b+2a, b+3a,...] has value 
>   
>                          I[b/a - 1](2/a)
>                          -------------------
>                            I[b/a](2/a)
>   
>  where the Bessel function  I[t](z) is 
>   
>                infinity
>                -----                (t + 2 m)
>                 \              (z/2)
>                  )    ---------------------------------
>                 /       GAMMA(m + 1) GAMMA(t + m + 1)
>                -----
>                m = 0
>   
>   
>   
>  The proof is based on the recursive formula for the convergents
>  of the continued fraction and the known recurrence for the Bessel
>  functions.  

Now you can argue about whether a quotient of Bessel functions
can be called "closed form".

An example with two interlaced arithmetic progressions:

  (1/2)(e+1)/(e-1) = [1, 12, 5, 28, 9, 44, 13, 60, 17, 76, ...]

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu
==============================================================================

From: "Iain Davidson" <Sttscitrans@tesco.net>
Subject: Re: e and pi, and continued fractions
Date: Mon, 29 Jan 2001 01:59:44 -0000
Newsgroups: sci.math

Will Self <wself@msubillings.edu> wrote in message
news:94n847$j5q$1@nnrp1.deja.com...
> In article <230120011131597497%edgar@math.ohio-state.edu>,
> Lehmer's result is quite interesting!  There are some things I'd like
> to understand better.  Let me report on what I learned yesterday.
> These Bessel functions are called Modified Bessel Functions of the
> First Kind, also called Hyperbolic Bessel Functions.  In Mathematica
> they are written BesselI[t,z].  For the fraction [1;3,5,7,9,...], we
> have b=1, a=2, so we get BesselI[-1/2,1]/BesselI[1/2,1].  Now the
> question remains, how can you convert this value to (E^2+1)/(E^2-1) ?
> I can get a value for BesselI[1/2,1] but not for the other.
>
> And there is also the fraction [1;2,3,4,...].  Here a=1, b=1, so the
> value is BesselI[0,2]/BesselI[1,2].  One can check this numerically.
> However, I cannot seem to make any progress trying to write this
> quotient as a simple expression involving E, yet I have a feeling that
> [1;2,3,4,...] ought to be expressible this way.
>
> If anybody gets a different expression for [1;2,3,4,...], I sure would
> like to see it.

You can derive the expression for [1;2,3,4,...] more directly from
the recurrence relation using generating functions.

a(0) = 1*a(1) + a(2)
a(1) = 2*a(2) + a(3)
.
.a(n-1) -na(n) - a(n+1) =0

Assume generating function
y = a(0) + SUM(over Z/0)  a(n)x^n

The a(n-1) and a(n+1) terms are obtained from (1 - x^(-2))y
and na(n) from the derivative of y

The differential equation
y' = y(1 - x^(-2))
can be solved to give

y = exp(x + 1/x)

I don't see how to extract expressions for a(0) and a(1)
in terms of exp from this.

The CF [1;8,27,64,125,.....]
can be tackled in the same way
but the differential eqn is more difficult to solve

y(1- 1/x^2) = y' + 3xy'' + x^2y'''