Some Observations on the Lah and Laguerre Transforms of Integer Sequences

We study the Lah and related Laguerre transforms within the context of exponential Riordan arrays. Links to the Stirling numbers are explored. Results for finite matrices are generalized, leading to a number of useful matrix factorizations. In this note, we shall consider integer sequences a : N 0 → Z with general term a n = a(n). Normally, sequences will be described by either by their ordinary generating function (o.g.f.), that is, the function g(x) such that g(x) = ∞ n=0 a n x n , or by their exponential generating function (e.g.f.) f (x), where f (x) = ∞ n=0 a n x n n!. We shall encounter transformations that operate on integer sequences during the course of this note. An example of such a transformation that is widely used in the study of integer 1