Distribution of Shifted Discrete Random Walk and Vandermonde matrices

In this work we set up the generating function of the ultimate time survival probability $\varphi(u+1)$, where $$\varphi(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-\kappa\right)<u\right)$$ and $u\in\mathbb{N}_0,\,\kappa\in\mathbb{N}$, and the random walk $\left\{\sum_{i=1}^{n}X_i,\,n\in\mathbb{N}\right\}$ consists of independent and identically distributed random variables $X_i$, which are non-negative and integer valued. We also give expressions of $\varphi(u)$ via the roots of certain polynomials. Based on the proven theoretical statements, we give several examples on $\varphi(u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, Geometric and some other distributions.