In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta funct...In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.展开更多
Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity o...Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity on the constant Πq is also settled.展开更多
Gosper introduced the functions sinq z and cosq z as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with...Gosper introduced the functions sinq z and cosq z as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by Πqn （n ∈ N）. Gosper noticed that all his formulas on these constants have more than two of the Πqn. So, it is natural to raise the question of establishing identities involving only two of the Πqn. In this paper, our main goal is to give examples of such formulas in only two Πqn.展开更多
基金Supported by Innovation Program of Shanghai Municipal Education Commission and PCSIRT
文摘In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.
基金supported by National Natural Science Foundation of China(Grant No.11801451).supported by National Natural Science Foundation of China(Grant No.11371184)the Natural Science Foundation of Henan Province(Grant Nos.162300410086,2016B259 and 172102410069)。
文摘Applying an addition formula of Liu(2007),we deduce certain Jacobi theta function identities.From these results we confirm several q-trigonometric identities con.jectured by Gosper(2001).Another conjectured identity on the constant Πq is also settled.
文摘Gosper introduced the functions sinq z and cosq z as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by Πqn （n ∈ N）. Gosper noticed that all his formulas on these constants have more than two of the Πqn. So, it is natural to raise the question of establishing identities involving only two of the Πqn. In this paper, our main goal is to give examples of such formulas in only two Πqn.
