# What is the value of the power series with rising factorial coefficients?

The power series given by

$$S(\alpha,n,x) = \sum_{k=0}^\infty a_k(\alpha,n)x^k,\quad n\in \pmb{N}_0,\quad x\in\pmb{R}$$ where $$a_k(\alpha,n)=\left[ \alpha k+\alpha \atop n \right] = \frac{(\alpha k+\alpha)(\alpha k+\alpha +1)\cdots (\alpha k+\alpha +n-1)}{n!},\quad \quad 0<\alpha<2$$

To find its convergence range, I tried $$\begin{split} \lim_{k\to\infty}\frac{a_k(\alpha,n)}{a_{k-1}(\alpha,n)}&=\lim_{k\to\infty}\frac{(\alpha k+\alpha)(\alpha k+\alpha +1)\cdots (\alpha k+\alpha +n-1)}{(\alpha k)(\alpha k +1)\cdots (\alpha k +n-1)}\\ &=\lim_{k\to\infty}(1+\frac{1}{k})(1+\frac{\alpha}{\alpha k+1})\cdots(1+\frac{\alpha}{\alpha k+n-1})\\ &=1 \end{split}$$

Thus, I conclude that it converges when $|x|<1$. However, I find it's hard to test its convergence at $x=\pm 1$. I surfed the internet for hours but couldn't find a solution.

Besides, for fixed values of $\alpha$ and $n$, to find an closed form expression to the sum is even beyond my knowledge. I even don't know where to start.

I guess there may not be an explicit expression for the infinite sum, thus I tried to find approximation of it such that it can be evaluated numerically. But still I didn't succeed.

Another more challenging problem for me is to obtain the value or approximation of

$$\sum_{n=0}^\infty [S(\alpha,n,x)]^2$$

Can you give some reference on the above questions?

• The square-brackets $\left[ n \atop k\right]$ is rising factorial. On the other hand the usual binomial coefficient is falling factorial. The notation used in the above post is the generalized rising factorial in which $n$ is allowed to be a real number. – ecook Aug 22 '14 at 6:17
• Ah. So it's the same as multi-set coefficients, generalized to noninteger values. Note that your coefficient can be written in terms of the usual binomial coefficient as $\binom{\alpha+ k\alpha+n-1}{n}$ (which looks uglier, but may be more accessible for simplification.) – Semiclassical Aug 22 '14 at 12:40