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Mathematics of Computation

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Bernoulli related polynomials and numbers

Author: Ch. A. Charalambides
Journal: Math. Comp. 33 (1979), 794-804
MSC: Primary 10A40; Secondary 62E15
MathSciNet review: 521294
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Abstract: The polynomials $ {\varphi _n}(x;a,b)$ of degree n defined by the equations

$\displaystyle {\Delta _a}{\varphi _n}(x;a,b) = \frac{{{{(x)}_{n - 1,b}}}}{{{b^{... {\text{and}}\quad {\Delta _b}{\varphi _n}(x;a,b) = {\varphi _{n - 1}}(x;a,b)$

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.

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Keywords: Difference operator, generalized factorial, Stirling polynomials, Stirling numbers of the first and second kind, Bernoulli polynomials and numbers of the first and second kind, generating functions, probability factorial moments
Article copyright: © Copyright 1979 American Mathematical Society