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Formulas evaluating differences of integer partitions according to the parity of the parts are referred to as Legendre theorems. In this paper we give some formulas of Legendre type for overpartitions. [ABSTRACT FROM AUTHOR]

In this paper we prove supercongruence relations for truncated N-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogues of some Ramanujan-type supercongruences. [ABSTRACT FROM AUTHOR]

In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$ -hypergeometric series. [ABSTRACT FROM AUTHOR]

We prove some Lambert series identities which were stated by Gosper without proof or reference. As an application we shall evaluate integrals involving Ramanujan theta function ψ(q). Furthermore, motivated by Ramanujan identities for qψ4(q²) and ψ³(q)/ψ(q³), we shall evaluate the squares of qψ4(q²) and ψ³(q)/ψ(q³) in terms of Lambert series. [ABSTRACT FROM AUTHOR]

Gosper introduced the functions sinq z and cosq z as q-analogues for the trigonometric functions sin z and cos z respectively. He stated but did not prove a variety of identities involving these two q-trigonometric functions. In this paper, we shall use the theory of elliptic functions to prove three formulas from the list of Gosper on the functions sinq z and cosq z. [ABSTRACT FROM AUTHOR]

Gosper introduced the functions sinqz and cosqz 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. [ABSTRACT FROM AUTHOR]

Gosper introduced the functions sinqz and cosqz 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. [ABSTRACT FROM AUTHOR]

In one of his notebooks, Ramanujan stated but did not prove an identity for a ratio of gamma functions involving two separate variables. Such an identity was referred to as a reciprocity theorem due to the symmetry relations between the variables. The two variable result was later proved and generalized to several variables by Berndt and Koukoulopoulos (Proc Am Math Soc 601:2369-2373, 2007). In this paper, we state and prove similar reciprocity theorems for ratios of q-gamma functions involving two and more variables. [ABSTRACT FROM AUTHOR]

In this paper we shall give a method to generate new divisibility results involving the binomial coefficients. We shall apply this approach to some well-known examples to derive new divisibility properties for the binomial coefficients. [ABSTRACT FROM AUTHOR]