There are two kinds of Bell polynomials. A Bell polynomial B_n(x), also called an exponential polynomial and denoted ϕ_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 B_n(-1) | = | B^~_n. 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].
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