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A formula relating Bell polynomials and Stirling numbers of the first kind

Abstract: In this paper, we prove a general convolution formula involving the Bell polynomials and the Stirling numbers of the first kind. Our proof of the formula is algebraic and establishes an equivalent identity involving the associated exponential generating function, where we make use of induction, manipulation of finite sums and several identities to demonstrate the latter. A bijective proof that draws upon a sign-changing involution on the related combinatorial structure is given for a special case of the formula.