Hi.

I just managed to put together the hard proof of the explicit, non-recursive formula for the solution of general recurrences of the form

that I mentioned here:

http://math.eretrandre.org/tetrationforu...42#pid5542

The note with the proof is attached to this post. Comments, critique, etc. would be welcomed.

This proves the existence of the explicit non-recursive expression for the coefficients of the regular Schroder function of , and indeed, of the coefficients of regular Schroder functions in general.

So the question now becomes: is there a more "elegant" expression? I'll give some attempts at trying to take a whack at it here.

1. The regular Schroder function (at the natural fixed point 0) coefficients are given by the above recurrence formula, or the general explicit formula, with

.

(where is a Stirling number of the 2nd kind.)

Let the resulting coefficients "" be denoted . Then the Schroder function is

.

2. Now, we go and expand these out. The first few coefficients look like:

.

...

Factoring the denominators and pattern recognition suggests

where are the so-called "Schroder function magic numbers".

3. We want to turn this into a recurrence on the numerators. Let denote the numerator of the nth term . Then we have

.

Now insert the recurrent expression for :

.

But beyond here we run out of luck. The next step would be to set the sum for in this and try to solve for a recurrence for the and then try to come up with an explicit non-recursive formula for that, but it gets hairy and we don't have an expression for the expanded-out product there. So the first order of business here would be to find the explicit, non-recursive formula for the coefficients of

.

Any ideas? The degree of the resulting polynomial is .

Also, . We don't set it to 1, since if you look at the denominator formula, you'll notice it is equal to -1 there and we have .

I just managed to put together the hard proof of the explicit, non-recursive formula for the solution of general recurrences of the form

that I mentioned here:

http://math.eretrandre.org/tetrationforu...42#pid5542

The note with the proof is attached to this post. Comments, critique, etc. would be welcomed.

This proves the existence of the explicit non-recursive expression for the coefficients of the regular Schroder function of , and indeed, of the coefficients of regular Schroder functions in general.

So the question now becomes: is there a more "elegant" expression? I'll give some attempts at trying to take a whack at it here.

1. The regular Schroder function (at the natural fixed point 0) coefficients are given by the above recurrence formula, or the general explicit formula, with

.

(where is a Stirling number of the 2nd kind.)

Let the resulting coefficients "" be denoted . Then the Schroder function is

.

2. Now, we go and expand these out. The first few coefficients look like:

.

...

Factoring the denominators and pattern recognition suggests

where are the so-called "Schroder function magic numbers".

3. We want to turn this into a recurrence on the numerators. Let denote the numerator of the nth term . Then we have

.

Now insert the recurrent expression for :

.

But beyond here we run out of luck. The next step would be to set the sum for in this and try to solve for a recurrence for the and then try to come up with an explicit non-recursive formula for that, but it gets hairy and we don't have an expression for the expanded-out product there. So the first order of business here would be to find the explicit, non-recursive formula for the coefficients of

.

Any ideas? The degree of the resulting polynomial is .

Also, . We don't set it to 1, since if you look at the denominator formula, you'll notice it is equal to -1 there and we have .