Calculate $\sum_{n=1}^\infty\frac{n^x}{n!}$ I want to evaluate function defined by following sum: $$\sum_{n=1}^\infty\frac{n^x}{n!}$$ I was thinking about writing Taylor series expansion for it. However my try resulted in sum that looks even harder to calculate:
$$\sum_{n=1}^{\infty}\frac{\ln^k(n)}{n!}$$Thanks for all the help in solving this problem.

We can rewrite the sum as
$$
\eqalign{
  & F(x) = \sum\limits_{1\, \le \,n} {{{n^{\,x} } \over {n!}}}  = \sum\limits_{0\, \le \,n} {{{\left( {n + 1} \right)^{\,x} } \over {\left( {n + 1} \right)!}}}  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,j} {\left( \matrix{
  x \cr 
  j \cr}  \right){{n^{\,j} } \over {\left( {n + 1} \right)!}}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,j} {{{x^{\,\underline {\,j\,} } } \over {j!}}{{n^{\,j} } \over {\left( {n + 1} \right)!}}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,j} {\left( {\sum\limits_{0\, \le \,n} {{{n^{\,j} } \over {\left( {n + 1} \right)!}}} } \right)} {{x^{\,\underline {\,j\,} } } \over {j!}} \cr} 
$$
which gives the expansion of $F(x)$ as a Newton series, the inner sum being "near-Bell" numbers
If we replace the falling Factorial with
$$
x^{\,\underline {\,j\,} }  = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,j} \right)} {\left( { - 1} \right)^{\,j - k} \left[ \matrix{
  j \cr 
  k \cr}  \right]x^{\,k} } 
$$
we unfortunately get an alternating diverging series for the coefficients of $x^k$.
However the actual Bell numbers intevene into
$$
\eqalign{
  & F(x + 1) = \sum\limits_{0\, \le \,n} {{{\left( {n + 1} \right)^{\,x} } \over {n!}}}  = \sum\limits_{0\, \le \,k} {\left( {\sum\limits_{0\, \le \,n} {{{n^{\,k} } \over {n!}}} } \right)\left( \matrix{
  x \cr 
  k \cr}  \right)}  =   \cr 
  &  = e\sum\limits_{0\, \le \,k} {B_{\,k} \left( \matrix{
  x \cr 
  k \cr}  \right)}  = e\sum\limits_{0\, \le \,k} {{{B_{\,k} } \over {k!}}x^{\,\underline {\,k\,} } }  \cr} 
$$
wherefrom we also get the interesting recurrence
$$
\eqalign{
  & F(x + 1) = \sum\limits_{0\, \le \,k} {\left( {\sum\limits_{0\, \le \,n} {{{n^{\,k} } \over {n!}}} } \right)\left( \matrix{
  x \cr 
  k \cr}  \right)}  =   \cr 
  &  = \sum\limits_{0\, \le \,k} {\left( {\left[ {k = 0} \right] + \sum\limits_{1\, \le \,n} {{{n^{\,k} } \over {n!}}} } \right)\left( \matrix{
  x \cr 
  k \cr}  \right)}  =   \cr 
  &  = \sum\limits_{0\, \le \,k} {\left( {\left[ {k = 0} \right] + F(k)} \right)\left( \matrix{
  x \cr 
  k \cr}  \right)}  =   \cr 
  &  = 1 + \sum\limits_{0\, \le \,k} {\left( \matrix{
  x \cr 
  k \cr}  \right)F(k)}  \cr} 
$$
