Lacunary power series

A series

(*)

with gaps (lacunas), so that the exponents do not run through all the natural numbers. Depending on the properties of the sequence one obtains many properties of the series (*). Thus, if

and the series (*) converges in the disc , , then all points of the circle are singular for (Hadamard's gap theorem). A strengthening of this theorem is Fabry's gap theorem (cf. Fabry theorem). If the lower density

then is a single-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also Over-convergence.

References

[1]	L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3

References

[a1]	E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)
[a2]	P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957)