Sum of a power series with a parameter

Question

Does anybody know the sum

$$ \sum_{n=0}^{\infty}\frac{x^{n}}{(n+a)n!}=f(x,a)$$

here $a$ is a number $ a>0 $. A hint please ? :D

If $ a=1$ I believe $ f(x,1)=(e^{x}-1)/x $

Answer 1

For $a\gt0$, $$ \begin{align} af(x,a) &=a\sum_{n=0}^\infty\frac{x^n}{(n+a)n!}\\ &=1+\sum_{n=1}^\infty\left(\frac1n-\frac1{n+a}\right)\frac{x^n}{(n-1)!}\\ &=e^x-x\sum_{n=0}^\infty\frac{x^n}{(n+a+1)n!}\\ &=e^x-xf(x,a+1)\tag{1} \end{align} $$ Reversing $(1)$ yields $$ f(x,a)=\frac{e^x-(a-1)f(x,a-1)}{x}\tag{2} $$ Evaluating $(1)$ directly gives $$ \begin{align} f(x,1) &=\sum_{n=0}^\infty\frac{x^n}{(n+1)n!}\\ &=\frac1x\sum_{n=0}^\infty\frac{x^{n+1}}{(n+1)!}\\ &=\frac{e^x-1}{x}\tag{3} \end{align} $$ $f(x,a)$ can be computed for higher $a\in\mathbb{Z}$ using recursion and $(2)$.

Answer 2

I think you should rewrite your function in a better way... $$ f(x,a) = \sum \frac{x^n}{(n+a)n!} = \sum \frac{x^n}{n \cdot n!} + \sum \frac{x^n}{a \cdot n!} = S + \frac{1}{a}\sum \frac{x^n}{n!} = S + \frac{e^x}{a}. $$ If you differentate $S$, you'll find that $S' = \sum \frac{n \cdot x^{n-1}}{n \cdot n!} = \sum \frac{x^{n-1}}{n!}$, so $xS' = \sum \frac{x^n}{n!} = e^x$. Now ... since $xS' = e^x, S = \int e^x / x = Ei(x)$. So $f(x,a) = \sum \frac{x^n}{(n+a)n!} = e^x/a + Ei(x)$.