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While researching some control and optimization algorithms I encountered power series in the form $$\sum_{k=1}^\infty z^{k^2}, \text{ and also } \sum_{k=1}^\infty k z^{k^2}.$$ I know nothing about series in which not all terms appear, only those whose orders are perfect squares. Would anyone kindly suggest elementary references on such series?

Pait
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    Maybe relevant: https://en.wikipedia.org/wiki/Lacunary_function. – Martín-Blas Pérez Pinilla Oct 13 '19 at 10:37
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    Some results to almost similar problems from a Russian table book of integrals & series (Интегралы и ряды. В 3 т. Прудников А.П., Брычков Ю.А., Маричев О.И.) https://drive.google.com/file/d/1nLg7fTLbXKMvKj5krylCctqwPI9-zRfz/view page 571 (in book pages) – M.P Oct 13 '19 at 10:45
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    Related: https://en.wikipedia.org/wiki/Theta_function – Ivan Neretin Oct 13 '19 at 11:03
  • These answers are lovely! Thanks Martin, M.P., and Ivan! – Pait Oct 13 '19 at 12:42
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    If it is not a secret, could you share more details about the context where these series are appearing? :) – shamisen Oct 14 '19 at 15:24
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    @shamisen Last year I wrote a paper about derivative-free optimization, it's at https://arxiv.org/abs/1801.10533.

    I want to derive approximate formulas for the rate of convergence of the function whose minimum we wish to find, under simplifying assumptions about the shape of the function and the distributions of the random search and measurement noise. A series of this type appears.

    Thanks for asking!

     – Pait Oct 14 '19 at 18:29

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