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Mathematics of Computation

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A special class of Bell polynomials

Author: F. T. Howard
Journal: Math. Comp. 35 (1980), 977-989
MSC: Primary 10A40; Secondary 05A15
MathSciNet review: 572870
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Abstract: We examine the integers $ V(n,k)$ defined by means of

$\displaystyle k!\sum\limits_{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},} $

and, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal of $ {e^x} - x - 1$. We prove that the $ V(n,k)$ are also related to the numbers $ W(n,k)$ defined by

$\displaystyle k!\sum\limits_{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}} $

in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomials $ {B_{n,k}}({a_1},{a_2},3 - \alpha ,4 - \alpha ,5 - \alpha , \ldots )$ and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.

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Keywords: Exponential partial Bell polynomial, Stirling number of the second kind, associated Stirling number of the second kind, Bernoulli number, Genocchi number, van der Pol number
Article copyright: © Copyright 1980 American Mathematical Society

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