In this paper, the distribution function
and the generating function of
$\varphi (u+1)$ are set up. We assume that
$u\in \mathbb{N}\cup \{0\}$,
$\kappa \in \mathbb{N}$, the random walk
$\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves
$N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables
${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where
$\{\kappa =1,N\in \mathbb{N}\}$ and
$\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums
$\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability
$1-\varphi (u)$ or survival probability
$\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability
$\varphi (u)$ and its generating function are provided when
$\{\kappa =2,\hspace{0.1667em}N=2\}$,
$\{\kappa =3,\hspace{0.1667em}N=2\}$,
$\{\kappa =5,\hspace{0.1667em}N=10\}$ and
${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.
Open Access