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In terms of the creative microscoping method, El Bachraoui's lemmas, and the Chinese remainder theorem for coprime polynomials, we establish some q-supercongruences for double and triple basic hypergeometric series. Several related conjectures are also proposed for further research. [ABSTRACT FROM AUTHOR]

Let $ \beta (z) $ β (z) be the Dirichlet beta function. Series for $ \beta (4) $ β (4) are very rare in the historical development of mathematics. With the help of the operator method and a transformation formula for hypergeometric series, we prove a surprising conjectural series of Z.-W Sun for the mathematical constant $ \beta (4) $ β (4). Furthermore, we find five new series for the same constant in this paper. [ABSTRACT FROM AUTHOR]

We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin's method of creative microscoping ['A q-microscope for supercongruences', Adv. Math.346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including $$ \begin{align*}\sum_{k=0}^{(p-2)/3}\frac{(\frac{2}{3})_k^3}{(1)_k^3}\equiv\frac{p}{2}\frac{(1)_{(p-2)/3}}{(\frac{4}{3})_{(p-2)/3}}\pmod{p^3},\end{align*} $$ where p is a prime with $p\equiv 5\pmod {6}$. [ABSTRACT FROM AUTHOR]

In terms of the q-Saalschütz identity, the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), pp. 329–358], and the Chinese remainder theorem for coprime polynomials, we establish some q-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a q-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), pp. 773–808]. [ABSTRACT FROM AUTHOR]

In 1981, Andrews gave a four-variable generalization of Ramanujan's 1 ψ 1 summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two 8 ϕ 7 series and Bailey's transformation formula for three 8 ϕ 7 series. Then, it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two 3 ψ 3 summation formulas in terms of two limiting relations and Bailey's another transformation formula for three 8 ϕ 7 series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo–Schlosser transformation formula and other two transformation formulas. [ABSTRACT FROM AUTHOR]

We give a short proof of the Hou-Sun identity using a q-hypergeometric summation. Furthermore, q-analogues of several other π-formulas are also established according to the q-hypergeometric summations and transformations and the telescoping method. [ABSTRACT FROM AUTHOR]

Recently, Wang and Ma proposed a conjecture, embedding the Andrews–Warnaar partial theta function identity in an infinite family of such identities. In this paper we use q-series methods to give a proof of the Wang–Ma conjecture. We also present a result which may be regarded as the inverse of the Wang–Ma conjecture. [ABSTRACT FROM AUTHOR]

By means of inversion techniques and four known hypergeometric series identities, eight summation formulas for the Fox-Wright function are established. They can give numerous summation formulas for 2-balanced hypergeometric series when the parameters are specified. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to Whipple-type 3 F 2 -series identities and utilizing L'Hôspital rule, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to Watson-type -series identities and using the bijection method of two-term difference as well as L'Hôspital rule, three families of summation formulae involving generalized harmonic numbers are established. [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 means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established. [ABSTRACT FROM PUBLISHER]

By applying the derivative operator to Minton and Karlsson's hypergeometric series identities, several interesting summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM PUBLISHER]

In terms of the derivative operator and a binomial sum from the telescoping method, we establish a family of summation formulae involving harmonic numbers. Meanwhile, a general recurrence relation is also deduced in accordance with Abel's lemma on summation by parts. [ABSTRACT FROM AUTHOR]

In terms of the integral method, we establish two Bailey-Borwein-Plouffe-type (BBP-type) formulas for π with a free parameter. Some BBP-type formulas for π without free parameters are also derived in the same way. [ABSTRACT FROM AUTHOR]

In terms of the hypergeometric method, we establish the extension of a series for 1/π due to Ramanujan [Modular equations and approximations to π. Quart J Math (Oxford). 1914;45:350–372] with two free parameters and a beautiful series for π with one free parameter. Meanwhile, two series for 1/π with three free parameters are also derived in the same way. [ABSTRACT FROM AUTHOR]

In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results. [ABSTRACT FROM PUBLISHER]

Abstract: We use the method of series rearrangement to establish two families of summation formulae for -Watson type -series in this paper. [Copyright &y& Elsevier]

Abstract: In terms of the hypergeometric method, we establish the extensions of two formulas for due to Ramanujan (1914) [32]. Further, other five summation formulas for with free parameters are also derived in the same way. [Copyright &y& Elsevier]

By applying the derivative operators to Chu–Vandermonde convolution, several general harmonic number identities are established. [ABSTRACT FROM PUBLISHER]

By means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity. [ABSTRACT FROM AUTHOR]