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82
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 581 (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...
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 205 (15 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
The Classical Moment Problem as a SelfAdjoint Finite Difference Operator
, 1998
"... This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong r ..."
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Cited by 150 (8 self)
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This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger.
ChernSimons matrix models and StieltjesWigert polynomials
 J. Math. Phys
, 2007
"... Abstract. Employing the random matrix formulation of ChernSimons theory on Seifert manifolds, we show how the StieltjesWigert orthogonal polynomials are useful in exact computations in ChernSimons matrix models. We construct a biorthogonal extension of the StieltjesWigert polynomials, not availa ..."
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Cited by 20 (7 self)
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Abstract. Employing the random matrix formulation of ChernSimons theory on Seifert manifolds, we show how the StieltjesWigert orthogonal polynomials are useful in exact computations in ChernSimons matrix models. We construct a biorthogonal extension of the StieltjesWigert polynomials, not available in the literature, necessary to study ChernSimons matrix models when the geometry is a lens space. We also discuss several other results based on the properties of the polynomials: the equivalence between the StieltjesWigert matrix model and the discrete model that appears in q2D YangMills and the relationship with RogersSzegö polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the StieltjesWigert ensemble. 1.
SUSLOV: The qharmonic oscillator and the AlSalam and Carlitz polynomials
 Letters in Mathematical Physics
, 1993
"... Abstract. One more model of a qharmonic oscillator based on the qorthogonal polynomials of AlSalam and Carlitz is discussed. The explicit form of qcreation and qannihilation operators, qcoherent states and an analog of the Fourier transformation are established. A connection of the kernel of t ..."
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Cited by 18 (2 self)
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Abstract. One more model of a qharmonic oscillator based on the qorthogonal polynomials of AlSalam and Carlitz is discussed. The explicit form of qcreation and qannihilation operators, qcoherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of selfdual biorthogonal rational functions is observed.
On powers of Stieltjes moment sequences
, 2005
"... For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment s ..."
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Cited by 17 (5 self)
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For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment sequence (n!) c, corresponding to the Bernstein function f(s) = s. Nevertheless there always exists a unique product convolution semigroup (ρc)c>0 such that ρc has moments s c n. We apply the indeterminacy of (n!) c for c> 2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p ≥ 3.
The Atiyah–Hitchin bracket and open Toda Lattice
 Journal of Geometry and Physics
, 2003
"... The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three–diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one–to–one correspondence with their Weyl functions. These are rational functions mapping the upper half–plane i ..."
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Cited by 17 (6 self)
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The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three–diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one–to–one correspondence with their Weyl functions. These are rational functions mapping the upper half–plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic–geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear–fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.
Some Aspects of Hankel Matrices in Coding Theory and Combinatorics
 J. Comb
, 2001
"... Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be # 1#i#j#k i+j+2n i+j and related them to the Bender  Knuth conjecture. The similar determinant formula # 1#i#j#k i+j1+2n i+j1 can be shown to ho ..."
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Cited by 16 (0 self)
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Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be # 1#i#j#k i+j+2n i+j and related them to the Bender  Knuth conjecture. The similar determinant formula # 1#i#j#k i+j1+2n i+j1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coe#cients # 2m+1 m # . Generalizing the Catalan numbers in a di#erent direction, it can be shown that determinants of Hankel matrices consisting of numbers 1 3m+1 # 3m+1 m # yield an alternate expression of two Mills  Robbins  Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan  like numbers. The well  known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp  Massey algorithm in Coding Theory, which can be applied in order to calculate the coe#cients in the three  term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.