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HYP and HYPQ
Abstract
We give a brief description of the main features of the Mathematica packages HYP and HYPQ. HYP allows a convenient handling of binomial sums and hypergeometric series, while its "q-analogue", the package HYPQ, allows a convenient handling of q-binomial sums and basic hypergeometric series. Both packages are available by anonymous ftp at pap.univie.ac.at or on WWW, http://radon.mat.univie.ac.at/people/kratt.
... At this point, for illustrative purposes, let us list the first ten half-integral and ten integral rational values (generalized separability probabilities), along with their approximate numerical values. To simplify the cumbersome (several-page) output yielded by the Mathematica FindSequenceFunction command, we employed certain of the 'contiguous rules' for hypergeometric functions listed by Krattenthaler in his package HYP [42] (cf [43]). Multiple applications of the rules C14 and C18 there, together with certain gamma function simplifications suggested by C Dunkl, led to the rather more compact formula displayed in figure 3. ...
We report major advances in the research program initiated in "Moment-Based
Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability
Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability
probability function P(alpha) is put forth, yielding for generic
(9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit
systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems,
P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and
colleagues by applying Zeilberger's algorithm ("creative telescoping") to a
fully equivalent--but considerably more complicated--expression containing six
7F6 hypergeometric functions (all with argument 27/64 = (3/4)^3). That
hypergeometric form itself had been obtained using systematic, high-accuracy
probability-distribution-reconstruction computations. These employed 7,501
determinantal moments of partially transposed four-by-four density matrices,
parameterized by alpha = 1/2, 1, 3/2,...,32. From these computations, exact
rational-valued separability probabilities were discernible. The
(integral/half-integral) sequences of 32 rational values, then, served as input
to the Mathematica FindSequenceFunction command, from which the initially
obtained hypergeometric form of P(alpha) emerged.
Computer algebra methods within the scope of the holonomic systems approach provide a versatile toolbox to integration problems in the context of Feynman diagrams. This is demonstrated with the aid of several benchmark problems, ranging from hypergeometric series evaluations to Bessel integrals of sunrise diagrams.
This article presents an algorithmic theory of contiguous relations. Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non-terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.
We report new hypergeometric constructions of rational approximations to Catalan's constant, , and , their connection with already known ones, and underlying `permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.
We present a further extension of the HYPERDIRE project, which is devoted to the creation of a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations. Specifically, we present the implementation of the differential reduction for the Lauricella function F-C of three variables. Program summary Program title: HYPERDIRE Catalogue identifier: AEPP_v4_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEPP_v4_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 243461 No. of bytes in distributed program, including test data, etc.: 61610782 Distribution format: tar.gz Programming language: Mathematica. Computer: All computers running Mathematica. Classification: 4.4. Does the new version supersede the previous version?: No, it significantly extends the previous version. Nature of problem: Reduction of hypergeometric function F-C of three variables to a set of basis functions. Solution method: Differential reduction. Reasons for new version: The extension package allows the user to handle the Lauricella function F-C of three variables. Summary of revisions: The previous version goes unchanged. Running time: Depends on the complexity of the problem.
This work considers the solution of atomic three-electron integrals that involve explicit inter-electronic separation factors. The traditional Sack expansion is replaced by an alternative expansion, which avoids the breakup of the radial integrals into a number of factors depending on the lesser and greater of the inter-electronic separation distances. The present approach avoids the N! increase in the number of required auxiliary functions, where N is the number of electrons. The new approach leads to additional infinite summations, but these summations either converge very quickly in a serial calculation, or can be effectively dealt with using the massively parallel architecture of a graphics processing unit. The new auxiliary functions that arise are discussed in detail.
We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over Q spanned by 1, zeta(m), zeta(m + 2),..., zeta(m + 2h), where m and h are integers such that m >= 2 and h >= 0. In particular, we immediately get the following results as corollaries: at least one of the eight numbers zeta(5), zeta(7),..., zeta(19) is irrational, and there exists an odd integer j between 5 and 165 such that 1, zeta(3) and zeta(j) are linearly independent over Q. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of zeta(4). The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers zeta(5),zeta(7),zeta(9),zeta(II) is irrational.
Integrals and sums involving special functions are in constant demand in
applied mathematics. Rather than refer to a handbook of integrals or to a
computer algebra system, we present a do-it-yourself systematic approach
that shows how the evaluation of such integrals and sums can be made as
simple as possible. Illustrating our method, we present several examples of
integrals of Poisson type, Fourier transform, as well as integrals involving
product of Bessel functions. We also obtain a new identity involving the
sums of .
We build a chainid=ϑ0ϑ1⋯ϑ⌊n2⌋−1ϑ⌊n2⌋ of nested involutions in the Bruhat ordering of Sn, with ϑ⌊n2⌋ the maximal element for the Bruhat order, and we study the cardinality of the Bruhat intervals [ϑj,ϑk] for all 0⩽jk⩽⌊n2⌋, and the number of permutations incomparable with ϑt, for all 0⩽t⩽⌊n2⌋.
Although most of the symmetry groups or “invariance groups” associated with two term transformations between (basic) hypergeometric series have been studied and identified, this is not the case for the most general transformation formulae in the theory of basic hypergeometric series, namely Bailey's transformations for φ910-series. First, we show that the invariance group for both Bailey's two term transformations for terminating φ910-series and Bailey's four term transformations for non-terminating φ910-series (rewritten as a two term transformation of a so-called Φ-series) is isomorphic to the Weyl group of type E6. We continue our recent research concerning the group structure underlying three term transformations [S. Lievens, J. Van der Jeugt, Invariance groups of three term transformations for basic hypergeometric series, J. Comput. Appl. Math. 197 (2006) 1–14] and demonstrate that the group associated with a three term transformation between these Φ-series, each admitting Bailey's two term transformation, is the Weyl group of type E7. We do this by giving a description of the root system of type E7 that allows to find a transformation between equivalent three term identities in an easy way. A computation shows that there are five, essentially different, three term transformations between these Φ-series; we give an explicit form of each of these five transformations in an elegant way. To our knowledge only one of these transformations has appeared in the literature.
We show that certain two-term transformation formulas between basic hypergeometric series can easily be described by means of invariance groups. For the transformations of nonterminating 3ϕ2 series, and those of terminating balanced 4ϕ3 series, these invariance groups are symmetric groups. For transformations of 2ϕ1 series the invariance group is the dihedral group of order 12. Transformations of terminating 3ϕ2 series are described by means of some subgroup of S6, and finally the invariance group of transformations of very-well-poised nonterminating 8ϕ7 series is shown to be isomorphic to the Weyl group of a root system of type D5. © 1999 American Institute of Physics.
The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-order differential equations satisfied by the involved polynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized linearization formulae involving these families. The connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions, which often reduce to a single function of the same type; when possible, these functions are evaluated in closed form. In some cases, sign properties of the coefficients such as positivity or non-negativity conditions are derived as a by-product from their resulting explicit representations.
We study the Schrödinger equation of the hydrogen atom in the (arbitrarily) strong magnetic field in two dimensions, which is an integrable and separable system. The energy spectrum is very interesting as it has infinitely many accumulation points located at the values of the Landau energy levels of a free electron in the uniform magnetic field. In the polar coordinates, the canonical (not kinetic!) angular momentum has a precise eigenvalue and we have the one-dimensional radial Schrödinger equation, which is an ordinary second-order differential equation whose analytical exact solution is unknown. We describe the qualitative properties of the energy spectrum, and we propose a semi-analytical method to numerically calculate the eigenenergies (the representation matrix of the Hamiltonian in the Landau basis is analytically calculated and is exactly known). Also, we use a number of useful analytical approximation methods, such as semiclassical approximations, the perturbation method, the variational method and the Taylor power expansion of the potential around the minimum to estimate the ground-state energy and the higher levels.
We study the Schrödinger equation for the hydrogen atom in an arbitrarily strong magnetic field in two dimensions, which is an integrable and separable system. The energy spectrum is very interesting as it has infinitely many accumulation points located at the values of the Landau energy levels of a free electron in the uniform magnetic field. In the polar coordinates the canonical (not kinetic!) angular momentum has a precise eigenvalue and we have the one dimensional radial Schrödinger equation which is an ordinary second order differential equation whose analytic exact solution is unknown. The problem is reduced to a linear three-term recurrence difference equation whose solution is unknown. We describe the qualitative properties of the energy spectrum and propose a semi-analytic method to numerically calculate the eigenenergies.
The study of invariance groups associated with two term transformations between (basic) hypergeometric series has received its fair share of attention, and indeed, for most two term transformations between (basic) hypergeometric series, the underlying invariance group is explicitly known. In this article, we study the group structure underlying some three term transformation formulae, thereby giving an explicit and simple realization that is helpful in determining whether two of these transformation formulae are equivalent or not.
This paper is an exposition of different methods for computing closed forms of definite sums. The focus is on recently-developed results on computing closed forms of definite sums of hypergeometric terms. A design and an implementation of a software package which incorporates these methods into the computer algebra system Maple are described in detail.
In this article, hypergeometric identities (or transformations) for p+1 F p -series and for Kampà e de Fà eriet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series, respectively, using the beta integral method in an automated manner, based on the Mathematica package HYP. As a result, we obtain some known and some identities which seem to not have been recorded before in literature.
In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding and Lecture Notes, Vol. 22, American Mathematical Society, 1999, pp. 183–200) have established a continuous orthogonality relation and some other properties of a 21-Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the Bessel-type orthogonality found in Ismail et al. (1999) at the 21-level has really a general character and can be extended up to the 87-level. Very-well-poised 87-functions are known as a nonterminating version of the classical Askey–Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008–1016; Memoirs Amer. Math. Soc. Number
319 (1985)). Askey's conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877–5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal 87-functions. Another type of the orthogonality relation for a very-well-poised 87-function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25–55.
In the paper, an implementation in the computer algebra system Maple of an approach to summation based on hypergeometric pattern
matching is considered. In spite of a number of existing modern summation algorithms, this approach remains to be valuable,
since, if applicable, it allows one to find the result much faster than other approaches. The implementation is performed
in the form of package Pattern Matching Summation. The structure of the package, pattern representation, possibility of extending
the list of patterns, and matching mechanism are described. Examples of using the package are presented. Specific features
of using summation formulas from handbooks as the patterns are considered. To obtain correct results, additional analysis
is often required to determine the cases where the given formula is applicable.
We construct integral bases for the SO(3)SO(3)-TQFT-modules
of surfaces in genus one and two at roots of unity of
prime order and show that the corresponding mapping class group
representations preserve a unimodular Hermitian form over a ring
of algebraic integers. For higher genus surfaces the Hermitian form
sometimes must be non-unimodular. In one such case,
genus three at a fifth root of unity,
we still give an explicit basis.
We compute the number of rhombus tilings of a hexagon with side lengths a, b, c, a, b, c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a, b, c, a, b, c which contain the “almost central” rhombus above the centre.
The purpose of this paper is to introduce the RRtools and recpf Maple packages which were developed by the author to assist in the discovery and proof of finitizations of identities of the Rogers–Ramanujan type.
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: one, pfq, is relevant for manipulations of hypergeometric functions F-p+1(p), and the other, AppellF1F4, for manipulations with Appell hypergeometric functions F-1, F-2, F-3, F-4 of two variables. Program summary Program title: HYPERDIRE Catalogue identifier: AEPP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEPP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence No. of lines in distributed program, including test data, etc.: 3428 No. of bytes in distributed program, including test data, etc.: 394247 Distribution format: tar.gz Programming language: Mathematica. Computer: All computers running Mathematica. Operating system: Operating systems running Mathematica. Classification: 4.4. Nature of problem: Reduction of hypergeometric functions F-p(p-1), F-1, F-2, F-3, F-4 to sets of basis functions. Solution method: Differential reduction Running time: Dependent on the complexity of problem. The examples provided take less than a minute to run. (c) 2013 Elsevier B.V. All rights reserved.
We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving
explicit closed formulas for Whitney numbers of the distributive lattices of
order ideals of the fence poset and crown poset. Moreover, we get explicit
closed formulas for Whitney numbers of lattices of order ideals of fences with
higher asymmetric peaks.
In 1879, Thomae discussed the relations between two generic hypergeometric -series with argument 1. It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the hypergeometric series). More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over between two convergent -series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many -relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain -transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and -series.
A hypergeometric identity equating a triple sum to a single sum, originally found by Gelfand, Graev and Retakh [Russian Math. Surveys 47 (1992), 1-88] by using systems of differential equations, is given hypergeometric proofs. As a bonus, several q-analogues can be derived.
In the beginning of the 1950's, Wigner introduced a fundamental deformation from the canonical quantum mechanical harmonic oscillator, which is nowadays sometimes called a Wigner quantum oscillator or a parabose oscillator. Also, in quantum mechanics the so-called Wigner distribution is considered to be the closest quantum analogue of the classical probability distribution over the phase space. In this article, we consider which definition for such distribution function could be used in the case of non-canonical quantum mechanics. We then explicitly compute two different expressions for this distribution function for the case of the parabose oscillator. Both expressions turn out to be multiple sums involving (generalized) Laguerre polynomials. Plots then show that the Wigner distribution function for the ground state of the parabose oscillator is similar in behaviour to the Wigner distribution function of the first excited state of the canonical quantum oscillator.
The application of basic hypergeometric functions to partitions is briefly discussed. It is shown how bilateral basic hypergeometric series have application in number theory (apart from partitions). The relationship between finite vector spaces and basic hypergeometric functions is discussed; the theory of Eulerian differential operators, which arose from the combinatorial theory of finite vector spaces, and which has interesting applications to basic hypergeometric series, is briefly described. Finally, the usefulness of both ordinary and basic hypergeometric functions in the proof of and classification of combinatorial identities, and some of the recent applications of basic hypergeometric functions in physics are discussed.
The problem of proving a particular binomial identity is taken as an opportunity to discuss various aspects of this field and to discuss various proof techniques in an exemplary way. In particular, the unifying role of the hypergeometric nature of binomial identities is underlined. This aspect is also basic for combinatorial models and techniques, developed during the last decade, and for the recent algorithmic proof procedures.“Much of mathematics comes from looking at very simple examples from a more general perspective. Hypergeometric functions are a good example of this.”
An algorithm for definite hypergeometric summation is given. It is based, in a non-obvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems.
We observe that many special functions are solutions of so-called holonomic systems. Bernstein's deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps. This is partially substantiated by an algorithm that proves terminating hypergeometric series identities, and that is given both in English and in MAPLE.
Given a summand a(n), we seek the "indefinite sum" S(n) determined (within an additive constant) by [Formula: see text] or, equivalently, by [Formula: see text] An algorithm is exhibited which, given a(n), finds those S(n) with the property [Formula: see text] With this algorithm, we can determine, for example, the three identities [Formula: see text] [Formula: see text] and [Formula: see text] and we can also conclude that [Formula: see text] is inexpressible as S(m) - S(0), for any S(n) satisfying Eq. 2.
Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35
- G Gasper
- M Rahman
G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its
Applications 35, Cambridge University Press, Cambridge, 1990.