These numbers and corresponding polynomials were introduced by Korobov, see second edition of his book "Number theory methods in numerical analysis". He found some examples, where these "discrete" polynomials are more useful than "continuous" ones. They work in some boundary cases where infinite constructions do not converge.

Similar construction is also known as degenerate Bernoulli polynomials, which were introduced earlier by Carlitz.

Some related links are

On summation and interpolation formulas,

A Discrete Analog of Euler's Summation Formula

On one generalization of Stirling numbers

Korobov polynomials and umbral calculus.

For a fixed $p$, *Korobov numbers* $K_n$
and *Korobov polynomials* $K_n(x)$ can be defined via generating functions
\begin{gather}
\label{Pr_Fu_P}F_K(t)=\sum\limits_{n=0}^{\infty}
K_n\,\dfrac{t^n}{n!} =
\dfrac{pt}{(t+1)^p -1},\\
\label{Pr_Fu_Px}F_K(x,t)=\sum\limits_{n=0}^{\infty}
K_n(x)\,\dfrac{t^n}{n!}=\dfrac{pt(t+1)^x}{(t+1)^p-1}.
\end{gather}
In particular
$$K_0=1,\quad K_1=-\dfrac{p-1}{2},\quad
K_2=\dfrac{p^2-1}{6},\quad
K_3=-\dfrac{p^2-1}{4},\\
\label{ExK} K_0(x)=1,\quad K_1(x)=x-\dfrac{p-1}{2},\quad
K_2(x)=x^2-px+\dfrac{p^2-1}{6},\\
\nonumber
K_3(x)=x^3-\dfrac{3(p+1)}{2}x^2+\dfrac{p(p+3)}{2}x-\dfrac{p^2-1}{4}.
$$
Probably the main feature of Korobov numbers and polynomials is that they connect real and $p$-adic worlds. In the real world they become close to usual Bernoulli numbers and polynomials as $p\to\infty$:
$$\lim\limits_{p\to\infty}p^{-n}K_n=B_n,\qquad\lim\limits_{p\to\infty}p^{-n}K_n(px)=B_n(x)\qquad(n\ge0).$$
If $p=0$ then Korobov numbers and polynomials coinside with
Bernoulli numbers and polynomials of the second kind
\begin{gather*}
b_0=1,\quad b_1=\dfrac{1}{2},\quad b_2=\dfrac{1}{6},\quad
b_3=\dfrac{1}{4},\\
b_0(x)=1,\quad b_1(x)=x+\dfrac{1}{2},\quad
b_2(x)=x^2-\dfrac{1}{6},\\
b_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}.
\end{gather*}
They can be defined by generating functions
\begin{gather}
\label{Def_b}
F_b(t)=\sum_{n=0}^{\infty} b_n \frac{t^n}{n!}=\dfrac{t}{\ln(1+t)},\\
\label{Def_b(x)}F_b(x,t)=\sum_{n=0}^{\infty} b_n(x)
\frac{t^n}{n!}=\dfrac{t(1+t)^x}{\ln(1+t)}.
\end{gather}
It means that $K_n(x)$ and $b_n(x)$
are close to each other in $p$-adic sense.

The same is true for corresponding Stirling numbers $\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{\!p}$ and $\left[\displaystyle{\vphantom{M}n\atop k}\right]_{\!p}$ are defined by
\begin{gather*}\label{St9}
x^{\underline{n}}=\sum\limits_{k=0}^{n}\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{\!p}p^k\left(\dfrac{x}{p}\right)^{\underline{k}},\\
\label{St10}p^n\left(\dfrac{x}{p}\right)^{\overline{n}}
=\sum\limits_{k=0}^{n}\left[\displaystyle{\vphantom{M}n\atop k}\right]_{\!p}x^{\overline{k}},
\end{gather*}
where
$$\begin{array}{c}
x^{\underline{n}}=x(x-1)\ldots(x-n+1),\\
x^{\overline{n}}=x(x+1)\ldots(x+n-1).
\end{array}$$
These numbers are dual to each other
$$\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{\!\frac{1}{p}}=\left(-\dfrac{1}{p}\right)^{n-k}\left[\displaystyle{\vphantom{M}n\atop k}\right]_{\!p},\qquad \left[\displaystyle{\vphantom{M}n\atop k}\right]_{\!\frac{1}{p}}=\left(-\dfrac{1}{p}\right)^{n-k}\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{\!p}\qquad(p\ne0),$$
and lie somewhere between usual Stirling numbers
\begin{align*} \label{St16}&\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{0}=(-1)^{n-k}\left[\displaystyle{\vphantom{M}n\atop k}\right],&&\left[\displaystyle{\vphantom{M}n\atop k}\right]_{0}=(-1)^{n-k}\left\{\displaystyle{\vphantom{M}n\atop k}\right\},\\
\label{St17}&\lim\limits_{p\to\infty}\dfrac{1}{p^{n-k}}\left\{\displaystyle{\vphantom{M}n\atop k}\right\}_{\!p}=\left\{\displaystyle{\vphantom{M}n\atop k}\right\},&&\lim\limits_{p\to\infty}\dfrac{1}{p^{n-k}}\left[\displaystyle{\vphantom{M}n\atop k}\right]_{\!p}=\left[\displaystyle{\vphantom{M}n\atop k}\right].\end{align*}