Recent zbMATH articles in MSC 33Dhttps://www.zbmath.org/atom/cc/33D2021-04-16T16:22:00+00:00WerkzeugTerminating basic hypergeometric representations and transformations for the Askey-Wilson polynomials.https://www.zbmath.org/1456.330132021-04-16T16:22:00+00:00"Cohl, Howard S."https://www.zbmath.org/authors/?q=ai:cohl.howard-s"Costas-Santos, Roberto S."https://www.zbmath.org/authors/?q=ai:costas-santos.roberto-s"Ge, Linus"https://www.zbmath.org/authors/?q=ai:ge.linusSummary: In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.A-type quiver varieties and ADHM moduli spaces.https://www.zbmath.org/1456.140172021-04-16T16:22:00+00:00"Koroteev, Peter"https://www.zbmath.org/authors/?q=ai:koroteev.peterThis paper studies relation between equivariant \(K\)-theories of the framed \(A_n\)-type quiver variety \(X_n\) and of the rank \(r\) ADHM moduli space \(M_r= \bigsqcup_{k \ge 0}M_{r,k}\), where \(k\) denotes the instanton number. The \(K\)-theories considered are the equivariant quantum \(K\)-theory \(H_n:=K_{T^{n+2}}(QM(\mathbb{P}^1,X_n))\) and the equivariant \(K\)-theory \(K_{r,k}:=K_{T^2}(M_{r,k})\), where \(T\) denotes the one-dimensional torus.
The main results are Theorem 3.3 and Theorem 4.6, where an embedding \(\bigoplus_{l=0}^k K_{r,l} \hookrightarrow H_{n r}\) is constructed. The embedding is designed to map the \(T^2\)-fixed point classes in \(K_{r,l}\) to the coefficients of vertex functions in \(H_{n r}\). The construction is done by explicit calculations using Macdonald symmetric polynomials.
The paper also proposes an interesting Conjecture 5.4 which relates the eigenvalues of quantum multiplication operators in \(K\)-theory of \(M_r\) with those of the elliptic Ruijsenaars-Schneider model.
Reviewer: Shintaro Yanagida (Nagoya)Sklyanin-like algebras for \((q\)-)linear grids and \((q\)-)para-Krawtchouk polynomials.https://www.zbmath.org/1456.812652021-04-16T16:22:00+00:00"Bergeron, Geoffroy"https://www.zbmath.org/authors/?q=ai:bergeron.geoffroy"Gaboriaud, Julien"https://www.zbmath.org/authors/?q=ai:gaboriaud.julien"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://www.zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: S-Heun operators on linear and \(q\)-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big \(q\)-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and \(q\)-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators.
{\copyright 2021 American Institute of Physics}Partial theta function identities from Wang and Ma's conjecture.https://www.zbmath.org/1456.050182021-04-16T16:22:00+00:00"Wei, Chuanan"https://www.zbmath.org/authors/?q=ai:wei.chuananThe author considers partial theta functions, i.~e. functions of the form \(\sum_{n=0}^{\infty}q^{n^2+Bn}x^n\), where \(x\) and \(q\) are complex numbers, \(|q|<1\). An identity about such functions (attributed to Andrews and Warnaar) has been recently considered by \textit{J. Wang} and \textit{X. Ma} [Adv. Appl. Math. 97, 36--53 (2018; Zbl 1384.05051)] who have proposed a conjecture, embedding this identity in an infinite family of such identities. In the present paper, the author uses \(q\)-series methods to give a proof of the Wang-Ma conjecture and presents a result which may be regarded as the inverse of the Wang-Ma conjecture.
Reviewer: Vladimir P. Kostov (Nice)Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)The power of \(q\). A personal journey.https://www.zbmath.org/1456.110012021-04-16T16:22:00+00:00"Hirschhorn, Michael D."https://www.zbmath.org/authors/?q=ai:hirschhorn.michael-dPublisher's description: This unique book explores the world of \(q\), known technically as basic hypergeometric series, and represents the author's personal and life-long study -- inspired by Ramanujan -- of aspects of this broad topic. While the level of mathematical sophistication is graduated, the book is designed to appeal to advanced undergraduates as well as researchers in the field. The principal aims are to demonstrate the power of the methods and the beauty of the results. The book contains novel proofs of many results in the theory of partitions and the theory of representations, as well as associated identities. Though not specifically designed as a textbook, parts of it may be presented in course work; it has many suitable exercises.
After an introductory chapter, the power of \(q\)-series is demonstrated with proofs of Lagrange's four-squares theorem and Gauss's two-squares theorem. Attention then turns to partitions and Ramanujan's partition congruences. Several proofs of these are given throughout the book. Many chapters are devoted to related and other associated topics. One highlight is a simple proof of an identity of Jacobi with application to string theory. On the way, we come across the Rogers-Ramanujan identities and the Rogers-Ramanujan continued fraction, the famous ``forty identities'' of Ramanujan, and the representation results of Jacobi, Dirichlet and Lorenz, not to mention many other interesting and beautiful results. We also meet a challenge of D.H. Lehmer to give a formula for the number of partitions of a number into four squares, prove a ``mysterious'' partition theorem of H. Farkas and prove a conjecture of R.Wm. Gosper ``which even Erdős couldn't do.'' The book concludes with a look at Ramanujan's remarkable tau function.A note on \(q\)-Gaussians and non-Gaussians in statistical mechanics.https://www.zbmath.org/1456.820032021-04-16T16:22:00+00:00"Hilhorst, H. J."https://www.zbmath.org/authors/?q=ai:hilhorst.hendrik-jan"Schehr, G."https://www.zbmath.org/authors/?q=ai:schehr.gregoryWeighted partition identities and divisor sums.https://www.zbmath.org/1456.111992021-04-16T16:22:00+00:00"Garvan, F. G."https://www.zbmath.org/authors/?q=ai:garvan.frank-gConsider the partitions of a positive integer \(n\) into unequal parts. \textit{Z. B. Wang} et al. [Am. Math. Mon. 102, No. 4, 345--347 (1995; Zbl 0831.11056)] proved that the sum of the smallest parts in the above partitions of odd length minus the sum of the smallest parts in the above partitions of even length equals the number of positive divisors of \(n\). \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] gave another proof for this result by showing the equality of the corresponding \(q\)-generating functions. In the paper under review, the author obtains some identities similar to this \(q\)-identity. The author proves two one-parameter \(q\)-series identities which specialize to four weighted partition identities. Three of the weighted partition identities have coefficients which are divisor sums. The other is an identity originally due to \textit{K. Alladi} [Ramanujan J. 29, No. 1--3, 339--358 (2012; Zbl 1256.05011)] (involving triangular numbers). Some of the proofs depend on \(q\)-series results in \(Q(\sqrt 2)\) due to \textit{J. Lovejoy} [J. Number Theory 106, No. 1, 178--186 (2004; Zbl 1050.11085)] and due to \textit{D. Corson} et al. [J. Number Theory 107, No. 2, 392--405 (2004; Zbl 1056.11056)].
For the entire collection see [Zbl 1407.33001].
Reviewer: Mihály Szalay (Budapest)Free-fermion entanglement and orthogonal polynomials.https://www.zbmath.org/1456.810582021-04-16T16:22:00+00:00"Crampé, Nicolas"https://www.zbmath.org/authors/?q=ai:crampe.nicolas"Nepomechie, Rafael I."https://www.zbmath.org/authors/?q=ai:nepomechie.rafael-i"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.lucSums of partial theta functions through an extended Bailey transform.https://www.zbmath.org/1456.050172021-04-16T16:22:00+00:00"El Bachraoui, Mohamed"https://www.zbmath.org/authors/?q=ai:el-bachraoui.mohamedSummary: 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 \textit{G. Gasper} and \textit{M. Rahman} [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)] on \(q\)-hypergeometric series.Multiple Askey-Wilson polynomials and related basic hypergeometric multiple orthogonal polynomials.https://www.zbmath.org/1456.330142021-04-16T16:22:00+00:00"Nuwacu, Jean Paul"https://www.zbmath.org/authors/?q=ai:nuwacu.jean-paul"van Assche, Walter"https://www.zbmath.org/authors/?q=ai:van-assche.walterThe purpose of this paper is to use special transformations to create some new and some old \(q\)-hypergeometric polynomials, in particular `multiple polynomials'. Only the special \(q\)-Hermite polynomials from Gasper-Rahman book are used, but there are several other \(q\)-Hermite polynomials, as was observed by Johann Cigler and the reviewer. The key of the proofs is the Askey-Wilson integral, which does not use \(\Gamma_q\) functions, but infinite \(q\)-shifted factorials. These polynomials are schematically described in an Askey tableau. The multiple polynomials, which are proved by induction on the Rodrigues formula, are given as multiple sums and no reference is given to the multiple Laguerre polynomials which already exist in the literature. The proofs also use \(q\)-binomial theorem and Hankel determinant.
It should be pointed out that it is quite hard to determine the convengence regions of these functions of several real or complex variables.
Reviewer: Thomas Ernst (Uppsala)\((q,c)\)-derivative operator and its applications.https://www.zbmath.org/1456.050192021-04-16T16:22:00+00:00"Zhang, Helen W. J."https://www.zbmath.org/authors/?q=ai:zhang.helen-w-jSummary: In this paper, we introduce new concept of \((q,c)\)-derivative operator of an analytic function, which generalizes the ordinary \(q\)-derivative operator. From this definition, we give the concept of \((q,c)\)-Rogers-Szegö polynomials, and obtain the expanded theorem involving \((q,c)\)-Rogers-Szegö polynomials. In addition, we construct two kinds \((q,c)\)-exponential operators, apply them to \((q,c)\)-exponential functions, and establish some new identities. At last, some properties of \((q,c)\)-Rogers-Szegö polynomials are discussed.