We consider the Lommel functions ,() for different values of the parameters ( , ). We show that if ( , ) are half integers, then it is possible to describe these functions with an explicit combination of polynomials a...
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We consider the Lommel functions ,() for different values of the parameters ( , ). We show that if ( , ) are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Pade approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, whenis an integer, ,() can be written as an integral involving an explicit ( 1 combination of trigonometric functions. A closed formula for 2 1 2 +,12-;+12;sin(2 )21JJ withan integer is given.
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for ...
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HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions F-D of r variables;and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions F-S of three variables. Both functions are related with one-loop Feynman diagrams. Program summary Program title: HYPERDIRE Catalogue identifier: AEPP_v3_0 Program summary URL: http://***/summaries/AEPP_v3_*** Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 310 No. of bytes in distributed program, including test data, etc.: 7666 Distribution format: *** Programming language: Mathematica Computer: All computers running Mathematica Operating system: All operating systems running Mathematica Classification: 4.4. Catalogue identifier of previous version: AEPP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 184 (2013) 2332 Does the new version supersede the previous version?: No. It is an extension to the previous program, which performs reductions of hypergeometric functions F-p(p,) F-1, F-2, F-3 and F-4 Nature of problem: Reduction of hypergeometric functions F-D and F-S to the set of basis functions Solution method: Differential reduction Reasons for new version: New algorithms for the reduction of multivariable Lauricella functions, Horn functions, (hypergeometric functions F-D and F-S) Summary of revisions: HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions F-D of r variables;and the second one
In this lecture, uses and influences of hypergeometric functions (both Kummer's and confluent hypergeometric functions) in the study of geometric function theory and its generalizations are discussed, as a survey ...
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In this lecture, uses and influences of hypergeometric functions (both Kummer's and confluent hypergeometric functions) in the study of geometric function theory and its generalizations are discussed, as a survey of the author's work. (C) 2006 Published by Elsevier Inc.
We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, resp...
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We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively.
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We st...
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We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport-Hasse multiplication formula for Gauss sums.
In this work we present an explicit relation between the number of points on a family of algebraic curves over F-q and sums of values of certain hypergeometric functions over F-q. Moreover, we show that these hypergeo...
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In this work we present an explicit relation between the number of points on a family of algebraic curves over F-q and sums of values of certain hypergeometric functions over F-q. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over F-q in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.
We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order q = 1 (mod 4) in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric p...
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We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order q = 1 (mod 4) in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's p-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.
It is shown that the roots of the trinomial equation x(n) - x + t = 0 are finite sums of generalized hypergeometric functions for each positive integer n. As a consequence, we demonstrate a class of algebraic identiti...
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It is shown that the roots of the trinomial equation x(n) - x + t = 0 are finite sums of generalized hypergeometric functions for each positive integer n. As a consequence, we demonstrate a class of algebraic identities satisfied by certain of these functions. (C) 2000 Elsevier Science B.V. All rights reserved.
We introduce hypergeometric functions related to projective Schur functions Q(lambda) and describe their properties. Linear equations, integral representations, and Pfaffian representations are obtained. These hyperge...
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We introduce hypergeometric functions related to projective Schur functions Q(lambda) and describe their properties. Linear equations, integral representations, and Pfaffian representations are obtained. These hypergeometric functions are vacuum expectations of free fermion fields and are therefore tau functions of the so-called BKP hierarchy of integrable equations.
A method is proposed of evaluation of symbol and/or bit error probabilities for coherent diversity receiving of multipositional signal constructions in communication channel with fadings, which are described with the ...
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A method is proposed of evaluation of symbol and/or bit error probabilities for coherent diversity receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.
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