c_j c_{k-j}  + \sum_{j=1}^{\infty} \left(e^{-i \omega (j+k) }  +e^{i \omega j } \right) c_j^* c_{j+k}.
\label{eq:FourierSequenceEquation_2} \end{aligned}$$ Dividing through by $\alpha$ ensures that the parameter dependence in $F$ is solely concentrated in the linear part. In this manner $y$ is a periodic solution with frequency $ \omega$ to Wright’s equation at parameter $ \alpha$ if and only if $ [F(\alpha,\omega, c)]_k=0$ for all $ k \in {\mathbb{Z}}$ [@BergJaquette; @jlm2016Floquet].

To more succinctly express the functional $F$ we introduce additional notation. For a sequence $ c=\{ c_k\}_{k=1}^\infty$ we denote the projection onto the $k$-coefficient by $ [c]_k := c_k$. We define unnormalized basis elements $ e_j \in \ell^1,{\Omega}^s$ for $ j\in {\mathbb{N}}$ by: $$\begin{aligned}
    [e_j]_k = 
    \begin{cases}
    1 & \mbox{ if } k = j, \\
    0 & \mbox{ if } k \neq j.
    \end{cases}\end{aligned}$$ We define the discrete convolution $a*b$ for $ a,b \in \ell^1$ component-wise by: $$\begin{aligned}
    \left[ a * b \right]_k &:= \sum_{|k_1| + |k_2| = k} a_{k_1} b_{k_2}  & 
    & = \sum_{j=1}^{k-1} a_j b_{k-j} + \sum_{j=1}^\infty a_j^* b_{k+j} + a_{k+j} b_j^*,\end{aligned}$$ where $  a_{-k} = a_k^*$ and $ b_{-k}=b_k^*$, and the sum is taken over $ k_1 , k_2 \in {\mathbb{Z}}$. The space $\ell^1$ is a Banach 