We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb {H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realisations of a natural family of cyclic Y-parabolically induced $\mathbb {H}$-representations. We recover Cherednik’s well-known polynomial representation as a special case.

The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalisations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action.

We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.

We find an Lascoux–Leclerc–Thibon (LLT)-type formula for a general power of the nabla operator of [BG99] applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$ [HHL+05a, CM18, Mel21], and the formula for $(\nabla^k p_1^n,e_n)$ from [EH16, GH22] as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\nabla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\mathbb{P}^1$ due to the second author [Mel20]. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson in [GKM04], and also to Stanley’s chromatic symmetric functions.

We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$, which follows from our recent formula for $\nabla $ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde {H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde {H}_{\mu }(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to ${\mathbb N}[q,t]$, generalizing Macdonald positivity.

Our primary result concerns the positivity of specific kernels constructed using the q-ultraspherical polynomials. In other words, it concerns a two-parameter family of bivariate, compactly supported distributions. Moreover, this family has a property that all its conditional moments are polynomials in the conditioning random variable. The significance of this result is evident for individuals working on distribution theory, orthogonal polynomials, q-series theory, and the so-called quantum polynomials. Therefore, it may have a limited number of interested researchers. That is why, we put our results into a broader context. We recall the theory of Hilbert–Schmidt operators and the idea of Lancaster expansions (LEs) of the bivariate distributions absolutely continuous with respect to the product of their marginal distributions. Applications of LE can be found in Mathematical Statistics or the creation of Markov processes with polynomial conditional moments (the most well-known of these processes is the famous Wiener process).

We establish a q-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a q-congruence of Guo and Zeng [‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory 145 (2014), 301–316]. The important ingredients in the proof include Andrews’ $_4\phi _3$ terminating identity.

Liu [‘Supercongruences for truncated Appell series’, Colloq. Math. 158(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series $F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.

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

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

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].

Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$. Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$. In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

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.

We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$, where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

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.

Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to $2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$) and odd parts at most half the peak. We prove that two-variable generating functions for $\unicode[STIX]{x1D707}(m,n)$ and $\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single $C^{\infty }$ function in $\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables $w$ and $q$, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak.

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

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

$q$

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.