Definitions

# Lidstone series

In mathematics, certain types of entire functions can be expressed as a certain polynomial expansion known as the Lidstone series.

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

$f\left(z\right)=sum_\left\{n=0\right\}^infty left\left[A_n\left(1-z\right) f^\left\{\left(2n\right)\right\}\left(0\right) + A_n\left(z\right) f^\left\{\left(2n\right)\right\}\left(1\right) right\right] + sum_\left\{k=1\right\}^N C_k sin \left(kpi z\right)$.

Here An(z) is a polynomial in z of degree n, Ck a constant, and f(n)(a) the derivative of f at a.

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

$h\left(theta; f\right) = lim sup frac\left\{1\right\}\left\{r\right\} log |f\left(r e^\left\{itheta\right\}\right)|,$

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

$t= lim sup h\left(theta; f\right),$

with

$Npi leq t < \left(N+1\right)pi.,$

## References

• Ralph P. Boas, Jr. and C. Creighton Buck, Polynomial Expansions of Analytic Functions, (1964) Academic Press, NY. ISBN 3-540-03123-5

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