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Bell Polynomial

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There are two kinds of Bell polynomials.

ExponentialPolynomials

A Bell polynomial B_n(x), also called an exponential polynomial and denoted phi_n(x) (Bell 1934, Roman 1984, pp. 63-67) is a polynomial B_n(x) that generalizes the Bell number B_n and complementary Bell number B^~_n such that

B_n(1)=B_n
(1)
B_n(-1)=B^~_n.
(2)

These Bell polynomial generalize the exponential function.

Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted B_n(x).

Bell polynomials are implemented in the Wolfram Language as BellB[n, x].

The first few Bell polynomials are

B_0(x)=1
(3)
B_1(x)=x
(4)
B_2(x)=x^2+x
(5)
B_3(x)=x^3+3x^2+x
(6)
B_4(x)=x^4+6x^3+7x^2+x
(7)
B_5(x)=x^5+10x^4+25x^3+15x^2+x
(8)
B_6(x)=x^6+15x^5+65x^4+90x^3+31x^2+x
(9)

(OEIS A106800).

{B_n(x)} forms the associated Sheffer sequence for

f(t)=ln(1+t),
(10)

so the polynomials have that exponential generating function

sum_(k=0)^infty(B_k(x))/(k!)t^k=e^((e^t-1)x).
(11)

Additional generating functions for B_n(x) are given by

B_n(x)=e^(-x)sum_(k=0)^infty(k^nx^k)/(k!)
(12)

or

B_n(x)=xsum_(k=1)^n(n-1; k-1)B_(k-1)(x),
(13)

with B_0(x)=1, where (n; k) is a binomial coefficient.

The Bell polynomials B_n(x) have the explicit formula

B_n(x)=sum_(k=0)^nS(n,k)x^k,
(14)

where S(n,k) is a Stirling number of the second kind.

A beautiful binomial sum is given by

B_n(x+y)=sum_(k=0)^n(n; k)B_k(x)B_(n-k)(y),
(15)

where (n; k) is a binomial coefficient.

The derivative of B_n(x) is given by

d/(dx)B_n(x)=(B_(n+1)(x))/x-B_n(x),
(16)

so B_n(x) satisfies the recurrence equation

B_(n+1)(x)=x[B_n(x)+B_n^'(x)].
(17)

The second kind of Bell polynomials B_(n,k)(x_1,x_2,...) are defined by

B_(n,k)(x_1,x_2,...) =sum_(j_1+j_2+...=k; j_1+2j_2+...=n)(n!)/(j_1!j_2!...)((x_1)/(1!))^(j_1)((x_2)/(2!))^(j_2)....
(18)

They have generating function

sum_(k=0)^infty(B_k(x;x_1,x_2,...))/(k!)t^k=e^x(sum_(k=1)^infty(x_k)/(k!)t^k).
(19)

See also

Actuarial Polynomial, Bell Number, Complementary Bell Number, Dobiński's Formula, Idempotent Number, Lah Number, Sheffer Sequence, Stirling Number of the Second Kind

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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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Bell Polynomial

Cite this as:

Weisstein, Eric W. "Bell Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BellPolynomial.html

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