The sum of a series

Question

Let $0< \alpha <1$ and $q>1.$ Consider the (alternating) series: $$ \sum_{k=1}^\infty (-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$ Denote its sum by $f(q,\alpha).$

Prove (or disprove) that $f(q,\alpha)\neq 0$ for all $\alpha\in(0,1)$ and all $q> 1.$

Computer computation confirms that there are no zeros for $ 1<q<2$ and $0<\alpha<1.$ For $\alpha\in N_0,$ we obtain 0 by Euler's formula. (Added on September 5 after the discussion, thanks to all who participated).

Remark. For $0<\alpha <1/2,$ and $q\geq 2$ the proof is rather simple, but, for $\alpha$ close to 1,I have not succeeded. The problem is related to the study of the convergence for the $q$-Bernstein polynomials.

Answer 1

I have no doubt that you have figured this exercise out by now but let me post the solution just for the fun of it.

Denote $U_k=\prod_{m=1}^k\frac 1{q^m-1}$ with the usual convention $U_0=1$.

We have $\prod_{k\ge 1}(1-q^{-k}y)=\sum_{k\ge 0}(-1)^kU_ky^k$. In particular, it implies that $$ \sum_{k\ge 1}(-1)^k U_{k-1}y^k<0\text{ for all } 0<y<q $$ The sum you are interested in is $$ \sum_{k\ge 1}(-1)^k U_{k-1}\frac{q^k}{(q^k-1)^\beta} $$ where $\beta=1-\alpha$.

Now just write $$ \frac{q^k}{(q^k-1)^\beta}=(q^{1-\beta})^k(1-q^{-k})^{-\beta}=(q^{1-\beta})^k\sum_{m\ge 0}c_{m}(\beta)q^{-mk}=\sum_{m\ge 0}c_m(\beta)(q^{1-m-\beta})^k $$ and observe that all $c_m(\beta)$ (the Taylor coefficients of $x\mapsto (1-x)^{-\beta}$ at $x=0$) are positive.

Answer 2

Mathematica claims that your sum at $q=2, \alpha=1/2$ is $-0.519219,$ and in fact this is the minimal value when $1<q<2$ and $1/2<\alpha < 1.$ It seems to indicate that the sum at $a=1, q=2$ IS equal to zero (but that is the max over the same region). However, what is even better, is that the sum at $a=3/2, q=2$ is claimed to be $0.307852,$ so the sum definitely vanishes somewhere (since the sum converges very quickly, I don't have reason to mistrust the evaluation).