Amy M. Fu, Alain Lascoux
We give a Newton type rational interpolation formula (Theorem \ref{theo}). It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu, which allows to recover many important classical $q$-series identities. We show in particular that some bibasic identities are a consequence of our formula.
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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 use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dim…
We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
By telescoping method, Sun gave some hypergeometric series whose sums are related to $π$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper summability of $r(k)t_k$ for $r(k)$ being a rational function of $k$. We give an upper bound and a lower bound on the degree of the numerator of $r(k)$ such that $r(k)t_k$ is Gosper summable. We also show that the denominator of the $r(k)$ can read from the Gosper representation of $t_{k+1}/t_k$. Based on these results, we give a systematic method to construct series…
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).
In this paper, we present a $q$-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the $q$-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given $q$-hypergeometric term. The structure theorem enables us to generalize the $q$-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of $q$-identities. As applications, several $q$-analogues of series for $π$ are presented.
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function, a hypergeometric term, and a q-hypergeometric term. We outline an algorithm for computing this product, and present an application.
A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A.
We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R) equivalence and deduce that the only gkz-rational toric four-folds in complex projective space P^6 are those varieties associated with an essential Cayley configuration. In this case, we study a suitable hyperplane arrangement and show that all rational A-hypergeometric functions may be described in terms of toric residues.
The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and ``$q$'' integral/sums.
The quantum spectral curve equation associated to KP $τ$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer $G$-functions, or their asymptotic series. Using their Mellin integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.
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