Here is a lovely generalized continued fraction!

While working with Stephen Glasby on some problems (see Some nice combinatorics for your viewing pleasure), I found a different way to write a partial sum of binomial coefficients. (Here’s my chance to use LaTeX in a post.) The problem involved the following function of r and m:

Stephen and I were optimizing certain functions involving this term. After checking some examples, I came up with some terms for a_i and b_i and conjectured the following relation:

The notation with the curly K stands for a continued fraction. I think it looks better than b_1/[a_1 + b_2/[a_2 + b_3/[a_3 + … + b_r/a_r] …] . I found some nice things to put in for a_i (m -2r+3i) and b_i (2i(r+1-i)).

It took about three weeks and over thirty sheets of paper (both sides) before I found a proof. This may be in the literature, but it was new to me and Stephen. A similar statement,

almost follows from the proof and gives a representative form which works for many values of r. I hope to use this in other problems involving partial sums of binomial coefficients.

Update 2022.03.09: Work on the earlier problem got accepted for publication in the Electronic Journal of Combinatorics. (Yay!) I’ve learned a little more about generalized continued fractions. However, the easy route (Euler’s continued fraction theorem) seems not to lead to the representation here. Perhaps it really is new?