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Some identities involving exponential functions and Stirling numbers and applications. (English) Zbl 1293.05011

Summary: B.-N. Guo and F. Qi [J. Comput. Appl. Math. 255, 568–579 (2014; Zbl 1291.11051)] posed a problem asking to determine the coefficients \(a_{k,i-1}\) for \(1 \leq i \leq k\) such that \(1/(1 - e^{-t})^k = 1 + \sum_{i=1}^k a_{k,i-1}(1/(e^t - 1))^{(i-1)}\). The authors answer this question alternatively by Faà di Bruno’s formula, unify the eight identities due to Guo and Qi to two identities involving two parameters, and apply them to obtain an explicit expression for the Apostol-Bernoulli numbers and the Fubini numbers, respectively.

MSC:

05A19 Combinatorial identities, bijective combinatorics
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
33B10 Exponential and trigonometric functions

Citations:

Zbl 1291.11051
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Full Text: DOI

References:

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