# Lidstone series

Appearance

In mathematics, a **Lidstone series**, named after George James Lidstone, is a kind of polynomial expansion that can express 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 *n*th 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

## References

[edit]- Ralph P. Boas, Jr. and C. Creighton Buck,
*Polynomial Expansions of Analytic Functions*, (1964) Academic Press, NY. Library of Congress Catalog 63-23263. Issued as volume 19 of*Moderne Funktionentheorie*ed. L.V. Ahlfors, series*Ergebnisse der Mathematik und ihrer Grenzgebiete*, Springer-Verlag ISBN 3-540-03123-5