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290
A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities
, 1993
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Generalized Hypergeometric Functions and Rational Curves on CalabiYau complete intersections in Toric Varieties
 COMMUN. MATH. PHYS
, 1995
"... We formulate general conjectures about the relationship between the Amodel connection on the cohomology of a ddimensional CalabiYau complete intersection V of r hypersurfaces V1,...,Vr in a toric variety PΣ and the system of differential operators annihilating the special generalized hypergeometr ..."
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Cited by 75 (14 self)
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We formulate general conjectures about the relationship between the Amodel connection on the cohomology of a ddimensional CalabiYau complete intersection V of r hypersurfaces V1,...,Vr in a toric variety PΣ and the system of differential operators annihilating the special generalized hypergeometric function Φ0 depending on the fan Σ. In this context, the mirror symmetry phenomenon can be interpreted as the twofold characterization of the series Φ0. First, Φ0 is defined by intersection numbers of rational curves in PΣ with the hypersurfaces Vi and their toric degenerations. Second, Φ0 is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the homolomorphic differential dform on an another CalabiYau dfold V ′ called the mirror of V. Using this generalized hypergeometric series, we propose a general construction for mirrors V ′ of V and canonical qcoordinates on the moduli spaces of CalabiYau manifolds.
A Mathematica qAnalogue of Zeilberger's Algorithm for Proving qHypergeometric Identities
, 1995
"... Besides an elementary introduction to qidentities and basic hypergeometric series, a newly developed Mathematica implementation of a qanalogue of Zeilberger's fast algorithm for proving terminating qhypergeometric identities together with its theoretical background is described. To illustrate t ..."
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Cited by 62 (11 self)
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Besides an elementary introduction to qidentities and basic hypergeometric series, a newly developed Mathematica implementation of a qanalogue of Zeilberger's fast algorithm for proving terminating qhypergeometric identities together with its theoretical background is described. To illustrate the usage of the package and its range of applicability, nontrivial examples are presented as well as additional features like the computation of companion and dual identities.
Special Values of Multiple Polylogarithms
 Sém. Bourbaki, 53 e année, 2000–2001, n ◦ 885, Mars 2001; Astéisque 282 (2002
"... Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recen ..."
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Cited by 59 (18 self)
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Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier. 1.
Arithmetic and Attractors
, 2003
"... We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the “attractor mechanism ” of N = 2 supergravity. In IIB string compactification this mechanism singles out certain “attractor varieties. ” We show that these attractor varieties are ..."
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Cited by 55 (2 self)
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We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the “attractor mechanism ” of N = 2 supergravity. In IIB string compactification this mechanism singles out certain “attractor varieties. ” We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N = 4, 8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N = 4 theories Uduality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general CalabiYau compactifications and explore further connections to arithmetic including connections to Kronecker’s Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled “Attractors and Arithmetic,” hepth/9807056.
A new matrix inverse
 Proc. Amer. Math. Soc
, 1996
"... Abstract. We compute the inverse of a specific infinitedimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type. 1. ..."
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Cited by 33 (2 self)
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Abstract. We compute the inverse of a specific infinitedimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type. 1.
HYP and HYPQ: Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively qbinomial sums and basic hypergeometric series
 Journal of Symbolic Computation
, 1995
"... Introduction Binomial series and qbinomial series, such as 1 X k=0 ` M k '` N R \Gamma k ' ; respectively 1 X k=0 q (M \Gammak)(R\Gammak) M k q N R \Gamma k q ; where the qbinomial coefficient is defined by n k q = (1 \Gamma q n )(1 \Gamma q n\Gamma1 ) \Delta ..."
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Cited by 30 (10 self)
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Introduction Binomial series and qbinomial series, such as 1 X k=0 ` M k '` N R \Gamma k ' ; respectively 1 X k=0 q (M \Gammak)(R\Gammak) M k q N R \Gamma k q ; where the qbinomial coefficient is defined by n k q = (1 \Gamma q n )(1 \Gamma q n\Gamma1 ) \Delta \Delta \Delta (1 \Gamma q n\Gammak+1 ) (1 \Gamma q k )(1 \Gamma q k\Gamma1 ) \Delta \D
Pricing equity derivatives subject to bankruptcy
 Mathematical Finance
, 2006
"... We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation who ..."
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Cited by 27 (4 self)
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We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.
THE ASSOCIATED ASKEYWILSON POLYNOMIALS
, 1991
"... We derive some contiguous relations for very wellpoised 8<^7 series and use them to construct two linearly independent solutions of the threeterm recurrence relation of the associated AskeyWilson polynomials. We then use these solutions to find explicit representations of two families of associa ..."
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Cited by 26 (3 self)
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We derive some contiguous relations for very wellpoised 8<^7 series and use them to construct two linearly independent solutions of the threeterm recurrence relation of the associated AskeyWilson polynomials. We then use these solutions to find explicit representations of two families of associated AskeyWilson polynomials. We identify the corresponding continued fractions as quotients of two very wellpoised 8^>7 series and find the weight functions.
Hypergeometrics and the Cost Structure of Quadtrees
, 1995
"... Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral repr ..."
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Cited by 25 (2 self)
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Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral representations akin to Mellin transforms leads to explicit values for various structure constants related to path length, retrieval costs, and storage occupation.