Partial sums of falling factorials

Solution 1:

I like Knuth's notation for falling factorials better:

$$ \alpha^{\underline{h}} = \alpha \cdot (\alpha - 1) \cdot \dotsm \cdot (\alpha - h + 1) $$

First note that:

$$ \Delta \alpha^{\underline{h}} = (\alpha + 1)^{\underline{h}} -\alpha^{\underline{h}} = (\alpha + 1) \cdot \alpha^{\underline{h - 1}} - \alpha^{\underline{h - 1}} \cdot (\alpha - h + 1) = h \alpha^{\underline{h - 1}} $$

This suggests:

$$ \sum_{0 \le k \le n} (\alpha + k)^{\underline{h}} = \frac{(\alpha + n + 1)^{\underline{h + 1}}}{h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} $$

This we prove by induction.

Base: $n = 0$ gives:

$\begin{align} \sum_{0 \le k \le 0} (\alpha + k)^{\underline{h}} &= \alpha^{\underline{h}} \\ \frac{(\alpha + 1)^{\underline{h + 1}}}{h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} &= \frac{1}{h + 1} \Delta \alpha^{\underline{h + 1}} \\ &= \alpha^{\underline{h}} \end{align}$

This checks out.

Induction: Assume it is true for $n$, look at $n + 1$:

$\begin{align} \sum_{0 \le k \le n + 1} (\alpha + k)^{\underline{h}} &= \sum_{0 \le k \le n} (\alpha + k)^{\underline{h}} + (\alpha + n + 1)^{\underline{h}} \\ &= \frac{(\alpha + n + 1)^{\underline{h + 1}}}{h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} + (\alpha + n + 1)^{\underline{h}} \\ &= \frac{(\alpha + n + 1)^{\underline{h}} \cdot (\alpha + n + 1 - h)} {h + 1} + \frac{(\alpha + n + 1)^{\underline{h}} \cdot (h + 1)} {h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} \\ &= \frac{(\alpha + n + 1)^{\underline{h}} \cdot (\alpha + n + 2)} {h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} \\ &= \frac{(\alpha + n + 2)^{\underline{h + 1}}}{h + 1} - \frac{\alpha^{\underline{h + 1}}}{h + 1} \end{align}$

This is exactly as claimed.

For your second sum, note that:

$$ (\alpha + k)^{\underline{h}} x^{\alpha + k - h} = \frac{\mathrm{d}^h}{\mathrm{d} x^h} x^{\alpha + k} $$

Thus the sum is essentially:

$\begin{align} \sum\limits_{k=0}^n (\alpha + k)^{\underline{h}} r^k &= r^{h - \alpha} \sum\limits_{k=0}^n (\alpha + k)^{\underline{h}} r^{\alpha + k - h} \\ &= r^{h - \alpha} \left. \frac{\mathrm{d}^h}{\mathrm{d} x^h} \sum\limits_{k=0}^n x^{\alpha + k} \right|_{x = r} \\ &= r^{h - \alpha} \left. \frac{\mathrm{d}^h}{\mathrm{d} x^h} \left(x^\alpha \sum\limits_{k=0}^n x^k\right) \right|_{x = r} \\ &=\left.r^{h-\alpha} \frac{d^h}{d x^h} \left( \frac{x^\alpha - x^{\alpha+n+1}}{1- x} \right)\right|_{x=r} \\ &=r^{h-\alpha} \sum\limits_{l=0}^h \binom{h}{l} \left(\alpha^{\underline{h-l}} r^{\alpha-(h-l)} - (\alpha+n+1)^{\underline{h-l}} r^{\alpha+n+1-(h-l)}\right) \cdot \frac{l!}{(1-r)^{l+1}} \\ &= \frac{1}{(1-r)^{h+1}} \sum\limits_{l=0}^h \binom{h}{l} \left(\alpha^{\underline{h-l}} r^{l} - (\alpha+n+1)^{\underline{h-l}} r^{n+1+l}\right) \cdot l! (1-r)^{h-l} \\ &= \frac{h!}{(1-r)^{h+1}} \sum\limits_{l=0}^h \frac{1}{(h-l)!} \left(\alpha^{\underline{h-l}} r^{l} - (\alpha+n+1)^{\underline{h-l}} r^{n+1+l}\right) \cdot (1-r)^{h-l} \\ &= \frac{h!}{(1-r)^{h+1}} \sum\limits_{l=0}^h \left(\binom{\alpha}{h-l} r^{l} - \binom{\alpha+n+1}{h-l} r^{n+1+l}\right) \cdot (1-r)^{h-l} \\ &= \frac{h!}{(1-r)} \left\{ \binom{\alpha}{h} \cdot F_{2,1} \left[\begin{array}{ll}1 & -h \\ \alpha-h+1 \end{array};\frac{r}{r-1}\right] - r^{n+1} \binom{\alpha+n+1}{h} \cdot F_{2,1} \left[\begin{array}{ll}1 & -h \\ \alpha+n-h+2 \end{array};\frac{r}{r-1}\right] \right\} \end{align}$

This is quite ugly. The remaining sum is geometric, and can be expressed as a fraction. Leibnitz' formula for multiple derivatives of a product reduce that somewhat, but it is still a mess. If $h$ is small integer, perhaps a CAS gives somewhat manageable.

Note: It is a matter of taste whether an expression is ugly or not. In my opinion the final result has a closed form if we use hypergeometric functions.