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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]

q -Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q -supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ϕ 7 8 summation, Watson's ϕ 7 8 transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a q -analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773–808] and two q -supercongruences involving double series. [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]

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]

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 Baily's F 1 2 ( 1 2 ) -series identity, two kinds of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

In 2017, He (2017) [9] established two supercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu (2017) [11] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several q -supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for basic hypergeometric series. As conclusions, He's first conjecture is confirmed and a more general form of He's second conjecture is proved. [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]

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]

Abstract: In terms of the hypergeometric method, we establish ten general -formulas with free parameters which include several known results as special cases. [Copyright &y& Elsevier]

Abstract: By applying the triple form of Gould–Hsu inversions to the Pfaff–Saalschütz theorem, a family of summation formulas on the Fox–Wright function, which implies several known -series identities due to Chen and Chu (2011) and numerous new hypergeometric series identities, is established. [Copyright &y& Elsevier]

Abstract: By applying the duplicate form of Carlitz inversions to three special cases of the -Saalschütz theorem respectively, we establish three classes of interesting summation formulae. [Copyright &y& Elsevier]

Abstract: By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities. [Copyright &y& Elsevier]

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q -supercongruence with two parameters modulo [ n ] Φ n (q) 3. Here [ n ] = (1 − q n) / (1 − q) and Φ n (q) is the n -th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q -analogue of Long's supercongruence. The latter is also a generalization of a recent q -supercongruence obtained by Guo and Schlosser. [ABSTRACT FROM AUTHOR]