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Let $m$ and $n$ are positive integers, then find the sum of the infinite series defined as $$\sum_{k=0}^\infty \frac{(-1)^k\Gamma(m+k) }{\Gamma(m)k!(k+m)^n}.$$ I was managed to the sum with $m=2$ and $...

The harmonic numbers are defined by $$ H_n=\sum_{j=1}^n\frac{1}{j} $$ I have come across the following sum: $$ g(z)=\sum_{n=1}^\infty z^{H_n} $$ Clearly it converges only for $z<1/e$. Is it a ...