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A Bivariate Circular Summation Formula for Cubic Theta Functions and its Implications
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In this paper, a general bivariate circular summation formula and its dual form are established. They connect Ramanujan’s theta function with the cubic analogue, ∑m,n=−∞ qm2+mn+n2 xm, of the classical theta functions introduced by M. Hirschhorn, F. Garvan and J. Borwein [18]. As applications, many new theta function identities are found and some well-known results are recovered from this summation formula as well as its dual form.
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