Here is the transformation pair I've been working with.
$\hat{f}(n)=\displaystyle\lim_{a\to1}\sum_{j=0}^{\lfloor\log_a n\rfloor}(-1)^j\binom{k}{j}a^j f( a^{-j} n)$
$f(n)=\displaystyle\lim_{a\to1}\sum_{j=0}^{\lfloor\log_a n\rfloor}\binom{k+j-1}{k-1}a^j \hat{f}( a^{-j} n)$
with $n$ a positive integer, $k$ some arbitrary complex argument, and $f(n)$ the original function, of course. This transform actually seems to work for any $a > 1$, but I'm specifically interested in what happens as $a$ approaches 1.
Can these sums be rewritten as more conventional integrals? Are they variants of some standard transform?