The Kronecker theta function and a decomposition theorem for theta functions I

Abstract

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan’s \({}_{1}\psi _{1}\) summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved.

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Acknowledgements

I would like to express my deep appreciation to Li-Chien Shen for his invaluable suggestions. I am grateful to the referee for many helpful criticisms and suggestions to improve an earlier version of this paper. I also thank Dandan Chen for pointing out several misprints of an earlier version of this paper.

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Correspondence to Zhi-Guo Liu.

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In memory of Professor Richard Askey.

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This work was supported by the National Science Foundation of China (Grant Nos. 11971173 and 11571114) and Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400)

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Liu, ZG. The Kronecker theta function and a decomposition theorem for theta functions I. Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00376-6

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Keywords

  • Theta function
  • Elliptic function
  • Kronecker theta function
  • Ramanujan \({}_{1}\psi _{1}\) summation

Mathematics Subject Classification

  • 33D15
  • 11F37
  • 11F27