M-Fold Hypergeometric Solutions of Linear Recurrence Equations Revisited

Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K of characteristic zero, a term a(n) is called hypergeometric with respect to K, if the ratio a(n+1)/a(n) is a rational function over K. The solutions space of holonomic recurrence equations gained more interest in the 1990s from the well known Zeilberger's algorithm. In particular, algorithms computing the subspace of hypergeometric term solutions which covers polynomial, rational, and some algebraic solutions of these equations were investigated by Marko Petkov\v{s}ek (1993) and Mark van Hoeij (1999). The algorithm proposed by the latter is characterized by a much better efficiency than that of the other; it computes, in Gamma representations, a basis of the subspace of hypergeometric term solutions of any given holonomic recurrence equation, and is considered as the current state of the art in this area. Mark van Hoeij implemented his algorithm in the Computer Algebra System (CAS) Maple through the command $LREtools[hypergeomsols]$. We propose a variant of van Hoeij's algorithm that performs the same efficiency and gives outputs in terms of factorials and shifted factorials, without considering certain recommendations of the original version. We have implementations of our algorithm for the CASs Maxima and Maple. Such an implementation is new for Maxima which is therefore used for general-purpose examples. Our Maxima code is currently available as a third-party package for Maxima. A comparison between van Hoeij's implementation and ours is presented for Maple 2020. It appears that both have the same efficiency, and moreover, for some particular cases, our code finds results where $LREtools[hypergeomsols]$ fails.

In 1992, Koepf proposed a symbolic approach to compute power series. This algorithm was extended for a larger family of expressions thanks to Petkovsek’s and van Hoeij’s algorithms (1993 and 1998) which compute hypergeometric term solutions of any given holonomic recurrence equation (RE). Mark van Hoeij’s algorithm whose outputs are bases is available in Maple through the command LREtools[hypergeomsols], and Koepf’s algorithm through convert and the built-in module FormalPowerSeries. LREtools[hypergeomsols] is internally used by convert/FormalPowerSeries.However, using van Hoeij’s algorithm one cannot compute m-fold hypergeometric term solutions of holonomic REs, for integers \(m>1\). Given a field K of characteristic zero, a term a(n) is said to be m-fold hypergeometric if the term ratio \(a(n+m)/a(n)\) is rational over K. Note that the hypergeometric term case corresponds to \(m=1\). If one adds for example an odd hypergeometric function, like \(\arcsin (z)\), and an even hypergeometric function, like \(\cos (z)\) (which both are two-fold hypergeometric), then van Hoeij’s algorithm cannot find those by solving the resulting recurrence equation. Due to this limitation, the computation of many power series is missed by Maple, in particular, linear combinations of power series having m-fold hypergeometric term coefficients are generally not detected.We overcome these issues by using a new algorithm called mfoldHyper, proposed in the first author’s Ph.D. thesis to compute bases of the subspace of m-fold hypergeometric term solutions of holonomic REs. It turns out that mfoldHyper linearizes the computation of hypergeometric type power series, i.e. every linear combination of hypergeometric type power series is detected. This paper describes our Maple implementation of an algorithm that conclusively extends Maple’s capabilities regarding the computation of hypergeometric type power series.KeywordsHypergeometric type power seriesm-fold hypergeometric termHolonomic recurrence equation

In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. The first algorithm for this was given by Hendriks in 1998, and Hendriks and Singer in 1999. Several improvements have been published, including a paper by Cha and van Hoeij that reduces the combinatorial problem. But the number of combinations still depended exponentially on the number of singularities. For irreducible second order equations, we give a short and very efficient algorithm; the number of combinations is 1.

Liouvillian Solutions of Linear Differential Equations of Order Three and Higher

q-Deformation of meromorphic solutions of linear differential equations

Previous article Next article Chebyshev and $l^1 $-Solutions of Linear Equations Using Least Squares SolutionsC. S. Duris and V. P. SreedharanC. S. Duris and V. P. Sreedharanhttps://doi.org/10.1137/0705040PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. Ya. Remez, General computation methods for Čebyšev approximation. Problems with real parameters entering linearly, Izdat. Akad. Nauk Ukrainsk. SSR. Kiev, 1957, 454–, A translation is available through the Office of Technical Services, Department of Commerce, Washington, D. C. MR0088788 Google Scholar[2] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964xi+257 MR0175290 0161.12101 Google Scholar[3] H. S. Wilf, A. Ralston and , Herbert S. Wilf, Matrix inversion by the method of rank annihilationMathematical methods for digital computers, Wiley, New York, 1960, 73–77 MR0117911 Google Scholar[4] Eduard L. Stiefel, An introduction to numerical mathematics, Translated by Werner C. Rheinboldt and Cornelie J. Rheinboldt, Academic Press, New York, 1963x+286 MR0181077 0154.16702 Google Scholar[5] D. Jackson, The Theory of Approximation, American Mathematical Society, Providence, 1930 56.0936.01 Google Scholar[6] M. S. Bartlett, An inverse matrix adjustment arising in discriminant analysis, Ann. Math. Statistics, 22 (1951), 107–111 MR0040068 0042.38203 CrossrefISIGoogle Scholar[7] C. De La Vallée Poussin, Sur la méthode de l'approximation minimum, Ann. Soc. Sci. Bruxelles, 35 (1911), 1–16, Seconde Partie, Mémoires 42.0255.02 Google Scholar[8] David Moursund, Chebyshev solution of $n+1$ linear equations in n unknowns, J. Assoc. Comput. Mach., 12 (1965), 383–387 MR0182139 0142.11505 CrossrefISIGoogle Scholar[9] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York, 1966xii+259 MR0222517 0161.25202 Google Scholar[10] John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964xi+203 MR0166520 0114.27001 Google Scholar[11] Allen A. Goldstein and , Ward Cheney, A finite algorithm for the solution of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations, Pacific J. Math., 8 (1958), 415–427 MR0101505 0084.01902 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Theoretical Upperbound of the Spurious-Free Dynamic Range in Direct Digital Frequency Synthesizers Realized by Polynomial Interpolation MethodsIEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 54, No. 10 Cross Ref Minimum ℓ1, ℓ2, and ℓ∞ Norm Approximate Solutions to an Overdetermined System of Linear EquationsDigital Signal Processing, Vol. 12, No. 4 Cross Ref An algorithm for estimating the parameters in multiple linear regression model with linear constraintsComputers & Industrial Engineering, Vol. 28, No. 4 Cross Ref Minimization technique for a convex function with application to multiple regression model27 June 2007 | Optimization, Vol. 19, No. 2 Cross Ref THE CHEBYSHEV ADJUSTMENT OF A GEODETIC LEVELLING NETWORK19 July 2013 | Survey Review, Vol. 28, No. 220 Cross Ref The Chebyshev solution of the linear matrix equationAX+YB=CNumerische Mathematik, Vol. 46, No. 3 Cross Ref On a particular case of the inconsistent linear matrix equation AX+YB=CLinear Algebra and its Applications, Vol. 66 Cross Ref Discrete Chebyshev Approximation with Linear ConstraintsMichael Brannigan14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 22, No. 1AbstractPDF (1337 KB)The strict Chebyshev solution of overdetermined systems of linear equations with rank deficient matrixNumerische Mathematik, Vol. 40, No. 3 Cross Ref Computational methods of linear algebraJournal of Soviet Mathematics, Vol. 15, No. 5 Cross Ref Least absolute values estimation: an introduction27 June 2007 | Communications in Statistics - Simulation and Computation, Vol. 6, No. 4 Cross Ref An overdetermined linear systemJournal of Approximation Theory, Vol. 18, No. 3 Cross Ref Annotated Bibliography on Generalized Inverses and Applications Cross Ref Chebyshev solution of overdetermined systems of linear equationsBIT, Vol. 15, No. 2 Cross Ref A new algorithm for the Chebyshev solution of overdetermined linear systems1 January 1974 | Mathematics of Computation, Vol. 28, No. 125 Cross Ref A Finite Step Algorithm for Determining the “Strict” Chebyshev Solution to $Ax=b$C. S. Duris and M. G. Temple14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 10, No. 4AbstractPDF (787 KB)Least squares algorithms for finding solutions of overdetermined linear equations which minimize error in an abstract normNumerische Mathematik, Vol. 17, No. 5 Cross Ref Note on the fitting of non-equispaced two-dimensional dataGeoexploration, Vol. 8, No. 1 Cross Ref Solutions of overdetermined linear equations which minimize error in an abstract normNumerische Mathematik, Vol. 13, No. 2 Cross Ref Volume 5, Issue 3| 1968SIAM Journal on Numerical Analysis History Submitted:14 November 1966Accepted:06 February 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705040Article page range:pp. 491-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

SECTION 6 - Solutions of Linear Equations

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; Ş. Nas, S. Yalçinbaş, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.

Valuations of rational solutions of linear difference equations at irreducible polynomials

Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $$P \subseteq {\mathbb{R}}^m$$. As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $$m \ge 2$$. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.

A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bv(f), Bv+1(f). For second order equations, with rational function coefficients, f must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if f is a rational function. In this paper we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions.

For a prime number $$p$$ p , Bergman (Israel J Math 18:257---277, 1974) established that $$\mathrm {End}(\mathbb {Z}_{p} \times \mathbb {Z}_{p^{2}})$$ End ( Z p × Z p 2 ) is a semilocal ring with $$p^{5}$$ p 5 elements that cannot be embedded in matrices over any commutative ring. In an earlier paper Climent et al. (Appl Algebra Eng Commun Comput 22(2):91---108, 2011), the authors presented an efficient implementation of this ring, and introduced a key exchange protocol based on it. This protocol was cryptanalyzed by Kamal and Youssef (Appl Algebra Eng Commun Comput 23(3---4):143---149, 2012) using the invertibility of most elements in this ring. In this paper we introduce an extension of Bergman's ring, in which only a negligible fraction of elements are invertible, and propose to consider a key exchange protocol over this ring.

In Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) a novel method was established to estimate the minimum distance of primary affine variety codes and a thorough treatment of the Klein quartic led to the discovery of a family of primary codes with good parameters, the duals of which were originally treated in Kolluru et al (Appl Algebra Eng Commun Comput 10(6):433-464, 2000)[Ex. 3.2, Ex. 4.1]. In the present work we translate the method from Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) into a method for also dealing with dual codes and we demonstrate that for the considered family of dual affine variety codes from the Klein quartic our method produces much more accurate information than what was found in Kolluru et al (Appl Algebra Eng Commun Comput 10(6):433-464, 2000). Combining then our knowledge on both primary and dual codes we determine asymmetric quantum codes with desirable parameters.

An algorithmic approach to exact power series solutions of second order linear homogeneous differential equations with polynomial coefficients

Generating solutions of a linear equation and structure of elements of the Zelisko group