Find the error in $\displaystyle\sum_{k=0}^{\infty}\frac{a^k}{p(k)}$ where $|a|< 1$ and $p(z)$ is a polynomial

Question

Premise

I wanted to try to find the closed formula of the following series:

$$\sum_{k=0}^{\infty}\frac{a^k}{p(k)}\qquad |a|<1$$

Where $p(z)=(z-z_1)\cdot...\cdot(z-z_n)$ is an $n$ degree polynomial with $n$ distinct roots and $z_k\not\in\mathbb{N}$

My work

Using partial fraction decomposition: $$\frac{1}{p(z)}=\sum_{i=1}^{n}\frac{1}{p'(z_i)}\frac{1}{z-z_i}$$ So $$\begin{align}\sum_{k=0}^{\infty}\frac{a^k}{p(k)}=&\sum_{k=0}^{\infty}a^k\sum_{i=1}^{n}\frac{1}{p'(z_i)}\frac{1}{k-z_i}\\ =&\sum_{i=1}^{n}\frac{1}{p'(z_i)}\sum_{k=0}^{\infty}\frac{a^k}{k-z_i}\\ =&\sum_{i=1}^{n}\frac{\Phi(a,1,-z_k)}{p'(z_i)} \end{align}$$

Where $\Phi(z,s,a)$ is the Lerch Phi function

Question

I tried to see if the formula was correct by putting random values ​​but it doesn't add up ($a=0.915, z_1=3.7, z_2=5.2$ and $z_3=-1.1$) in fact I get that: $$\sum_{k=0}^{\infty} \frac{0.915^k}{(k-3.7)(k-5.2)(k+1.1)}\approx-0.46321$$ $$\neq$$ $$\frac{\Phi(0.915,1,-3.7)}{-7.2}+\frac{\Phi(0.915,1,-5.2)}{9.45}+\frac{\Phi(0.915,1,1.1)}{-30.24}\approx -0.22423$$ Could anyone tell me where I went wrong?