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Date: 12-8-2018
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Date: 13-8-2018
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Date: 25-6-2019
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There are two kinds of Bell polynomials.
A Bell polynomial , also called an exponential polynomial and denoted
(Bell 1934, Roman 1984, pp. 63-67) is a polynomial
that generalizes the Bell number
and complementary Bell number
such that
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(1) |
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(2) |
These Bell polynomial generalize the exponential function.
Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted .
Bell polynomials are implemented in the Wolfram Language as BellB[n, x].
The first few Bell polynomials are
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
(OEIS A106800).
forms the associated Sheffer sequence for
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(10) |
so the polynomials have that exponential generating function
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(11) |
Additional generating functions for are given by
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(12) |
or
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(13) |
with , where
is a binomial coefficient.
The Bell polynomials have the explicit formula
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(14) |
where is a Stirling number of the second kind.
A beautiful binomial sum is given by
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(15) |
where is a binomial coefficient.
The derivative of is given by
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(16) |
so satisfies the recurrence equation
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(17) |
The second kind of Bell polynomials are defined by
They have generating function
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(18) |
REFERENCES:
Bell, E. T. "Exponential Polynomials." Ann. Math. 35, 258-277, 1934.
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 133, 1974.
Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. pp. 35-38, 49, and 142, 1980.
Roman, S. "The Exponential Polynomials" and "The Bell Polynomials." §4.1.3 and §4.1.8 in The Umbral Calculus. New York: Academic Press, pp. 63-67 and 82-87, 1984.
Sloane, N. J. A. Sequence A106800 in "The On-Line Encyclopedia of Integer Sequences."
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السعودية.. نجاح أول عملية استبدال صمام للقلب بآخر دون غرز جراحية
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هيونداي تحقق نجاحًا كبيرًا في السوق الأمريكي
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بالصور: جهود كبيرة يقدمها المتطوعون ضمن قسم حفظ النظآم في العتبة الحسينية وفي عدة مفاصل خدمية خلال زيارة الأربعين
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