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38
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...
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 98 (9 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 89 (7 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.
Optimal inequalities in probability theory: A convex optimization approach
 SIAM Journal of Optimization
"... Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provi ..."
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Cited by 66 (10 self)
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Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NPhard to find tight bounds for k ≥ 4 and Ω = Rn and for k ≥ 2 and Ω = Rn +, when the data in the problem is rational. For k =1andΩ=Rn + we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Ω are unions of convex sets, over which linear functions can be optimized efficiently. For the case k =2andΩ=Rn, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently. Key words. optimization probability bounds, Chebyshev inequalities, semidefinite optimization, convex
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 14 (5 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.
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 12 (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.
Spectral theory and special functions
"... Abstract. A short introduction to the use of the spectral theorem for selfadjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on ℓ 2 (Z≥0), leading to a proof of Favard’s theorem statin ..."
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Cited by 11 (6 self)
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Abstract. A short introduction to the use of the spectral theorem for selfadjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on ℓ 2 (Z≥0), leading to a proof of Favard’s theorem stating that polynomials satisfying a threeterm recurrence relation are orthogonal polynomials. We discuss the link to the moment problem. In the second example, the spectral theorem is applied to Jacobi operators on ℓ 2 (Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operator on ℓ 2 (Z) to those of two Jacobi operators on ℓ 2 (Z≥0). For two examples of Jacobi operators on ℓ 2 (Z), namely for the Meixner, respectively MeixnerPollaczek, functions, related to the associated Meixner, respectively MeixnerPollaczek, polynomials, and for the second order hypergeometric qdifference operator, we calculate the spectral measure explicitly. This gives explicit (generalised) orthogonality relations for hypergeometric and basic hypergeometric series. Contents. 1.
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 11 (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.
A transformation from Hausdorff to Stieltjes moment sequences
 Ark. Mat
, 2004
"... Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on e ..."
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Cited by 10 (6 self)
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Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on exponential functionals of L'evy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to qseries. 2000 Mathematics Subject Classification: primary 44A60; secondary 33D65. Keywords: moment sequence, qseries. 1 Introduction and main results In his fundamental memoir [23] Stieltjes characterized sequences of the form sn = Z 1