Journal of the Ramanujan Mathematical Society

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A Bivariate Circular Summation Formula for Cubic Theta Functions and its Implications


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1 Department of Mathematics, Soochow University, Suzhou, P. O. Box 173, Suzhou-215006, China
     

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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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  • A Bivariate Circular Summation Formula for Cubic Theta Functions and its Implications

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Authors

Tianzhu Dai
Department of Mathematics, Soochow University, Suzhou, P. O. Box 173, Suzhou-215006, China
Xinrong Ma
Department of Mathematics, Soochow University, Suzhou, P. O. Box 173, Suzhou-215006, China

Abstract


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.