It is known that the random variable $X$ only takes integer values, and its pmf is: $$ \mathbb{P}(X = k) = p_k, k ∈ \mathbb{Z}, - ν ≤ k < + ∞, $$ where we assume $ν > 0, p_{- ν} > 0, \mathbb{E}(X) > 0$.
Let $g(t) = \mathbb{E}(t^X) = \sum_{k = - ν}^{+ ∞}{p_k t^k}$ be the generating function of $X$.
Because $∀ k ≥ 1, ∀ t ∈ (0, 1]$, then $\lvert p_k t^k \rvert ≤ p_k, \sum_{k = - ν}^{+ ∞}{p_k} = 1$. According to the Weirstras discriminant method, then the power series $\sum_{k = - ν}^{+ ∞}{p_k t^k}$ converges uniformly on $t ∈ (0, 1]$.
Because $∀ k ≥ - ν$, then $p_k t^k$ is a continuous and convex function on $t ∈ (0, 1]$, so $g(t)$ is a continuous and convex function on $t ∈ (0, 1]$.
What's more, because $$ \begin{align} \lim_{t → 0 + 0}{g(t)} & = + ∞ \\ g(1) & = 1 \\ g'(1) & = \mathbb{E}(X) > 0 \end{align} $$ And $g(t)$ is a continuous and convex function on $t ∈ (0, 1]$, so only one $∃ η_1 ∈ (0, 1)$, then $g(η_1) = 1$, and $ ∀ t ∈ (η_1, 1)$, with $g(t) < 1 \quad (1)$.
$η_2, η_3, \ldots, η_ν$ are other roots of $g(t) = 1$ in complex unit circle. Assuming $\lvert η_2 \rvert ≥ \lvert η_3 \rvert ≥ \ldots ≥ \lvert η_ν \rvert$, it can be shown that $\lvert η_1 \rvert > \lvert η_2 \rvert$:
Let $z ∈ \mathbb{C}, \lvert z \rvert < 1, z ∉ (0, 1)$ is a root of $g(z) = 1$, then $$ \begin{align} g(\lvert z \rvert) & = \sum_{k = - ν}^{+ ∞}{p_k \lvert z \rvert^k} \\ & = \sum_{k = - ν}^{+ ∞}{\lvert p_k z^k \rvert} \\ & ≥ \left\lvert \sum_{k = - ν}^{+ ∞}{p_k z^k} \right\rvert \\ & = \lvert g(z) \rvert \\ & = 1 \end{align} $$ iff $\{z^{- ν}, z^{- ν + 1}, \ldots\}$ have same argument ⇔ "=", which is obviously impossible.
If $η_1 ≤ \lvert z \rvert < 1$, according to $(1)$, then $g(\lvert z \rvert) ≤ 1$, which is a contradiction, so $\lvert z \rvert < η_1$.
Proof: $$ \lim_{ν → + ∞}{η_1} = 1, \\ i ≠ 1, \lim_{ν → + ∞}{\prod_{j = 1}^{ν}{\left(I(j = i) + \frac{1 - η_j}{η_i - η_j} I(j ≠ i)\right)}} = 0. \tag{*} $$
Note: When $ν$ changes, the pmf of $X$ may change accordingly, but always satisfy: $$ p_{- ν}> 0, \\ \sum_{k = - ν}^{+ ∞}{p_k} = 1. $$
I'm not sure if statements $(*)$ are true, so you can prove they false.
More information about this question: [https://arxiv.org/abs/1209.4203v5].