In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.

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 $p\equiv 5\pmod {6}$ .

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

We establish a family of q-supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Karlsson–Minton type summation due to Gasper, together with the ‘creative microscoping’ method introduced by the first author in recent joint work with Zudilin.

By making use of the ‘creative microscoping’ method, Guo and Zudilin [‘Dwork-type supercongruences through a creative $q$-microscope’, Preprint, 2020, arXiv:2001.02311] proved several Dwork-type supercongruences, including some conjectures of Swisher. In this paper, we apply the Guo–Zudilin method to prove a new Dwork-type supercongruence, which uniformly generalises several conjectures of Swisher.

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.

We give a $q$-analogue of the following congruence: for any odd prime $p$,

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$

$q$

A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author’s previous results on a transformation formula for Milne’s multivariate generalization of basic hypergeometric series of type $A$ with different dimensions and it can be considered as a generalization of the Whipple–Sears transformation formula for terminating balanced $_{4}{{\phi }_{3}}$ series. As an application of the master formula, the one-variable cases of some transformation formulas for bilinear sums of basic hypergeometric series are given as examples. The bilinear transformation formulas seem to be new in the literature, even in the one-variable case.

We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. We also prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

We present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

A new type of basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new Selberg integral for Jack polynomials.

Nous étudions la nature arithmétique de $q$-analogues des valeurs $\zeta(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)=\sum_{k=1}^{\infty}q^k\sum_{d\mid k}d^{s-1}$, $s=1,2,\dots$, où $q$ est un nombre complexe, $|q|<1$ (ces fonctions sont intimement liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta_q(3),\zeta_q(5),\dots,\zeta_q(M)$ est au moins $c_1\sqrt{M}$, avec $c_1=0,3358$. Ce résultat peut être considéré comme un $q$-analogue du résultat de Rivoal et de Ball et Rivoal, qui affirme que la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta(3),\zeta(5),\dots,\zeta(M)$ est au moins $c_2\log M$, avec $c_2=0,5906$. Pour les mêmes valeurs de $q$, une minoration similaire pour les valeurs $\zeta_q(s)$ aux entiers $s$ pairs nous permet de redémontrer un cas particulier d’un résultat de Bertrand qui affirme la transcendance sur $\mathbb{Q}$ de l’une des deux séries d’Eisenstein $E_4(q)$ et $E_6(q)$ pour tout nombre complexe $q$ tel que $0<|q|<1$.

We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6ϕ5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.