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Lattice associated to 4th Jacobi theta function?


Reference request: theta functions for lattices which are not unimodular or evenconjectured identity of the product of two theta functionsJacobi Theta Functions?Implementation of Jacobi theta functions in MatlabClosed-form of an integral involving a Jacobi theta function, $ int_0^{infty} frac{theta_4^{n}left(e^{-pi x}right)}{1+x^2} dx $$E_8$ and theta functionsWhy do the Jacobi theta functions have a natural boundary?How interpret the dual lattice $Gamma^*$?power series expression of theta functionCalculating $Theta$ series of $E_8$ Lattice













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For a lattice (specifically the dual lattice of a torus) there is associated a theta function



$ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $



In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to



$theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,



and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.



These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.



My question is this: is there a lattice corresponding to the fourth Jacobi theta function
$theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?










share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    For a lattice (specifically the dual lattice of a torus) there is associated a theta function



    $ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $



    In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to



    $theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,



    and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.



    These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.



    My question is this: is there a lattice corresponding to the fourth Jacobi theta function
    $theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      For a lattice (specifically the dual lattice of a torus) there is associated a theta function



      $ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $



      In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to



      $theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,



      and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.



      These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.



      My question is this: is there a lattice corresponding to the fourth Jacobi theta function
      $theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?










      share|cite|improve this question











      $endgroup$




      For a lattice (specifically the dual lattice of a torus) there is associated a theta function



      $ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $



      In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to



      $theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,



      and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.



      These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.



      My question is this: is there a lattice corresponding to the fourth Jacobi theta function
      $theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?







      special-functions integer-lattices theta-functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited yesterday









      J. M. is not a mathematician

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      asked yesterday









      plebmaticianplebmatician

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