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572
The Askeyscheme of hypergeometric orthogonal polynomials and its qanalogue
, 1998
"... We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erent ..."
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Cited by 376 (4 self)
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We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erential or di#erence equation, the forward and backward shift operator, the Rodriguestype formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between di#erent classes of orthogonal polynomials listed in the Askeyscheme. In chapter 3 we list the qanalogues of the polynomials in the Askeyscheme. We give their definition, orthogonality relation, three term recurrence relation, second order di#erence equation, forward and backward shift operator, Rodriguestype formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally, in chapter 5 we...
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition
, 2003
"... ..."
private communication
"... The visibility representation(VRfor short) is aclassical representation of plane graphs. It has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. The trivial upper bound is (n−1)×(2n−5)(height × width). It is known that there exists a ..."
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Cited by 56 (3 self)
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The visibility representation(VRfor short) is aclassical representation of plane graphs. It has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. The trivial upper bound is (n−1)×(2n−5)(height × width). It is known that there exists a plane graph G with n vertices where any VR of G requires a grid of size at least 2 3n×(4 n−3). For upper bounds, it is known that 3 every plane graph has a VR with grid size at most 2 n×(2n −5), and a 3 VR with grid size at most (n − 1) × 4 n. It has been an open problem 3
Enumeration of totally positive Grassmann cells
 Harvard University, Cambridge
"... Abstract. Postnikov [7] has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr + k,n, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit g ..."
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Cited by 49 (6 self)
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Abstract. Postnikov [7] has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr + k,n, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Gr + k,n according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Gr + k,n is 1. Additionally, we use our result to produce a new qanalog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients. 1.
The Major Counting Of Nonintersecting Lattice Paths And Generating Functions For Tableaux  Summary
, 1995
"... A theory of counting nonintersecting lattice paths by the major index and generalizations of it is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the s ..."
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Cited by 46 (15 self)
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A theory of counting nonintersecting lattice paths by the major index and generalizations of it is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the starting points lie on a line parallel to x+ y = 0. In some cases these determinants can be evaluated to result into simple products. As applications we compute the generating function for tableaux with p odd rows, with at most c columns, and with parts between 1 and n. Besides, we compute the generating function for the same kind of tableaux which in addition have only odd parts. We thus also obtain a closed form for the generating function for symmetric plane partitions with at most n rows, with parts between 1 and c, and with p odd entries on the main diagonal. In each case the result is a simple product. By summing with respect to p we provide new proofs of the BenderKnuth and ...
SUMMATION AND TRANSFORMATION FORMULAS FOR ELLIPTIC HYPERGEOMETRIC SERIES
, 2000
"... Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, verywellpoised, elliptic hypergeometric series. ..."
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Cited by 45 (6 self)
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Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, verywellpoised, elliptic hypergeometric series.
Ramanujan’s theories of elliptic functions to alternative bases, Trans.Amer.Math.Soc.,347
, 1995
"... Abstract. In his famous paper on modular equations and approximations to n, Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories " in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recent ..."
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Cited by 44 (15 self)
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Abstract. In his famous paper on modular equations and approximations to n, Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories " in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.
Lie algebras and linear operators with invariant subspace
 in “Lie Algebras, Cohomologies and New Findings in Quantum Mechanics”, Contemporary Mathematics
, 1994
"... Abstract. A general classification of linear differential and finitedifference operators possessing a finitedimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as ..."
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Cited by 43 (13 self)
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Abstract. A general classification of linear differential and finitedifference operators possessing a finitedimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of some algebra of differential (difference) operators in finitedimensional representation plus an operator annihilating the finitedimensional invariant subspace. In low dimensions a classification is given by algebras sl2(R) (for differential operators in R) and sl2(R)q (for finitedifference operators in R), osp(2, 2) (operators in one real and one Grassmann variable, or equivalently, 2×2 matrix operators in R), sl3(R), sl2(R)⊕sl2(R) and gl2(R)⋉R r+1, r a natural number (operators in R 2). A classification of linear operators possessing infinitely many finitedimensional invariant subspaces with a basis in polynomials is presented. A connection to the recentlydiscovered quasiexactlysolvable spectral problems is discussed.
The cyclic sieving phenomenon
 J. Combin. Theory Ser. A
"... 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 qbinomial coefficients, PólyaRedfield theory, polygon dissections, n ..."
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Cited by 43 (11 self)
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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 qbinomial coefficients, PólyaRedfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field qanalogues. 1.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 42 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.