The Lah Numbers and the nth Derivative of Exp(1/x)
Mathematics and Computer Science
- We give five proofs that the coefficients in the nth derivative of exp(1/x) are the Lah numbers, a triangle of integers whose best-known applications are in combinatorics and finite difference calculus. Our proofs use tools from several areas of mathematics, including binomial coefficients, Faà di Bruno's formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions. The paper is based on the discussion to Siad's question about the nth derivative of exp(1/x) on Math Stack Exchange.
“The Lah Numbers and the nth Derivative of Exp(1/x),” with Siad Daboul, Jan Mangaldan, and Peter Taylor. Mathematics Magazine, 86 (1): 39–47, 2013.