Constant Term Identities Research Articles

On Elliptic Product Formulas for Jackson Integrals Associated with Reduced Root Systems

In this note, we state certain product formulae for Jackson integrals associated with any root systems, involved in elliptic theta functions which appear as connection coefficients. The fomulae arise naturally in case of arbitrary root systems by extending the connection problem which has been investigated in l1, 4r in case ofA type root system. This is also connected with the Macdonald-Morris constant term identity investigated by I. Cherednik l6r, and K. Kadell l15r on the one hand, and of the Askey-Habsieger-Kadell‘s q-Selberg integral formula and its extensions l4, 8, 12, 14, 15r on the other. This is also related with some of the results due to R.A. Gustafson l10, 11r, although our integrands are different from his.

Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions

We prove a constant term conjecture of Robbins and Zeilberger (J. Combin. Theory Ser. A 66 (1994), 17–27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a $(2n)\times(2n)\times(2n)$ box and that was used by Andrews (J. Combin. Theory Ser. A 66 (1994), 28–39) and Andrews and Burge (Pacific J. Math. 158 (1993), 1–14) to compute this number explicitly. The evaluation of the generalized determinant is independent of Andrews and Burge's computations, and therefore in particular constitutes a new solution to this famous enumeration problem. We also evaluate a related determinant, thus generalizing another determinant identity of Andrews and Burge (loc. cit.). By translating some of our determinant identities into constant term identities, we obtain several new constant term identities.

The Selberg–Jack Symmetric Functions

K. Aomoto has recently given a simple proof of an extension of A. Selberg's integral. We prove the following generalization of Aomoto's theorem. For each k⩾0, there exists a family { s k λ ( t)} of homogeneous symmetric functions such that if the integrand of Selberg's integral is multiplied by s k λ ( t 1, …, t n ), then the integral has a certain closed form. Since s k λ ( t) is a renormalized Jack symmetric function with α=1/ k, we call { s k λ ( t)} the Selberg–Jack symmetric functions. R. P. Stanley and I. G. Macdonald have extended many of the basic properties of the I. J. Schur functions to the Jack symmetric functions. Stanley's extensions of the Pieri formula, in a form obtained by Macdonald's extension of the duality property, and the combinatorial representation arise naturally in our proof. We give proofs of these and other results of Stanley and Macdonald. We give two constant-term identities which are equivalent to the integration formula. These results extend W. G. Morris' theorem for A n and the Macdonald–Morris constant term identity for BC n . By exploring Selberg's proof of his integral, we give a recurrence relation, outline a method for calculating the Selberg–Jack symmetric functions, and show that a constant term orthogonality relation, recently proved by Macdonald, arises naturally. We give an expansion which allows us to evaluate the normalization of this orthogonality, thus giving an alternative proof of the integration formula.

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - <x>)$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.

Constant term identities of Forrester-Zeilberger-Cooper

We give a different proof of constant term identities due to Forrester, Zeilberger and Cooper using our previous result [6].

Proof of a q-analog of a constant term identity conjectured by forrester

Proof of a q-analog of a constant term identity conjectured by forrester

A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, …

Andrews's recent proof of the Mills-Robbins-Rumsey conjectured formula for the number of totally symmetric self-complementary plane partitions is used to derive a new multi-variate constant term identity, reminiscent of, but not implied by, Macdonald's BC n -Dyson identity. The method of proof consists in translating to the language of constant terms an expression of Doran for the desired number in terms of sums of minors of a certain matrix. The question of a direct proof of the identity, which would furnish an alternative proof of the Mills-Robbins-Rumsey conjecture, is raised, and a prize is offered for its solution.

Some q-Beta Integrals on $SU(n)$ and $Sp(n)$ That Generalize the Askey–Wilson and Nasrallah–Rahman Integrals

$SU(n)$ and $Sp(n)$ generalizations of a q-beta integral of Nasrallah–Rahman are evaluated. Selberg’s beta integral can be deduced as a special limiting case of the $Sp(n)$ integral. Extensions of the q-Macdonald–Morris constant term identities for the affine root systems of types $S(BC_n )$, $S(B_n )$, $S(B_n )^v $, $S(C_n )$$SC(C_n )^v $, and $S(D_n )$ can also be obtained from the $Sp(n)$ integral. There are some additional integral evaluations for $Sp(2)$ and $Sp(3)$.

Constant term identities and hypergeometric functions on spaces of Hermitian matrices

We announce the generalization, to the setting of hypergeometric functions of matrix argument, of classical constant term identities. As special cases, we obtain some constant term identities which play a prominent role in the proof of Dyson's conjecture.

A constant term identity and its relationship to the log-gas and some quantum many body systems

The excess chemical potential of an impurity charge in the one-component log-gas on a line in periodic boundary conditions is calculated by using an identity for the constant term in the Laurent expansion of a certain multivariable function. The coefficient of the leading term in the large separation expansion of the ground state one-particle density matrix for the one-dimensional quantum many body system with the 1/ r 2 potential can be expressed in terms of this excess chemical potential and thus evaluated explicitly.

A character sum for root system 𝐺₂

A character sum analog of the Macdonald-Morris constant term identity for the root system G 2 {G_2} is proved. The proof is based on recent evaluations of Selberg character sums and on a character sum analog of Dixon’s summation formula. A conjectural evaluation is presented for a related sum.

A one-line proof of the Habsieger-Zeilberger G 2 constant term identity

A one-line proof of the Habsieger-Zeilberger G 2 constant term identity

Theta Function Generalizations of Some Constant Term Identities in the Theory of Random Matrices

The probability distribution for the location of the eigenvalues in Dyson’s unitary random matrix ensembles is generalized to involve theta functions. At three special values of a parameter, corresponding to Dyson’s orthogonal, unitary, and symplectic ensembles, the normalization constant of the probability distribution is calculated. The results can be viewed as giving the constant term in the Laurent expansion of the multivariable function $\prod _{k \ne l = 1}^N ({{1 - w_k } /{w_l }})^{{\Gamma /2}} ({{q^2 w_k } /{w_l ;q^2 }})_\infty ^\Gamma $ for $\Gamma = 1,2$, and 4.

A Proof of Askey’s Conjectured q-Analogue of Selberg’s Integral and a Conjecture of Morris

Aomoto has recently given an extension of Selberg’s integral. We extend his proof to the q-case and establish a conjecture of Askey. Our result is equivalent to a constant term identity for the root system $A_n $. This extends a conjecture of Morris.

A short proof of Macdonald’s conjecture for the root systems of type 𝐴

We give a new proof of I. G. Macdonald’s conjecture for the root systems of type A A (or equivalently, the equal parameter q q -Dyson Theorem) that is short, elementary and direct. We also give a short proof of the equal parameter case of a constant term identity due to D. Bressoud and I. Goulden.

Constant term identities extending the 𝑞-Dyson theorem

Andrews [1] has conjectured that the constant term in a certain product is equal to a q q -multinomial coefficient. This conjecture is a q q -analogue of Dyson’s conjecture [5], and has been proved, combinatorically, by Zeilberger and Bressoud [15]. In this paper we give a combinatorial proof of a master theorem, that the constant term in a similar product, computed over the edges of a nontransitive tournament, is zero. Many constant terms are evaluated as consequences of this master theorem including Andrews’ q q -Dyson theorem in two ways, one of which is a q q -analogue of Good’s [6] recursive proof.

Character sum analogues of constant term identities for root systems

Certain identities connected with root systems provide explicit constant terms in Laurent series expansions of multivariable functions. Character sum analogues of these identities are given.