Gabor Hetyei <ghetyei@uncc.edu> writes
While browsing the internet, I came accross your remarkable formula http://www.inwap.com/pdp10/hbaker/hakmem/cf.html#item99 expressing the value of a continued fraction with partial quotients increasing in arithmetic progression. I would be interested in the known proof of this formula. Is there any published paper or preprint on the issue that you could share with me or point me to? The reason of my interest is that I think it is possible to prove this formula using combinatorial enumeration, and I wonder if this has already been done or would be interesting to do.
I'd been interested in the value of the continued fraction 1+1/2+1/3+1/4+... since high school. In college, I came across the Dover reprint of the NIST tables by Abramowicz & Stegun, and spent a fair amount of time studying the special functions. I came across the Bessel-CF formula 9.1.73 for J, and recognized its relevance to my puzzle. Then I found the recurrence for I (first formula in section 9.6.26) and a litle manipulation does the rest. I was later able to find a direct proof using the power series for I, the one with a product of factorials in the denominator. I don't recall details, but it was pretty straightforward, just lining up terms to check the recurrence. You need to add something about convergence, I don't recall if I finished this step. I suspect "my" formula is actually well-known among continued fraction experts. I'm not a professional combinatorist, so I don't know if your work is new or not. I'd think that anything relating combinatorics to Bessel functions would count as interesting, since there's no obvious connection. (Although I think they turn up in the formula for partitions into unequal parts.) There are some people I can ask. I'm forwarding this reply to them. Rich Schroeppel rcs@cs.arizona.edu