We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.
Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in particular their modular properties, are studied.
A new type of sl_3 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 sl_3 basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl_3 Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl_3 series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl_3 Selberg integral for Jack p…
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6-phi-5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8-phi-7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexe…
The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equation…
We analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker-Akhiezer functions.
A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle and Schlosser's formula for partitions with length three.
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald q-difference operators. The c-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions…
In this paper we construct a discrete linear operator $K$ which transforms $A_2$ Macdonald polynomials into the product of two basic $3ϕ_2$ hypergeometric series with known arguments. The action of the operator $K$ on power sums in two variables can be reduced to a generalization of one particular case of the Bailey's summation formula for a very-well-poised $6ψ_6$ series. We also propose the conjecture for a transformation of $6ψ_6$ series with different arguments.
In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functio…
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this Note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n =1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.
We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if $q\neq 1$. Some relations for the basic hypergeometric function which follow from this fact are discussed.
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2ϕ_1$ transformation, the $q$-Pfaff-Kummer and Euler transformations, the $q$-Saalschütz summation formula and Sear's transformation for terminating, balanced ${}_4ϕ_3$ series. For bilateral series, we rederive Kaneko's analogue of the ${}_1ψ_1$ summation formula and give multivariable extensions of Bailey's ${}_2ψ_2$ transformations.
Simultaneous eigenfunctions of two Askey-Wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra U_q(sl(2,C)). These eigenfunctions are shown to be equal to Ruijsenaars' hypergeometric function under a proper parameter correspondence.
This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. It is the sequel to the manuscript "Hypergeometric functions I." The two manuscripts are very informal working papers, never intended for formal publication. Nevertheless, copies of the manuscripts have circulated widely, giving rise to quite a few citations in the subsequent 25 years. Therefore it seems justified to make the manuscripts available for the whole mathematical community. The author kindly gave his permission that typewritten versions be posted on arXiv.
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.
We give a new proof of the rearrangement lemma that works for all dimensions and all heat coefficients in the study of modular geometry on noncommutative tori. The building blocks of the spectral functions are landed in a hypergeometric family knowns as Lauricella functions of type $D$. We investigate the differential and recursive relations among the family and obtain a full reduction to Gauss hypergeometric functions. As for applications, we give phase one demonstration on how the hypergeometric features lead to new simplifications in computations involved in the modular geometry.
We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman-Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand-Tsetlin polytopes which appeared in the recent work of Bo…
We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of $U_q(\wideh…
Using the Baker-Akhiezer function technique we construct a separation of variables for the classical trigonometric 3-particle Ruijsenaars model (relativistic generalization of Calogero-Moser-Sutherland model). In the quantum case, an integral operator M is constructed from the Askey-Wilson contour integral. The operator M transforms the eigenfunctions of the commuting Hamiltonians (Macdonald polynomials for the root sytem A2) into the factorized form S(y1)S(y2) where S(y) is a Laurent polynomial of one variable expressed in terms of the 3phi2(y) basic hypergeometric series. The inversion of M …
Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by …
An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the …
We define the analogue of Jack's (Jacobi) polynomials, which were defined for finite-dimensional root system by Heckman and Opdam as eigenfunctions of trigonometric Sutherland operator for the affine root system $\hat A_{n-1}$. In the affine case, we define the polynomials as eigenfunctions of "affine Sutherland operator", which is Calogero-Sutherland operator with elliptic potential plus the term involving derivative with respect to the modular parameter. We show that such polynomials can be constructed explicitly as traces of certain intertwiners for affine Lie algebra. Also, we define the q…
We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation leads to a unified (works for arbitrary dimension) method of computing the modular curvature on toric noncommutative manifolds. We show that the spectral functions that define the quantum part of the curvature have closed forms in terms of hypergeometric functions. As a consequence, we are able to obtained explicit expressions (as functions in the dimension p…
We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the $n \air \infty$ limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function c…
There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.
As the second part of the sequel, we investigate the variation of rearrangement operators (more precisely, the spectral functions behind) arising in the study of modular geometry on noncommutative (two) tori. We initiate a systematic approach by introducing transformations corresponding to basic operations in calculus, like differentiation and integration by parts. As for applications, we extend, in a uniform way, the Connes-Moscovici's functional relations on noncommutative two tori attached to the variation of second heat coefficients to noncommutative tori of arbitrary dimension. Moreover, …
We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3,Z) transformation properties parallel to those of elliptic gamma functions.
We motivate and prove a series of identities which form a generalization of the Euler's pentagonal number theorem, and are closely related to specialized Macdonald's identities for powers of the Dedekind $η$--function. More precisely, we show that what we call ``denominator formula'' for the Virasoro algebra has ``higher analogue'' for all $c_{s,t}$-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as {\em fusion} and {\em modular invariance} that require all the irreducible modules of the series. In particular, in the case of …
A series of conjectures is obtained as further investigation of the integral transformation I(alpha) introduced in the previous paper. A Macdonald-type difference operator D is introduced. It is conjectured that D and I(alpha) are commutative with each other. Studying the series for the eigenfunctions under termination conditions, it is observed that a deformed Weyl group action appears as a hidden symmetry. An infinite product formula for the eigenfunction is found for a spacial case of parameters. A one parameter family of hypergeometric-type series F(alpha) is introduced. The series F(alpha…
We construct a family of continuous biorthogonal functions related to an elliptic analogue of the Gauss hypergeometric function. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. Relations to the Sklyanin algebra and elliptic analogues of the Faddeev modular double are discussed in detail.
In this paper we compute a $q$-hypergeometric expression for the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot $(2,2t+1)$ and use this to define a family of quantum modular forms which are dual to the generalized Kontsevich-Zagier series.
We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials. We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation.
This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner pr…
We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $κ=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $κ=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM…
In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the action of modular group on certain spaces of morphisms in MTC is unitary with respect to the natural inner product on these spaces. In a special case of category based on representations of the quantum group U_q sl_n at roots of unity we show that in some of these spaces of morphisms (for U_q sl_2, in all of them) the action of modular group can be written in ter…
We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of…
The Al-Salam & Carlitz polynomials are $q$-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the $q$-analogue of the type-$A$ Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a $q$-analogue of the Mehta integral, and the corresponding normalization of the multivariable Al-S…
We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, …
