Moment duality for orthogonal polynomials (with M. Ismail)
Abstract The Laguerre, Charlier, and Meixner polynomials are polynomials in two variables: in x they are classical orthogonal polynomials, and in a parameter b they are type R_I orthogonal polynomials. Thus they have two types of orthogonality relations. Remarkably, the type R_I moments are identified as the orthogonal polynomial moments for another set of classical polynomials. A general notion of moment duality is introduced for polynomials in two variables. This program is continued for two and three parameter Askey-Wilson polynomials. The equality of the moments is equivalent to the equality of two continued fractions.
Orthogonality of very well-poised series (with M. Ismail and E. Rains, pdf, after 15 years appearing in Indag. Math.)
Abstract Rodrigues formulas for very well-poised basic hypergeometric series of any order are given. Orthogonality relations are found for rational functions which generalize Rahman's 10\phi9 biorthogonal rational functions.
The Ariki-Koike algebras and q-Appell functions (with R. Li, S. Seo, and Ae Ja Yee, pdf)
Abstract:Two double sums originating from the Ariki–Koike algebras are evaluated and generalized using a transformation of F. H. Jackson. Two combinatorial approaches to these identities are given, using 2-cores and overpartitions.
Notes for Neighborly Partitions (with K. O'Hara, pdf)
Abstract:A proof of the first Rogers-Ramanujan identity is given using admissible neighborly partitions. This completes a program initiated by Mohsen and Mourtada. The admissible neighborly partitions involve an unusual mod 3 condition on the parts.
Generating functions and orthogonality (pdf)
Abstract: Askey used an idea of Hermite, the polynomiality of an integral, to prove Jacobi's generating function for Jacobi polynomials. The same idea is used for generating functions of the Al-Salam-Chihara, continuous dual q-Hahn, Wilson, and Askey-Wilson polynomials.
The Ariki-Koike algebras and Rogers-Ramanujan type partitions (with S. Chern, Z. Li, T. Xue, and Ae Ja Yee)
Three families of q-Lommel polynomials (with Jang Soo Kim, pdf)
AbstractThree q-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey-Wilson polynomials. Some combinatorial results are given, including a general result when RI moments coincide with orthogonal polynomial moments. The combinatorial results use weighted Motzkin paths, Schroder paths, and parallelogram polyominoes.
Orthogonal polynomials of type R_I (with Jang Soo Kim, pdf)
Abstract A combinatorial theory for type R_I orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are given.
Dyson's crank and the mex (with B. Hopkins and J. Sellers, pdf)
Abstract Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.
Ratios of Jackson's q-Bessel functions and q-Lommel polynomials (with Jang Soo Kim, pdf)
Abstract In 1993 Delest and Fedou showed that a generating function for connected skew shapes is given as a ratio J_v+1/J_v of Jackson's third q-Bessel functions when a parameter v is zero. They conjectured that when v is a nonnegative integer the coefficients of the generating function are rational functions whose numerator and denominator are polynomials in q with nonnegative integer coefficients, which is a q-analog of Kishore's 1963 result on Bessel functions. The first main result of this paper is a proof of the conjecture of Delest and Fedou. The second main result is a refinement of the result of Delest and Fedou: a generating function for connected skew shapes with bounded diagonals is given as a ratio of q-Lommel polynomials introduced by Koelink and Swarttouw. It is also shown that the ratio J_{v+1}/J_v has two different continued fraction expressions, which give respectively a generating function for moments of orthogonal polynomials of type R_I and a generating function for moments of usual orthogonal polynomials. Orthogonal polynomial techniques due to Flajolet and Viennot are used.
Some problems An informal summary of problems I have failed to do, but I have not given up. June 2019.
Marking and shifting a part in partition theorems (with K. O'Hara)
Abstract Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers-Ramanujan identities, the G"ollnitz-Gordon identities, Euler's odd=distinct theorem, and the Andrews-Gordon identities. Generalizations of each of these theorems are given where a single part is ``marked" or weighted. This allows a single part to be replaced by a new larger part, ``shifting" a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts.
Character sheaves for symmetric pairs I wrote an appendix to this paper by K. Vilonen and T. Xue
Orthogonal polynomials and Smith normal form (with Alex Miller, pdf)
Abstract Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schroder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q-double factorials, as well as generating functions for permutations with eight statistics.
On q-integrals over order polytopes (with Jang Soo Kim, pdf)
Abstract A combinatorial study of multiple q-integrals is developed. This includes a q-volume of a convex polytope, which depends upon the order of q-integration. A multiple q-integral over an order polytope of a poset is interpreted as a generating function of linear extensions of the poset. Specific modifications of posets are shown to give predictable changes in q-integrals over their respective order polytopes. This method is used to combinatorially evaluate some generalized q-beta integrals. One such application is a combinatorial interpretation of a q-Selberg integral. New generating functions for generalized Gelfand-Tsetlin patterns and reverse plane partitions are established. A q-analogue to a well known result in Ehrhart theory is generalized using q-volumes and q-Ehrhart polynomials.
Binomial Andrews-Gordon-Bressoud identities (pdf)
Abstract Binomial versions of the Andrews-Gordon-Bressoud identities are given.
On the cohomology of Fano varieties and the Springer correspondence (pdf) I wrote an appendix to this paper by T.-H. Chen, K. Vilonen, and T. Xue
Asymptotics of the number of involutions in finite classical groups (with Jason Fulman and Robert Guralnick, pdf)
Abstract Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new ``sum=product" identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.
Moments of orthogonal polynomials and combinatorics (with Sylvie Corteel and Jang Soo Kim, pdf)
Abstract: This paper is a survey on combinatorics of moments of orthogonal polynomials and linearization coefficients. This area was started by the seminal work of Flajolet followed by Viennot at the beginning of the 1980's. Over the last 30 years, several tools were conceived to extract the combinatorics and compute these moments. A survey of these techniques is presented, with applications to polynomials in the Askey scheme.
On the distribution of the number of fixed vectors for the finite classical groups (with Jason Fulman, pdf)
Abstract Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these distributions, and find exactly how large the rank of the group has to be in order for the moment to stabilize to its limiting value. The proofs require a subtle use of some q-series identities. We also point out connections with orthogonal polynomials.
The combinatorics of associated Laguerre polynomials (with Jang Soo Kim, pdf)
Abstract The explicit double sum for the associated Laguerre polynomials is derived combinatorially. The moments are described using certain statistics on permutations and permutation tableaux. Another derivation of the double sum is provided using only the moment generating function.
Refinements of the Rogers-Ramanujan identities The Official Journal link (with K. O'Hara, pdf)
Abstract Refinements of the classical Rogers-Ramanujan identities are given in which some parts are weighted. Combinatorial interpretations refining MacMahon's results are corollaries.
Invariants of GLn(F q) in polynomials mod Frobenius powers (with J. Lewis, V. Reiner, pdf)
Abstract Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups, one for invariants mod Frobenius powers of the irrelevant ideal, one for cofixed spaces of polynomials.
Expansions in Askey-Wilson polynomials (with M. Ismail, pdf)
Abstract We give a general expansion formula of functions in the Askey--Wilson polynomials and using the Askey--Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey--Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q-Jacobi type polynomials. We also give a new generating function for Askey--Wilson polynomials and a new evaluation for specialized Askey--Wilson polynomials.
What is cyclic sieving? (with V. Reiner and D. White, pdf)
(This is a self-explanatory blurb for the Notices of the AMS.)
The q=-1 phenomenon via homology concentration (with P. Hersh and J. Shareshian, pdf)
Abstract We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.
Bootstrapping and Askey-Wilson polynomials (with Jang Soo Kim, pdf)
Abstract The mixed moments for the Askey-Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey-Wilson generating function. Modified generating functions of orthogonal polynomials are shown to generate polynomials satisfying recurrences of known degree greater than three. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.
Reflection factorizations for Singer cycles (with Joel Lewis and Vic Reiner, pdf)
Abstract The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^{n-1}. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.
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.
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.
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.
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.
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.
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.
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.
(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 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)
q-analogues of Euler's Odd=Distinct Theorem (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.
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.
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.
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.
Summable sums of hypergeometric series (ps)
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.
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.
Enumeration and special functions (ps) pdf version(pdf is the latest version) dvi version
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.
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.
Fake Gaussian sequences (ps) pdf version
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.