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In terms of Andrews' series rearrangement method, we establish two generalizations of Jacobi's triple product identity. Their applications to mock theta functions and Rogers–Ramanujan identities are also discussed. [ABSTRACT FROM AUTHOR]

By applying the partial derivative operator to a summation of hypergeometric series, we prove two double series for π , which were conjectured by Guo and Lian recently. Employing the operator just mentioned and a summation formula for basic hypergeometric series, we also give q-analogues of these two double series. [ABSTRACT FROM AUTHOR]

Let denote the nth cyclotomic polynomial in q. Recently, Guo, Schlosser and Zudilin proved that for any integer n > 1 with n ≡ 1 (mod 4), where (a; q)m = (1 − a)(1 − aq) ... (1 − aqm−1). In this note, we give a generalization of the above q-congruence to the modulus case. Meanwhile, we give a corresponding q-congruence modulo for n ≡ 3 (mod 4). Our proof is based on the 'creative micro-scoping' method, recently developed by Guo and Zudilin, and a 4ϕ3 summation formula. [ABSTRACT FROM AUTHOR]

In terms of Dougall's 2 H 2 series identity and the series rearrangement method, we establish a symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschütz's theorem. Similarly, we also show that Bailey's 6 ψ 6 series identity implies the nonterminating form of Jackson's 8 ϕ 7 summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall's 2 H 2 series identity is equivalent to a known nonterminating form of Saalschütz's theorem and Bailey's 6 ψ 6 series identity is equivalent to the nonterminating form of Jackson's 8 ϕ 7 summation formula. [ABSTRACT FROM AUTHOR]

In terms of Abel's transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases. [ABSTRACT FROM AUTHOR]

By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $$_6\psi _6$$ -series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations of Ramanujan's reciprocity formula. Then we utilize also Milne's identity to deduce a multi-variable generalization of the Askey-Wilson integral. [ABSTRACT FROM AUTHOR]

By means of partial fraction decomposition, we give simple proofs of Mortenson's identities first. Then, inspired by them, we derive their q-generalizations and explore further identities of similar type. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to the corresponding hypergeometric form of a q-series transformation due to Andrews (see Theorem 4 in Theory and Application for Basic Hypergeometric Functions, pp. 191-224, Academic Press, New York, ), we establish a general harmonic number identity. As the special cases of it, several interesting Chu-Donno-type identities and Paule-Schneider-type identities are displayed. [ABSTRACT FROM AUTHOR]