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We have the elliptic theta function of third kind

$$ \vartheta_3(z)=\sum_{n=-\infty}^{\infty}z^{n^2} $$

Is there a generalization of this function to other exponents? I am looking for

$$ f_k(z)=\sum_{n=-\infty}^{\infty}z^{n^k} $$

for $k=4,6,8,\ldots$.

update: generalization and some explicit results

A more general form that can be evaluated for any value of $k$ is

$$ g_k(z)=\sum_{n=-\infty}^{\infty}z^{|n|^k} =1+2\sum_{n=1}^{\infty}z^{n^k}. $$

Some explicit results:

  • $g_0(z)=1-z$: see comment; using Dirichlet regularization of the sum
  • $g_1(z)=\frac{1+z}{1-z}$
  • $g_2(z)=\vartheta_3(z)$

Are there more such explicit results? Series expansions? Relationships to special functions? Integral representations?

Roman
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    For $k\to0^+$ the regularized sum seems to give $1-z$: Assuming $k>0$, for $n=0$ we have $z^{n^k}=1$ and for $n\neq0$ we have $z^{n^k}\approx z^{1+k \ln n}=z n^{k \ln z}$. Therefore the sum is approximately $\sum_{n=1}^{\infty}z(-1)^{k\ln z}n^{k\ln z}+1+\sum_{n=1}^{\infty}z n^{k\ln z}$, which can be summed analytically with some regularization and gives $z \left[(-1)^{k \ln z}+1\right] \zeta(-k \ln z)+1=(1-z)+\frac12 k z \left[2\ln(2\pi)-i\pi\right] \ln z+O(k^2)$. For $k\to0^+$ we get $1-z$, in some sense. Not sure this is useful, or if I regularized correctly. – Roman Aug 08 '24 at 20:30
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    I found some uses for this but I haven't yet found a name in the literature for this (as I commented in the linked s.e. post). – Mason Aug 08 '24 at 20:55
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    Also the interesting question might be in your comment. Is this regularization correct/meaningful? – Mason Aug 08 '24 at 23:19
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    So... what's your question? You have given a generalization – Mason Aug 10 '24 at 01:13
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    @Mason Fair point. I've tried to clarify the question now. – Roman Aug 10 '24 at 19:46
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    Here's a link to some identities of the theta function provided by wolfram. – Mason Aug 10 '24 at 22:34
  • OEIS A374016 does not give any closed form for $g_4(z)$ other than the direct sum (generating function). – Roman Aug 11 '24 at 09:48

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