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| In [[mathematics]], a '''Lidstone series''', named after [[George James Lidstone]], is a kind of polynomial expansion that can expressed certain types of [[entire function]]s.
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| 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''<sub>''n''</sub> as follows:
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| :<math>f(z)=\sum_{n=0}^\infty \left[ A_n(1-z) f^{(2n)}(0) + A_n(z) f^{(2n)}(1) \right] + \sum_{k=1}^N C_k \sin (k\pi z).</math> | |
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| Here ''A''<sub>''n''</sub>(''z'') is a [[polynomial]] in ''z'' of degree ''n'', ''C''<sub>''k''</sub> a constant, and ''ƒ''<sup>(''n'')</sup>(''a'') the ''n''th [[derivative]] of ''ƒ'' at ''a''.
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| A function is said to be of '''[[exponential type]] of less than ''t''''' if the function
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| :<math>h(\theta; f) = \underset{r\to\infty}{\limsup}\, \frac{1}{r} \log |f(r e^{i\theta})|\,</math>
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| is bounded above by ''t''. Thus, the constant ''N'' used in the summation above is given by
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| :<math>t= \sup_{\theta\in [0,2\pi)} h(\theta; f)\,</math>
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| with
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| :<math>N\pi \leq t < (N+1)\pi.\,</math>
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| ==References==
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| * 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
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| [[Category:Mathematical series]]
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| {{mathanalysis-stub}}
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