I was looking at some old questions of mine and stumbled upon this quesiton, which I could not solve:
Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy: $ \begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} = \begin{pmatrix}1 & 1 & 1\\1 & \omega & \omega^2 \\1 & \omega^2 & \omega\end{pmatrix} \cdot \begin{pmatrix}t_0(x) \\ t_1(x) \\ t_2(x)\end{pmatrix}$ where $\omega = \exp(2\pi i / 3)$
and might be defined this way.
As might be known, the matrix in the equation is the character table of the cyclic group $C_3$ and also a Vandermonde matrix. Using this last matrix equation one can prove the following addition theorem:
$\begin{pmatrix}t_0(x+y) \\ t_1(x+y) \\ t_2(x+y)\end{pmatrix} = \begin{pmatrix}t_0(x) & t_2(x) & t_1(x) \\ t_1(x) & t_0(x) & t_2(x)\\ t_2(x) & t_1(x) & t_0(x)\end{pmatrix} \cdot \begin{pmatrix}t_0(y)\\t_1(y) \\ t_2(y)\end{pmatrix}$
My question is:
Is it possible to give a product representation of these functions in terms of the roots of these functions, as is done for example with the $\sin$-e function?
Related questions:
https://mathoverflow.net/questions/227161/connection-between-cyclic-group-and-exponential-function