Truncated versions of three identities of Euler and Gauss
Olivia X. M. Yao
Abstract
Abstract In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.
Topics & Concepts
Euler's formulaMathematicsGaussSeries (stratigraphy)Pure mathematicsEuler summationIdentity (music)Algebra over a fieldCalculus (dental)Mathematical analysisBackward Euler methodEuler equationsPhysicsSemi-implicit Euler methodBiologyAcousticsMedicineQuantum mechanicsPaleontologyDentistryAdvanced Mathematical IdentitiesHistory and Theory of MathematicsAnalytic Number Theory Research