In this paper we present a unified approach to the spectral analysis of an hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well known formulas of Rodrigues type can be obtained directly. Moreover, other new Rodrigues type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.
We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.
Let $Γ$ be a simple graph and $I_Γ(x)$ its multivariate independence polynomial. The main result of this paper is the characterization of chordal graphs as the only $Γ$ for which the power series expansion of $I_Γ^{-1}(x)$ is Horn hypergeometric.
The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type g_n. Our main results include a new q-binomial theorem, new q-Gauss sum, and several transformation formulae for g_n series.
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.
In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x), $ every member of which is expressed by mean of a generalized hypergeometric function. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas …
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.
In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this pape…
Starting from a recently found branching formula for the six-parameter family of symmetric Macdonald-Koornwinder polynomials, we arrive by degeneration at corresponding branching rules for symmetric hypergeometric orthogonal polynomials of Wilson, continuous Hahn, Jacobi, Laguerre, and Hermite type.
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.
A one-parameter generalisation R_λ(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for R_λ(X;b). We also prove a new multiple q-Gauss summation formula and several further results for sl_n basic hypergeometric series based on R_λ(X;b).
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 investigate the solution space of hypergeometric systems of differential equations in the sense of Gelfand, Graev, Kapranov and Zelevinsky. For any integer $d \geq 2$ we construct a matrix $A_d \in \N^{d \times 2d}$ and a parameter vector $β_d$ such that the holonomic rank of the $A$-hypergeometric system $H_{A_d}(β_d)$ exceeds the simplicial volume $\vol(A_d)$ by at least $d-1$. The largest previously known gap between rank and volume was two. Our argument is elementary in that it uses only linear algebra, and our construction gives evidence to the general observation that rank-jumps seem …
The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics. We obtain new results about generating …
Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. As applications, we derived two infinite families of super-congruences.
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…
We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the…
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $π$-series involving Domb numbers and Franel numbers.
We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish…
