Bernoulli related polynomials and numbers
Author:
Ch. A. Charalambides
Journal:
Math. Comp. 33 (1979), 794804
MSC:
Primary 10A40; Secondary 62E15
MathSciNet review:
521294
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Abstract: The polynomials of degree n defined by the equations where is the generalized factorial and , are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients, , , of this representation are given explicitly or by a recurrence relation. The generating functions of and are obtained. The limits of as , or , and the limits of as or 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 in a form similar to that giving its usual moments in terms of the Bernoulli numbers.
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 J. R. ABERNETHY, "On the elimination of the systematic errors due to grouping," Ann. Math. Statist., v. 4, 1933, pp. 263277.
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 L. CARLITZ, "A degenerate StaudtClausen theorem," Arch. Math., v. 7, 1956, pp. 2833. MR 0074436 (17:586a)
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 CH. A. CHARALAMBIDES, "Some properties and applications of the differences of the generalized factorials," SIAM J. Appl. Math., v. 36, 1979. MR 524501 (80j:33024)
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 F. N. DAVID & D. E. BARTON, Combinatorial Chance, Griffin, London, 1962. MR 0155371 (27:5305)
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 C. C. GRAIG, "Sheppard's corrections for a discrete variable," Ann. Math. Statist., v. 7, 1936, pp. 5561.
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 CH. JORDAN, Calculus of Finite Differences, Chelsea, New York, 1960.
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 M. G. KENDALL & A. STUART, The Advanced Theory of Statistics, Vol. 1, Hafner, New York, 1961.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905212945
PII:
S 00255718(1979)05212945
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
