In a recent post I gave a forumla for the second diagonal of iterated-dxp (e^x-1) and this line of research has led to some interesting discoveries. I have since generalized the approach to "interpolating" the diagonals of these series, and I found generating functions for the first three diagonals of parabolic iteration. I would like to present my findings and see if there is anything like this already out there...

It all started with noticing that the first diagonal is:

where the function being iterated is of the form . This naturally lead to investigating the second diagonal, which I found is not too different than that of iterated-dxp:

but it involves a little more than just harmonic numbers. So I began looking at the third diagonal, and found some patterns, but I was only able to interpolate the coefficients up to a point, then I was stuck with a sequence of rational numbers I had no idea what to do with, then I eventually found A130894/A130895 which solved the problem I was having. Before I went to OEIS, I had found the coefficients of the third diagonal to be:

where A and D are constants (described later), and was the rational sequence [0, 1, 3/2, 71/36, 29/12, 638/225, 349/108, ...], which according to OEIS is equivalent to

where H is Conway and Guy's harmonic numbers (not the usual generalized harmonic numbers). Once I had the generating functions from OEIS, then I could being playing the game of generatingfunctionology. So I took this huge expression (D is "large" when written out) for the third diagonal, and played with derivatives and integrals until it was a recognizable function that generated the right coefficients. Maybe I'll post a more in-depth discussion of the techniques I used later on, but for now, I just want to show the results.

Going back to the first diagonal:

and according to OEIS the generating function of the 2nd degree harmonic numbers is , which means the second diagonal is:

where and using the new generating functions for , we find the generating function for the third diagonal is:

and finally, written out in full:

where .

The most fascinating part, though, is that t only appears in z.

Andrew Robbins

It all started with noticing that the first diagonal is:

where the function being iterated is of the form . This naturally lead to investigating the second diagonal, which I found is not too different than that of iterated-dxp:

but it involves a little more than just harmonic numbers. So I began looking at the third diagonal, and found some patterns, but I was only able to interpolate the coefficients up to a point, then I was stuck with a sequence of rational numbers I had no idea what to do with, then I eventually found A130894/A130895 which solved the problem I was having. Before I went to OEIS, I had found the coefficients of the third diagonal to be:

where A and D are constants (described later), and was the rational sequence [0, 1, 3/2, 71/36, 29/12, 638/225, 349/108, ...], which according to OEIS is equivalent to

where H is Conway and Guy's harmonic numbers (not the usual generalized harmonic numbers). Once I had the generating functions from OEIS, then I could being playing the game of generatingfunctionology. So I took this huge expression (D is "large" when written out) for the third diagonal, and played with derivatives and integrals until it was a recognizable function that generated the right coefficients. Maybe I'll post a more in-depth discussion of the techniques I used later on, but for now, I just want to show the results.

Going back to the first diagonal:

and according to OEIS the generating function of the 2nd degree harmonic numbers is , which means the second diagonal is:

where and using the new generating functions for , we find the generating function for the third diagonal is:

and finally, written out in full:

where .

The most fascinating part, though, is that t only appears in z.

Andrew Robbins