Bernoulli related polynomials and numbers
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- by Ch. A. Charalambides PDF
- Math. Comp. 33 (1979), 794-804 Request permission
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.References
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J. R. ABERNETHY, "On the elimination of the systematic errors due to grouping," Ann. Math. Statist., v. 4, 1933, pp. 263-277.
- L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7 (1956), 28–33. MR 74436, DOI 10.1007/BF01900520
- Ch. A. Charalambides, A new kind of numbers appearing in the $n$-fold convolution of truncated binomial and negative binomial distributions, SIAM J. Appl. Math. 33 (1977), no. 2, 279–288. MR 446989, DOI 10.1137/0133017
- Ch. A. Charalambides, Some properties and applications of the differences of the generalized factorials, SIAM J. Appl. Math. 36 (1979), no. 2, 273–280. MR 524501, DOI 10.1137/0136022
- F. N. David and D. E. Barton, Combinatorial chance, Hafner Publishing Co., New York, 1962. MR 0155371 C. C. GRAIG, "Sheppard’s corrections for a discrete variable," Ann. Math. Statist., v. 7, 1936, pp. 55-61. CH. JORDAN, Calculus of Finite Differences, Chelsea, New York, 1960. M. G. KENDALL & A. STUART, The Advanced Theory of Statistics, Vol. 1, Hafner, New York, 1961.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 794-804
- MSC: Primary 10A40; Secondary 62E15
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521294-5
- MathSciNet review: 521294