= Solution
Put $k=b/a>1$ and $\mathbf e_z=(0,0,1)$. The complete construction of <image charges between concentric grounded spheres> consists of two families, indexed by every integer:
$$
\boxed{\begin{aligned}
q_n&=qk^n,&\mathbf r_n&=r_0k^{2n}\mathbf e_z,\\
\widetilde q_n&=-\frac{qa}{r_0}k^n,&\widetilde{\mathbf r}_n&=\frac{a^2}{r_0}k^{2n}\mathbf e_z,
\qquad n\in\mathbb Z.
\end{aligned}}
$$
The term $(q_0,\mathbf r_0)$ is the physical <point charge>; every other term is an <image charge>. In the first family $n\leq-1$ lies inside $r=a$ and $n\geq1$ lies outside $r=b$. In the second family $n\leq0$ lies inside $r=a$ and $n\geq1$ lies outside $r=b$. Thus no <image charge> lies in the physical gap.
To derive and verify these families, reflection of an axial <point charge> $(Q,z)$ in a grounded sphere of radius $R$ replaces it by the <image charge> $(-RQ/z,R^2/z)$. Reflection in $a$ sends $(q_n,\mathbf r_n)$ to $(\widetilde q_{-n},\widetilde{\mathbf r}_{-n})$; reflection in $b$ sends it to $(\widetilde q_{1-n},\widetilde{\mathbf r}_{1-n})$. Successive reflections therefore generate exactly the listed families. The <method of images> gives
$$
\boxed{\phi(\mathbf r)=\frac1{4\pi\epsilon_0}\sum_{n\in\mathbb Z}
\left(\frac{qk^n}{|\mathbf r-r_0k^{2n}\mathbf e_z|}
-\frac{(qa/r_0)k^n}{|\mathbf r-(a^2/r_0)k^{2n}\mathbf e_z|}\right),\qquad a<r<b.}
$$
The physical <point charge> itself is excluded when evaluating this formula. On $r=a$, pair the first-family term $n$ with the second-family term $-n$; their <electric potentials> cancel by the single-sphere distance identity. On $r=b$, pair $n$ with $1-n$ instead. These reindexings are justified by <absolute convergence>, as follows.
For $n\to-\infty$, both source positions tend to the origin while their <electric charges> have size $O(k^n)$. On the closed gap their distances are eventually at least $a/2$, so both <electric potential> terms are $O(k^n)$. For $n\to+\infty$, both distances are at least a positive constant times $k^{2n}$ and the <electric charges> are $O(k^n)$, giving $O(k^{-n})$ terms. Both tails are bounded by convergent <geometric series>. This proves <absolute convergence> and <uniform convergence> of the tails on the closed gap. On any compact set avoiding the physical <point charge>, the differentiated tails obey the same type of bounds: for example, the large positive tail of the <electric field> is $O(k^{-3n})$. Thus termwise differentiation is valid and the sum is a <harmonic function> except at the physical <point charge>, where it has the correct <Coulomb's law> singularity. The boundary cancellations and <uniqueness of Poisson equation> complete the construction.
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