While working on my previous mathematical post on fractions, I stumbled across the following nice identities:
We can go further and raise both x and (x+1) to powers:
This suggests a connection with binomial coefficients. We conjecture
To prove this, we use the technique mentioned in my previous fractions post. To find the we multiply both sides by and move terms with powers of x in the denominator to the other side:
where p(x) is a polynomial. Taking the limit of both sides as requires (m-i) applications of l’Hôpital’s rule on the right side (using the fact that has root x=0 with multiplicity m-i). We end up with
as was to be shown.