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Dive into the research topics where Marko Petkovsek is active.

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Featured researches published by Marko Petkovsek.


Theoretical Computer Science | 1996

An introduction to pseudo-linear algebra

Manuel Bronstein; Marko Petkovsek

Abstract Pseudo-linear algebra is the study of common properties of linear differential and difference operators. We introduce in this paper its basic objects (pseudo-derivations, skew polynomials, and pseudo-linear operators) and describe several recent algorithms on them, which, when applied in the differential and difference cases, yield algorithms for uncoupling and solving systems of linear differential and difference equations in closed form.


Discrete Mathematics | 2000

Linear recurrences with constant coefficients: the multivariate case

Mireille Bousquet-Mélou; Marko Petkovsek

Abstract While in the univariate case solutions of linear recurrences with constant coefficients have rational generating functions, we show that the multivariate case is much richer: even though initial conditions have rational generating functions, the corresponding solutions can have generating functions which are algebraic but not rational, D-finite but not algebraic, and even non-D-finite.


international symposium on symbolic and algebraic computation | 1995

On polynomial solutions of linear operator equations

Sergei A. Abramov; Manuel Bronstein; Marko Petkovsek

The algorithm described here extends the algorithm to nd all polynomial solutions of di erential and di erence equations that was given in [1, 2] to more general operators. It also takes a more e cient approach that avoids using undetermined coe cients. This summary is based on [4]. Let K be a eld of characteristic 0 and L : K[x] ! K[x] a K-linear endomorphism of K[x]. A new algorithm is presented in [4] that nds all polynomial solutions of homogeneous equations of the form Ly = 0, of nonhomogeneous equations of the form Ly = f and of parametric nonhomogeneous equations of the form Ly = P m i=1 i f i . The endomorphisms L under consideration in the following are polynomials in one of the following operators, and with coe cients in K[x]: { the di erential operator D de ned by Df(x) = df=dx; { the di erence operator de ned by f(x) = f(x+ 1) f(x); { the q-dilation operator Q used for q-di erence equations and de ned by Qf(x) = f(qx). (In this case, q 2 K, is not zero and not a root of unity.) The interest of the new algorithm is twofold. First, numerous algorithms need to solve homogeneous, nonhomogeneous or parametric nonhomogeneous equations in K[x] as subproblems. Examples are algorithms to nd all rational, hyperexponential, geometric or Liouvillian solutions, to perform inde nite or de nite hypergeometric summation, to factorize linear operators, etc. (See for instance [5, 3, 7, 6].) Second, the algorithm that is described here has lower complexity than the usual algorithms, that are often based on undetermined coe cients. The approach here is to nd a degree bound on the solutions to be computed, and next nd recurrences to compute the coe cients of the solutions e ciently. The problem with undetermined coe cients arises with very concise equations having high degree solutions. Although the number of coe cients to be determined is high, the recurrences that are found by the new algorithm in [4] are of small order. The idea is to view the space K[x] as a subspace of a unusual space of formal power series, and to embed the space of polynomial solutions into a space of formal power series solutions.


Theoretical Computer Science | 2003

Walks confined in a quadrant are not always D-finite

Mireille Bousquet-Mélou; Marko Petkovsek

We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z2, and always stay in the quadrant x ≥ 0, y ≥ 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,-1), (-1, 2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis.


Discrete Mathematics | 1998

q -hypergeometric solutions of q -difference equations

Sergei A. Abramov; Peter Paule; Marko Petkovsek

Abstract We present algorithm qHyper for finding all solutions y ( x ) of a linear homogeneous q -difference equation, such that y ( qx )= r ( x ) y ( x ) where r ( x ) is a rational function of x . Applications include construction of basic hypergeometric series solutions, and definite q -hypergeometric summation in closed form.


Journal of Chemical Information and Computer Sciences | 2002

Three methods for calculation of the hyper-wiener index of molecular graphs

Gordon G. Cash; Sandi Klavzar; Marko Petkovsek

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.


Journal of Symbolic Computation | 1999

Multibasic and Mixed Hypergeometric Gosper-Type Algorithms

Andrej Bauer; Marko Petkovsek

Gosper?s summation algorithm finds a hypergeometric closed form of an indefinite sum of hypergeometric terms, if such a closed form exists. We extend his algorithm to the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. We also provide algorithms for finding polynomial as well as hypergeometric solutions of recurrences in the mixed case. We do not require the bases to be transcendental, but only that q1k1?qmkm?1 unless k1=?=km= 0. Finally, we generalize the concept of greatest factorial factorization to the mixed hypergeometric case.


Discrete Mathematics | 2002

Letter graphs and well-quasi-order by induced subgraphs

Marko Petkovsek

Given a word w over a finite alphabet and a set of ordered pairs of letters which define adjacencies, we construct a graph which we call the letter graph of w. The lettericity of a graph G is the least size of the alphabet permitting to obtain G as a letter graph. The set of 2-letter graphs consists of threshold graphs, unbounded-interval graphs, and their complements. We determine the lettericity of cycles and bound the lettericity of paths to an interval of length one. We show that the class of k-letter graphs is well-quasi-ordered by the induced subgraph relation, and that it has a finite set of minimal forbidden induced subgraphs. As a consequence, k-letter graphs can be recognized in polynomial time for any fixed k.


Advances in Applied Mathematics | 2002

On the structure of multivariate hypergeometric terms

Sergei A. Abramov; Marko Petkovsek

Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. We prove a slightly modified version of this conjecture in the case of several discrete variables.


international symposium on symbolic and algebraic computation | 2001

Minimal decomposition of indefinite hypergeometric sums

Sergei A. Abramov; Marko Petkovsek

We present an algorithm which, given a hypergeometric term <i>T</i>(<i>n</i>), constructs hypergeometric terms <i>T</i><subscrpt>1</subscrpt>(<i>n</i>) and <i>T</i><subscrpt>2</subscrpt>(<i>n</i>) such that <i>T</i>(<i>n</i>) = <i>T</i><subscrpt>1</subscrpt>(<i>n</i> + 1) -<i>T</i><subscrpt>1</subscrpt>(<i>n</i>) + <i>T</i><subscrpt>2</subscrpt>(<i>n</i>), and <i>T</i><subscrpt>2</subscrpt>(<i>n</i>) is minimal in some sense. This solves the decomposition problem for indefinite sums of hypergeometric terms: <i>T</i><subscrpt>1</subscrpt>(<i>n</i> + 1) - <i>T</i><subscrpt>1</subscrpt>(<i>n</i>) is the “summable part” and <i>T</i><subscrpt>2</subscrpt>(<i>n</i>) the “non-summable part” of <i>T</i>(<i>n</i>).

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Sergei A. Abramov

Russian Academy of Sciences

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Peter Zajec

University of Ljubljana

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