In 1985, Bressoud and Goulden derived the formula for the constant term in $\prod_{(i,j)\in T} \frac{x_j}{x_i}\\\prod_{0\le i<j \le n}(\frac{x_i}{x_j})_{a_i}(\frac{qx_j}{x_i})_{a_j-1}$, where $T \subseteq \{(i,j)\mid 0\le i<j \le n\}$. This result implies the Andrews' $q$-Dyson identity. In 2006, Gessel and Xin proved the $q$-Dyson identity by considering both sides of the equality as polynomials in $q^{a_0}$. We use this approach to determine the coefficients of $x_0/x_1$ and $x_0/x_2$ in Laurent polynomials studied by Bressoud and Goulden.
We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of partitions of $n$ is equal to the number of partitions of $2n+d{n \choose 2}$ with $n$ $d$-distant parts. We also provide a direct proof for this identity. This work is the result of our aim at finding a bijective proof for Rogers-Ramanujan identities.
For a fixed odd prime $\ell$, we present new families of identities defined on various subposets of the poset of isomorphism classes of finite abelian $\ell$-groups, generalizing identities of Hall and Cohen-Lenstra. We also present a $q$-series identity.
In $1944$, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function $p(n)$. The number of Garden of Eden partit…
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic relations.
In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald's constant term identities admit an extra set of free parameters, thereby linking them to Poincare polynomials. We then exploit these extra degrees of freedom in the case of type A to give the first proof of Kadell's orthogonality conjecture---a symmetric function generalisation of the q-Dyson conjecture or Zeilberger-Bressoud theorem. Key ingredients in our proof of Kadell's orthogonality conjecture are the polynomial lemma of Ka…
By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the $q$-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of $SL(n,\mathbb{C})$.
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.
We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.
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 indepen…
In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci and Lucas sequences we also provide a bijective proof. Both families are then treated by generating functions.
It is well-known that the Selberg integral is equivalent to the Morris constant term identity. More generally, Selberg type integrals can be turned into constant term identities for Laurent polynomials. In this paper, by extending the Gessel--Xin method of the Laurent series proof of constant term identities, we obtain an AFLT type $q$-Morris constant term identity. That is a $q$-Morris type constant term identity for a product of two Macdonald polynomials.
In recent work on the representation theory of vertex algebras related to the Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamovic and Milas arise naturally by differentiating as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove complex and logarithmic constant term id…
Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families of new identities for antimatroids, including trees, posets, chordal graphs and finite point sets in $\mathbb{R}^n$. It also gives a "new" linear relation for matroids that is implied by Brylawski's identities.
We prove a master identity for a class of sequences defined by full-history linear homogeneous recurrences with (non-negative) constant coefficients. The identity is derived in a combinatorial way, providing thus combinatorial proofs for many known and new identities obtained as its corollaries. In particular, we prove several interesting identities for the Pell, the Jacobsthal, and the m-nacci numbers.
We present proofs of two new families of sum-product identities arising from the cylindric partitions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured by Kanade-Russell in the spirit of Andrews-Schilling-Warnaar identities of the $A_2$ Rogers-Ramanujan type. We follow the footsteps of Kanade-Russell while we alter the computations heavily to accomplish our goals.
We generalize the zig-zag product construction to produce infinite families of regular graphs of any constant degree. We analyze the second largest eigenvalue of this new zig-zag product to show that the modified zig-zag product of good expanders is again a good expander (yet not Ramanujan).
Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of $_6ψ_6$ given by Slater. In particular, we obtain a compact form of two-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply…
We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.
Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero--Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of t…
In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and the relation with classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.
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.
This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky-Kottwitz-MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longe…
Baker and Rumely's tau lower bound conjecture claims that if the tau constant of a metrized graph is divided by its total length, this ratio must be bounded below by a positive constant for all metrized graphs. We construct several families of metrized graphs having small tau constants. In addition to numerical computations, we prove that the tau constants of the metrized graphs in one of these families, the hexagonal nets around a torus, asymptotically approach to $\frac{1}{108}$ which is our conjectural lower bound.
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…
Partition identities are often statements asserting that the set $\mathcal P_X$ of partitions of $n$ subject to condition $X$ is equinumerous to the set $\mathcal P_Y$ of partitions of $n$ subject to condition $Y$. A Beck-type identity is a companion identity to $|\mathcal P_X|=|\mathcal P_Y|$ asserting that the difference $b(n)$ between the number of parts in all partitions in $\mathcal P_X$ and the number of parts in all partitions in $\mathcal P_Y$ equals a $c|\mathcal P_{X'}|$ and also $c|\mathcal P_{Y'}|$, where $c$ is some constant related to the original identity, and $X'$, respectively…
What follows is a lightly edited version of the author's unpublished master's essay, submitted in partial fulfillment of the requirements of the degree of Master of Arts at the Pennsylvania State University, dated June 1994, written under the supervision of Professor George E. Andrews. It was retyped by the author on November 23, 2022. Obvious typographical errors in the original were corrected without comment; hopefully not too many new errors were introduced during the retyping. Explanatory text added by the author in 2022 is notated by \emph{Remark added in 2022}. After the initial posting …
In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $π$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.
Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in…
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apery's constant given by Ramanujan: $ζ(3)=\frac{7π^3}{180}-2\sum_{n=1}^\infty\frac{1}{n^3(e^{2πn}-1)}$ Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe's identities as corollaries. The resemblance of the general relation to the structure of theta functions…
Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gröbner basis and the Graver basis. While the former is generally a proper subset of the latter, there are cases for which the two sets coincide. The most prominent examples among them are toric ideals of unimodular matrices. Equality of universal Gröbner basis and Graver basis is a combinatorial property of the toric ideal (or, of the defining matrix), providing interesting information about ideals of higher Lawrence liftings of a mat…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $π,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclo…
We give unified modular proofs to all of Gosper's identities on the $q$-constant $Π_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,\cdots,n_m$ with $m\geq 3$, $Π_{q^{n_1}}$, $\cdots$, $Π_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on $Π_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gosper's identities using hauptmoduls for genus zero congruence subgroups.
We further develop the finite length path generating transforms introduced previously, and use them to obtain constant sign polynomial expressions that reduce, in the limit of infinite path lengths, to parafermion and ABF Virasoro characters. This provides us, in the ABF case, with combinatorial proofs of Melzer's boson-fermion polynomial identities.
For every odd p \geq 3, we investigate representation theory of the vertex algebra WW_{2,p} associated to (2,p) minimal models for the Virasoro algebras. We demonstrate that vertex algebras WW_{2,p} are C_2--cofinite and irrational. Complete classification of irreducible representations for WW_{2,3} is obtained, while the classification for p \geq 5 is subject to certain constant term identities. These identities can be viewed as "logarithmic deformations" of Dyson and Selberg constant term identities, and are of independent interest.
