Taylor Series of Composite Functions and Combinatorial Identities
David C. Vella
February 19, 2003
Bailey Hall 100
Bernoulli numbers, Euler numbers, and other famous sequences are defined in terms of their appearance in the Taylor series expansions of common functions. When these functions are composed with other functions, this naturally implies relationships between these numbers, which are combinatorial indentities of the title. Recent collaborative work with an undergraduate student has led to a more transparent way of computing the Taylor coefficients of these composite functions, resulting in a sort of machine for producing such identities. Among the consequences are new proofs of old results, including a short proof of a famous theorem of E.T. Bell, as well as new combinatorial identities. In this talk, after a brief review of Taylor series, I will illustrate this machine with several examples. No previous knowledge of Bernoulli numbers, Euler numbers or other combinatorial sequences is necessary.
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