## Recent papers

For papers which have already appeared, see my publication list where dvi or ps versions are given.

Abstract This is not a paper, but a report. Some positivity conjectures are made generalizing known results for Gaussian posets. A summary of partial results is given.

Applications of q-Taylor theorems (with Mourad Ismail) (ps) dvi version

Abstract We establish two new q-analogues of a Taylor series expansion for polynomials using special Askey-Wilson polynomial bases. Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization coefficients. Byproducts include new summation theorems, new results on a q-exponential function, and quadratic transformations for q-series.

Tribasic Integrals and identities of Rogers-Ramanujan type (with Mourad Ismail) (dvi)

Abstract Some general integrals involving three bases are evaluated as infinite products using complex analysis. Many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products. The resulting identities are identities of Rogers-Ramanujan type. Some integer partition interpretations of these identities are given. Generalizations of the Rogers-Ramanujan type identities involving polynomials are given, again as corollaries of integral evaluations.

Abstract These are notes for the 5 talks I gave at the August 12-16, 2002 Euro Summer School in OPSF at Leuven.

Proof of a monotonicity conjecture (with Thomas Prellberg, ps)

Abstract A monotonicity conjecture of Friedman, Joichi and Stanton is established.

q-Taylor theorems, polynomial expansions, and interpolation of entire functions (with Mourad Ismail, dvi) ps version

Abstract We establish q-analogues of Taylor series expansions in special polynomial bases for functions where ln M(r;f) grows like ln^2 r. This solves the problem of constructing such entire functions from their values at [aq^n+ q^{-n}/a]/2, for 0 < q < 1. Our technique is constructive and gives an explicit representation of the sought entire function. Applications to q-series identities are given.

The Charney-Davis quantity for certain graded posets (with V. Reiner and V. Welker, ps) dvi latex pdf

Abstract Given a naturally labelled graded poset P with r ranks, the alternating sum

W(P,-1):=\sum_{w \in \JH(P)} (-1)^{\des(w)}

is an instance of a quantity occurring in the Charney-Davis Conjecture on flag simplicial spheres. When |P|-r is odd it vanishes. When |P|-r is even and P satisfies the Neggers-Stanley Conjecture, it has sign (-1)^{\frac{|P|-r}{2}}. We interpret this quantity combinatorially for several classes of graded posets P, including certain disjoint unions of chains and products of chains. These interpretations involve alternating multiset permutations, Baxter permutations, Catalan numbers, and Franel numbers.

Abstract New expansions for certain 2F1's as a sum of r higher order hypergeometric series are given. When specialized to the binomial theorem, these r hypergeometric series sum. The results represent cubic and higher order transformations, and only Vandermonde's theorem is necessary for the elementary proof. Some q-analogues are also given.

The cyclic sieving phenomenon (with V. Reiner and D. White) (pdf)

Abstract The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge's q=-1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Polya theory, polygon dissections, non-crossing partitions, finite reflection groups, and some finite field q-analogues.

Springer's theorem for modular coinvariants of GL_n(F_q) (with V. Reiner and P. Webb) (ps)

Abstract Two related results are proven in the modular invariant theory of GL_n(F_q). The first is a finite field analogue of a result of Springer on coinvariants of the symmetric group in characteristic zero. The second result is a related statement about parabolic invariants and coinvariants.

Ramanujan Continued Fractions Via Orthogonal Polynomials (with M. Ismail) (pdf)

Abstract Some Ramanujan continued fractions are evaluated using asymptotics of polynomials orthogonal with respect to measures with absolutely continuous components.

Springer's regular elements over arbitrary fields (with V. Reiner and P. Webb) (pdf)

Abstract Springer's theory of regular elements in complex reflection groups is generalized to arbitrary fields. Consequences for the cyclic sieving phenomenon in combinatorics are discussed.

Block inclusions and cores of partitions (with B. Olsson) (pdf)

Abstract Necessary and sufficient conditions are given for an s-block of integer partitions to be contained in a t-block. The generating function for such partitions is found analytically, and also bijectively, using the notion of an (s, t)-abacus. The largest partition which is both an s-core and a t-core is explicitly given.

The combinatorics of the Al-Salam-Chihara q-Charlier polynomials (with D. Kim and J. Zeng) (pdf)

Abstract We describe various aspects of the Al-Salam-Chihara q-Charlier polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial proof of Anshelevich's recent result on the linearization coefficients.

Note on 1-crossing partitions (with M. Bergerson, A. Miller, A. Pliml, V. Reiner, P. Shearer, and N. Switala) (pdf)

Bimahonian distributions (with H. Barcelo and V. Reiner) (pdf)

Abstract Two q-analogues of Euler's theorem on integer partitions with odd or distinct parts are given. A q-lecture hall theorem is given.

(q,t)-analogues and GLn(Fq) (with V. Reiner, pdf)

Abstract We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's 7^{th} variation'' of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GLn(Fq).

The combinatorics of Al-Salam-Chihara q-Laguerre polynomials (with A. Kasraoui and J. Zeng, pdf)

Abstract We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients.

Formulae for Askey-Wilson moments and enumeration of staircase tableaux (with S. Corteel, R. Stanley, and L. Williams, pdf)

Abstract We explain how the moments of the (weight function of the) Askey Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula previously. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of permutations, and the number of matchings.

Some combinatorial and analytical identities (with M. Ismail)

Abstract We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, and Uchimura. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced 4\phi_3 to a very-well-poised 8\phi_7 is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials.

The negative q-binomial (with S. Fu, V. Reiner, and N. Thiem, pdf)

Abstract Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces.

Moments of Askey-Wilson polynomials (with Jang Soo Kim, pdf )

Abstract New formulas for the n^th moment \mu_n(a,b,c,d;q) of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.

Orthogonal basic hypergeometric Laurent polynomials (with Mourad Ismail, pdf)

Abstract The Askey-Wilson polynomials are orthogonal polynomials in x = \cos \theta, which are given as a terminating _4\phi_3 basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in z=e^{i\theta}, which are given as a sum of two terminating 4\phi_3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4\phi_3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.