Lidstone series

In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can expressed certain types of entire functions.

Let ƒ(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then ƒ(z) can be expanded in terms of polynomials A_n as follows:

Here A_n(z) is a polynomial in z of degree n, C_k a constant, and ƒ⁽ⁿ⁾(a) the nth derivative of ƒ at a.

A function is said to be of exponential type of less than t if the function

is bounded above by t. Thus, the constant N used in the summation above is given by

with

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