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Apparently, for any integers $m\ge 0,\ k\ge 1,$ and real $\alpha > 1$, the following series is convergent: $$S_{k,m}(\alpha) =\sum\limits_{m<i_1<\ldots<i_k<\infty}\alpha^{-(i_1+\ldots+i_k)}.$$

Examples (computed with Sage): $$\begin{align} S_{1,m}(\alpha) &= \frac{\alpha^{-m}}{\alpha-1}\\ S_{2,m}(\alpha) &= \frac{\alpha^{-2m}}{(\alpha-1)^2 (\alpha+1)}\\ S_{3,m}(\alpha) &= \frac{\alpha^{-3m}}{(\alpha-1)^3(\alpha+1)(\alpha^2+\alpha+1)}\\ S_{4,m}(\alpha) &= \frac{\alpha^{-4m}}{(\alpha - 1)^4(\alpha + 1)^2(\alpha^2 + 1)(\alpha^2 + \alpha + 1)} \end{align}$$

Question: Is there a general pattern in these fractions that allows a single formula for $S_{k,m}(\alpha)$ in terms of some standard functions? Is there a closed form for the alternating series $S_{1,m}(\alpha)-S_{2,m}(\alpha)+S_{3,m}(\alpha)-\ldots$?

Motivation: If $x_1 x_2 x_3 ...$ is an infinite string of random symbols i.i.d. uniform on a finite alphabet of size $\alpha\ge 2$, then, using [infinite inclusion-exclusion], the probability that the string has a square prefix of length greater than $m$ is
$$P(\exists i>m: x_1..x_i = x_{i+1}..x_{2i})=S_{1,m}(\alpha)-S_{2,m}(\alpha)+S_{3,m}(\alpha)-\ldots$$

(This is related to MSE postings [here], [here], and [here].)

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r.e.s.
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  • Your denominators simplify to $$\begin{align}&\alpha-1;,\ &(\alpha^2-1)(\alpha-1);,\&(\alpha^3-1)(\alpha^2-1)(\alpha-1);,\text{ and }\&(\alpha^4-1)(\alpha^3-1)(\alpha^2-1)(\alpha-1);,\end{align}$$ so it seems likely that $$S_{k,m}(\alpha)=\frac{\alpha^{-km}}{\prod_{i=1}^k(\alpha^i-1)}=\prod_{i=1}^k \frac1{\alpha^m(\alpha^i-1)};.$$ – Brian M. Scott Dec 16 '15 at 19:03
  • @BrianM.Scott - Thanks, I missed that. I'm hoping the alternating series might be expressible in terms of some standard function, at least in the case of $m=0$. – r.e.s. Dec 16 '15 at 19:25

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