Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures...Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures on congruences modulo powers of 3 for d_(2)(n).Their work has attracted the attention of da Silva,Hirschhorn,Sellers and Smoot.Very recently,Smoot proved a congruence family for d_(2)(n)which implies one conjecture due to Andrews and Paule by using the localization method.In this paper,we prove the rest two conjectures given by Andrews and Paule.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 the National Natural Science Foundation of China(Grant No.12371334)the Natural Science Foundation of Jiangsu Province(Grant No.BK20221383)。
文摘Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures on congruences modulo powers of 3 for d_(2)(n).Their work has attracted the attention of da Silva,Hirschhorn,Sellers and Smoot.Very recently,Smoot proved a congruence family for d_(2)(n)which implies one conjecture due to Andrews and Paule by using the localization method.In this paper,we prove the rest two conjectures given by Andrews and Paule.
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.
