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May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known modular form.

$$ {}_3 F_2\left(\begin{matrix}-\frac{3}{4}\quad -\frac{1}{2}\quad -\frac{1}{4}\\ \frac{1}{3}\quad \frac{2}{3}\end{matrix}\middle|-\frac{5120}{81}\frac{\alpha^{2}}{\tau^{4}}\right)$$

As a start, I would like to explore the possibility of casting such an expression in a manageable form and also check modular properties. I am not certain whether this can be done (writing these as Jacobi theta functions). Came across this expression in this paper (https://arxiv.org/pdf/1310.4410.pdf) - a partition function with spin three chemical potential. Please feel free to point me to a new direction if that would help. Thank you in advance.

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As far as I know, the class of hypergeometric functions, and of modular forms are disjoint. I don't know of any common intersection except the degenerate zero function.

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