In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Lidstone series, named after George James Lidstone

George James Lidstone

George James Lidstone was a British actuary who made several contributions to the field of statistics. He is known for Lidstone smoothing and Lidstone series.- References :...

, is a kind of polynomial expansion that can expressed certain types of entire function

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

s.

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

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 in z of degree n, C_k a constant, and ƒ⁽ⁿ⁾(a) the nth derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 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