Bernoulli related polynomials and numbers

Abstract:

The polynomials ${\varphi _n}(x;a,b)$ of degree n defined by the equations \[ {\Delta _a}{\varphi _n}(x;a,b) = \frac {{{{(x)}_{n - 1,b}}}}{{{b^{n - 1}} \cdot (n - 1)!}}\quad {\text {and}}\quad {\Delta _b}{\varphi _n}(x;a,b) = {\varphi _{n - 1}}(x;a,b)\] where ${(x)_{n,b}} = x(x - b)(x - 2b) \cdots (x - nb + b)$ is the generalized factorial and ${\Delta _a}f(x) = f(x + a) - f(x)$, are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients, $B(n,s)$, $s = a/b$, of this representation are given explicitly or by a recurrence relation. The generating functions of ${\varphi _n}(x;a,b)$ and $B(n,s)$ are obtained. The limits of ${\varphi _n}(x;a,b)$ as $a \to 1$, $b \to 0$ or $a \to 0$, $b \to 1$ and the limits of $B(n,s)$ as $s \to \pm \infty$ or $s \to 0$ are shown to be the Bernoulli polynomials and numbers of the first and second kind, respectively. Finally, the generalized factorial moments of a discrete rectangular distribution are obtained in terms of $B(n,s)$ in a form similar to that giving its usual moments in terms of the Bernoulli numbers.