A special class of Bell polynomials
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- by F. T. Howard PDF
- Math. Comp. 35 (1980), 977-989 Request permission
Abstract:
We examine the integers $V(n,k)$ defined by means of \[ k!\sum \limits _{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},} \] 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 \[ k!\sum \limits _{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}} \] 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.References
- L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 32 (1958/59), 247–260. MR 104845, DOI 10.2307/3029225
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
- F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J. 34 (1967), 599–615. MR 217035
- F. T. Howard, Associated Stirling numbers, Fibonacci Quart. 18 (1980), no. 4, 303–315. MR 600368
- F. T. Howard, Bell polynomials and degenerate Stirling numbers, Rend. Sem. Mat. Univ. Padova 61 (1979), 203–219 (1980). MR 569660
- F. T. Howard, Numbers generated by the reciprocal of $e^{x}-x-1$, Math. Comp. 31 (1977), no. 138, 581–598. MR 439741, DOI 10.1090/S0025-5718-1977-0439741-4
- F. T. Howard, Polynomials related to the Bessel functions, Trans. Amer. Math. Soc. 210 (1975), 233–248. MR 379348, DOI 10.1090/S0002-9947-1975-0379348-5
- F. T. Howard, Properties of the van der Pol numbers and polynomials, J. Reine Angew. Math. 260 (1973), 35–46. MR 318054, DOI 10.1515/crll.1973.260.35
- F. T. Howard, The van der Pol numbers and a related sequence of rational numbers, Math. Nachr. 42 (1969), 89–102. MR 258739, DOI 10.1002/mana.19690420107 C. JORDAN, Calculus of Finite Differences, Chelsea, New York, 1950.
- John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096594
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Math. Comp. 35 (1980), 977-989
- MSC: Primary 10A40; Secondary 05A15
- DOI: https://doi.org/10.1090/S0025-5718-1980-0572870-3
- MathSciNet review: 572870