Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases

Abstract:

We briefly review Ramanujan’s theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for $1/\pi$ similar to ones found by Ramanujan.