There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in terms of linear combinations of a $_{4}F_{3}$ hypergeometric function, and there's another which gives $v$ in terms of theta functions -- obviously for a given $v_{k}$, they must give the same value.

  • Is there a general identity (or class of identities) which relates linear combinations of the $_{4}F_{3}$ hypergeometric function to the appropriate sums of theta functions in question? -- one could produce one directly above as indicated above, but I would be interested in knowing why such an identity arises or what its implications would be, or what generalizations of it might exist. (I am aware that Chapter 5 of Berndt's /Number Theory in the Spirit of Ramanujan/ is "The Connection Between Hypergeometric Function and Theta Function" -- but it's only about $_{2}F_{1}$s)
  • Are their interpretations of such an identity in the context of analytic combinatorics?
  • $\begingroup$ Great question! Could you provide some references, that would be most welcomed! Thanks a lot! $\endgroup$ – user5831 Sep 26 '14 at 11:35

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