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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 558 (6 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
"... ..."
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 82 (21 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.
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 79 (6 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Applications of the superconformal index for protected operators and qhypergeometric identities to N=1 dual theories
, 2008
"... The results of Römelsberger for aN = 1 superconformal index counting protected operators, satisfying a BPS condition and which cannot be combined to form long multiplets, are analysed further. The index is expressible in terms of single particle superconformal characters forN = 1 scalar and vector m ..."
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Cited by 75 (2 self)
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The results of Römelsberger for aN = 1 superconformal index counting protected operators, satisfying a BPS condition and which cannot be combined to form long multiplets, are analysed further. The index is expressible in terms of single particle superconformal characters forN = 1 scalar and vector multiplets. For SQCD, involving SU(Nc) gauge groups and appropriate numbers of flavours Nf, the formula used to construct the index may be proved to give identical results for theories linked by Seiberg duality using recently proved theorems for qseries elliptic hypergeometric integrals. The discussion is also extended to KutasovSchwimmer dual theories in the large Nc, Nf limit and to dual theories with Sp(N) and SO(N) gauge groups. For the former, a transformation identity for elliptic hypergeometric integrals directly verifies that the index is the same for the electric and magnetic theory. For SO(N) theories the corresponding result may also be obtained from the same basic identity. An expansion of the index to several orders is also obtained in a form where the detailed protected operator content may be read off. Relevant mathematical results are reviewed.
Ramanujan’s theories of elliptic functions to alternative bases
 TRANS.AMER.MATH.SOC.,347 (1995)
, 1995
"... 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 67 (15 self)
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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.
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 66 (7 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.
Partition bijections, a survey
 Ramanujan J
"... Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. ..."
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Cited by 64 (13 self)
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Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.
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 61 (10 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.
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 59 (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.