# Bell polynomials

**Bell polynomials** are a family of polynomials introduced by the mathematician [Eric Temple Bell](https://www.edgepedia.org/eric-temple-bell) that encode the combinatorics of set partitions. In one common form, the single-variable (exponential) Bell polynomial \( B_n(x) \) generalizes the [Bell number](https://www.edgepedia.org/bell-number) \( B_n \), the number of ways to partition a set of \( n \) elements into nonempty blocks. In a second, multivariable form, the partial Bell polynomial \( B_{n,k}(x_1, x_2, \ldots) \) collects, over all partitions of \( n \) elements into \( k \) blocks, a sum of products of weights \( x_1, x_2, \ldots \) assigned to block sizes. Both forms appear throughout combinatorics, probability, and analysis, most prominently in the Faà di Bruno formula for derivatives of a composite function and in the moment–cumulant relations of probability theory.<sup>[1](https://mathshistory.st-andrews.ac.uk/Biographies/Bell/)</sup><sup> • </sup><sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by Eric Temple Bell in the 1934 papers "Exponential Numbers" (American Mathematical Monthly) and "Exponential Polynomials" (Annals of Mathematics); the name "Bell polynomials" was given by John Riordan in 1958<sup>[1](https://mathshistory.st-andrews.ac.uk/Biographies/Bell/)</sup><sup> • </sup><sup>[4](https://arxiv.org/abs/1402.4761)</sup> |
| Combinatorial meaning | The partial Bell polynomial \( B_{n,k} \) is a sum of products over all partitions of a set of \( n \) elements into \( k \) blocks<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup> |
| Relation to Bell numbers | \( B_n(1) = B_n \), the \( n \)th Bell number, whose exponential generating function is \( \exp(e^x - 1) \)<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup><sup> • </sup><sup>[5](https://dlmf.nist.gov/26.7)</sup> |
| Stirling connection | \( B_n(x) = \sum_{k=0}^{n} S(n,k)\, x^k \), where \( S(n,k) \) is a Stirling number of the second kind<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup> |
| First few polynomials | \( B_1(x) = x \), \( B_2(x) = x^2 + x \), \( B_3(x) = x^3 + 3x^2 + x \), \( B_4(x) = x^4 + 6x^3 + 7x^2 + x \)<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup> |
| Central analytic role | The coefficients in the Faà di Bruno formula for derivatives of a composite function are partial Bell polynomials<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup><sup> • </sup><sup>[6](https://www.tandfonline.com/doi/full/10.1080/23311835.2016.1220670)</sup> |
| Probabilistic role | Cumulants of a random variable are polynomial functions of its moments, with Bell polynomials supplying the coefficients<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup> |

## Two kinds of Bell polynomial

The term "Bell polynomial" refers to two related objects.<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup>

The **exponential Bell polynomial** \( B_n(x) \), also called an exponential polynomial, is a single-variable polynomial generalizing the Bell number: setting \( x = 1 \) recovers \( B_n \), and setting \( x = -1 \) gives the complementary Bell number \( \widetilde{B}_n \).<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup> Its exponential generating function is

\[ \sum_{k=0}^{\infty} \frac{B_k(x)}{k!}\, t^k = e^{(e^t - 1)x}, \]

and Dobiński's formula gives the explicit series

\[ B_n(x) = e^{-x} \sum_{k=0}^{\infty} \frac{k^n}{k!}\, x^k. \]

Setting \( x = 1 \) in this identity yields the corresponding formula for the [Bell numbers](https://www.edgepedia.org/bell-numbers).<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup><sup> • </sup><sup>[7](https://mathworld.wolfram.com/DobinskisFormula.html)</sup> The polynomials satisfy the recurrence \( B_{n+1}(x) = x \left( B_n(x) + B_n'(x) \right) \) with \( B_0(x) = 1 \), and the binomial-type identity \( B_n(x + y) = \sum_{k=0}^{n} \binom{n}{k} B_k(x) B_{n-k}(y) \).<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup> In the Sheffer classification, \( \{B_n(x)\} \) is the associated [Sheffer sequence](https://www.edgepedia.org/sheffer-sequence) for \( f(t) = \ln(1 + t) \).<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup><sup> • </sup><sup>[8](https://dlmf.nist.gov/18.2)</sup>

The **partial Bell polynomial** \( B_{n,k}(x_1, x_2, \ldots, x_{n-k+1}) \) is a homogeneous polynomial of degree \( k \) and weight \( n \) in the variables \( x_m \), with integer coefficients, defined by the exponential generating function

\[ \frac{1}{k!} \left( \sum_{m \geq 1} \frac{x_m}{m!}\, t^m \right)^{k} = \sum_{n \geq k} B_{n,k}(x_1, x_2, \ldots)\, \frac{t^n}{n!}. \]

Equivalently, it is a sum of products over all partitions of a set of \( n \) elements into \( k \) blocks, each block of size \( m \) contributing a factor \( x_m \).<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup><sup> • </sup><sup>[9](https://encyclopediaofmath.org/wiki/Bell_polynomial)</sup> The complete Bell polynomial \( B_n(x_1, \ldots, x_n) \) sums the partial polynomials over \( k \), and for particular choices of the weights \( x_m \) the partial polynomials reduce to various kinds of [Stirling numbers](https://www.edgepedia.org/stirling-numbers).<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup><sup> • </sup><sup>[9](https://encyclopediaofmath.org/wiki/Bell_polynomial)</sup>

## Faà di Bruno's formula

The best-known analytic appearance of Bell polynomials is in the **Faà di Bruno formula**, which gives the \( n \)th derivative of a composite function \( f(g(x)) \) as a sum over partitions of \( n \), with the partial Bell polynomials as coefficients. Because the coefficients \( A_{n,k} \) arising in such expansions are independent of the particular functions, they can be computed once and for all, for example by choosing \( f(x) = e^{ax} \); these coefficients coincide with the partial Bell polynomials \( B_{n,k} \).<sup>[6](https://www.tandfonline.com/doi/full/10.1080/23311835.2016.1220670)</sup> The same formula can be written in a partition-trace form, in which the complete Bell polynomial \( B_k \) is expressed as a trace over set partitions.<sup>[10](https://arxiv.org/abs/2605.30801)</sup>

This compositional role explains why Bell polynomials also govern the coefficients of power series compositions generally: they are, in Flajolet and Sedgewick's phrasing, a rephrasing of the multinomial expansion of powers of a generating function.<sup>[11](https://algo.inria.fr/flajolet/Publications/book070503.pdf)</sup>

## Moments and cumulants

In probability theory, the classical formula expressing the **cumulants** \( \kappa_n \) of a random variable as polynomial functions of its moments \( m_1, m_2, \ldots \), and the inverse relation expressing moments in terms of cumulants, are Bell-polynomial identities.<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup> If a random variable has a moment generating function finite in a neighborhood of zero, its cumulants are the coefficients of the logarithm of that generating function, and higher derivatives of the composite \( \log M_X \) are organized by the partial exponential Bell polynomials through Faà di Bruno's formula.<sup>[12](https://tesidottorato.depositolegale.it/bitstream/20.500.14242/199253/4/PhD_thesis_MARTINI_Tommaso.pdf)</sup> The moment–cumulant inversion can also be viewed as Möbius inversion on the partition lattice, the structure introduced into statistics by Speed's 1983 paper on cumulants and partition lattices.<sup>[10](https://arxiv.org/abs/2605.30801)</sup><sup> • </sup><sup>[13](https://www.stat.berkeley.edu/~sandrine/Docs/TerrySelectedWorksSpringer/Version1/McCullagh/McCullagh.pdf)</sup>

The same mechanism extends to **random sums**, sums of a random number \( N \) of independent and identically distributed variables, because the generating function of such a sum is the composition of two generating functions; this is the setting of branching processes.<sup>[2](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)</sup>

## Applications

**Compound Poisson moments.** Nadarajah, Withers, and Bakar gave expansions for the moments of the sample mean \( \overline{X} \) of a random sample, in both univariate and multivariate cases, whose coefficients are Bell polynomials. Applied to the compound Poisson variable \( S_N \), where \( S_n = n\overline{X} \) and \( N \) is Poisson and independent of the \( X_i \), the expansions are computationally more efficient than the Panjer recursion formula and the formula of Grubbström and Tang.<sup>[14](https://www.cambridge.org/core/journals/probability-in-the-engineering-and-informational-sciences/article/abs/expansions-for-moments-of-compound-poisson-distributions/075DA607F193BCA27957A09B3DA2E749)</sup>

**Wishart distributions.** Withers and Nadarajah expressed the moments of the central and noncentral complex [Wishart distribution](https://www.edgepedia.org/wishart-distribution) explicitly in terms of multivariate Bell polynomials, which they believed used there for the first time; multivariate Bell polynomials are easily written down from their univariate forms, which are available in most computer algebra packages.<sup>[15](https://www.sciencedirect.com/science/article/pii/S0047259X12001133)</sup>

**Other uses.** Bell polynomials are used to calculate moments of generalized Gaussian distributions,<sup>[16](https://www.mdpi.com/2075-1680/12/2/206)</sup> to write the \( n \)th Adomian polynomial of a nonlinear operator using partial exponential Bell polynomials,<sup>[17](https://arxiv.org/abs/1608.06880)</sup> and to supply a systematic algebra for the generating functions of moments belonging to combinatorial statistics — among them the biggest part of an integer partition, the crank and [rank of a partition](https://www.edgepedia.org/rank-of-a-partition), and the rank of a unimodal sequence.<sup>[10](https://arxiv.org/abs/2605.30801)</sup> In free probability, the volume polynomial of Pitman and Stanley plays the role that the complete Bell exponential polynomial plays in classical cumulant theory.<sup>[18](https://arxiv.org/abs/1002.4803)</sup>

## History and generalizations

Eric Temple Bell (1883–1960), a mathematician at the [California Institute of Technology](https://www.edgepedia.org/california-institute-of-technology), introduced the underlying concepts in the 1934 papers "Exponential Numbers" and "Exponential Polynomials"; the polynomials were named for him by John Riordan in *An Introduction to Combinatorial Analysis* (1958).<sup>[1](https://mathshistory.st-andrews.ac.uk/Biographies/Bell/)</sup><sup> • </sup><sup>[4](https://arxiv.org/abs/1402.4761)</sup> Bell also wrote 250 research papers and popular histories of mathematics, and the Bell numbers share the same origin.<sup>[1](https://mathshistory.st-andrews.ac.uk/Biographies/Bell/)</sup>

Noncommutative versions of the Bell polynomials were introduced by Schimming (1996) and by Munthe-Kaas (1995) in the setting of numerical integration on manifolds; they give rise to Faà di Bruno Hopf algebras, in which the Bell polynomials appear as rank polynomials of the underlying posets.<sup>[4](https://arxiv.org/abs/1402.4761)</sup>

The polynomials are standard in mathematical software: the [Wolfram Language](https://www.edgepedia.org/wolfram-language) implements the two kinds as `BellB` and `BellY`, and SymPy provides `bell(n, x)` for the exponential Bell polynomial together with the partial form.<sup>[3](https://mathworld.wolfram.com/BellPolynomial.html)</sup><sup> • </sup><sup>[19](https://docs.sympy.org/latest/modules/functions/combinatorial.html)</sup>

## References

1. [Eric Temple Bell (1883–1960), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Bell/)
2. [Jim Pitman, Combinatorial Stochastic Processes, Chapter 1: Bell polynomials, composite structures and Gibbs partitions](https://www.stat.berkeley.edu/~aldous/206-Exch/Papers/pitman_CSP.pdf)
3. [Bell Polynomial, Wolfram MathWorld](https://mathworld.wolfram.com/BellPolynomial.html)
4. [Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, arXiv:1402.4761](https://arxiv.org/abs/1402.4761)
5. [DLMF §26.7 Set Partitions: Bell Numbers, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/26.7)
6. [Remarks on Bell and higher order Bell polynomials and numbers, Cogent Mathematics (2016)](https://www.tandfonline.com/doi/full/10.1080/23311835.2016.1220670)
7. [Dobiński's Formula, Wolfram MathWorld](https://mathworld.wolfram.com/DobinskisFormula.html)
8. [DLMF §18.2 General Orthogonal Polynomials (Sheffer polynomials)](https://dlmf.nist.gov/18.2)
9. [Bell polynomial, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Bell_polynomial)
10. [Moments of partition statistics, Bell polynomials and Eisenstein-type series, arXiv:2605.30801](https://arxiv.org/abs/2605.30801)
11. [Flajolet & Sedgewick, Analytic Combinatorics](https://algo.inria.fr/flajolet/Publications/book070503.pdf)
12. [Tommaso Martini, PhD thesis, Appendix 2.A: Cumulants, Moments and Bell Polynomials](https://tesidottorato.depositolegale.it/bitstream/20.500.14242/199253/4/PhD_thesis_MARTINI_Tommaso.pdf)
13. [Introduction to Cumulants and partition lattices (P. McCullagh)](https://www.stat.berkeley.edu/~sandrine/Docs/TerrySelectedWorksSpringer/Version1/McCullagh/McCullagh.pdf)
14. [Nadarajah, Withers & Bakar, Expansions for moments of compound Poisson distributions, Probability in the Engineering and Informational Sciences](https://www.cambridge.org/core/journals/probability-in-the-engineering-and-informational-sciences/article/abs/expansions-for-moments-of-compound-poisson-distributions/075DA607F193BCA27957A09B3DA2E749)
15. [Withers & Nadarajah, Moments and cumulants for the complex Wishart, Journal of Multivariate Analysis 112 (2012)](https://www.sciencedirect.com/science/article/pii/S0047259X12001133)
16. [Approximating the Moments of Generalized Gaussian Distributions via Bell's Polynomials, Axioms 12(2), 206 (2023)](https://www.mdpi.com/2075-1680/12/2/206)
17. [Correlation between Adomian and Partial Exponential Bell Polynomials, arXiv:1608.06880](https://arxiv.org/abs/1608.06880)
18. [Cumulants and convolutions via Abel polynomials, arXiv:1002.4803](https://arxiv.org/abs/1002.4803)
19. [SymPy documentation: Combinatorial functions (bell)](https://docs.sympy.org/latest/modules/functions/combinatorial.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability distributions › Characteristic and generating functions › Cumulants and cumulant generating functions*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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