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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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Modularity of the Shimura-Shintani kernel
Following this paper by Bringmann, Kane, Kohnen, the kernel of the Shimura and Shintani lifts is the function $$\Omega(\tau,z) = \frac{1}{\binom{2k-2}{k-1}\pi} \sum_{D=1}^{\infty} \sum_{b^2 - 4ac = D} …