A generalized formula of Hardy (Q1326628)

The main result of the paper is the following identity, which generalizes a formula of Hardy. For integer \(m \geq 0\), \(\text{Re} x>0\) and \(\text{Re} a>1\), \[ \begin{multlined} \sum^ \infty_{k=0} k^ me^{-a^ kx}+\sum^ \infty_{k=1} (-k)^ m (e^{-a^{-k}x}-1) \\ ={1 \over (m+1) (\log a)^{m+1}} \sum^{m+1}_{j=0} {m+1 \choose j} \Gamma^{(m+1-j)} (1) \left( \log {1 \over x} \right)^ j-{B_{m+1} \over m+1} \\ -{1 \over (\log a)^{m+1}} \sum' \left( \log {t \over x} \right)^ m \left( {t \over x} \right)^{{2ki \pi \over \log a}} {dt \over te^ t} \end{multlined} \] where \(\Gamma^{(j)}\) is the \(j\)-th derivative of the gamma function, \(B_ m\) is the \(m\)-th Bernoulli number, \(\sum'\) is the sum over nonzero integers, and \(0^ 0=1\). The formula has significance to the partition of numbers into powers of \(a\), and also to mathematical physics.