Sergei A. Abramov

On polynomial solutions of linear operator equations

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.

Rational solutions of linear difference and q -difference equations with polynomial coefficients

A simple algorithm is suggested for the construction of a polynomial divisible by the denominator of any rational solution of the linear difference equation.

Rational solutions of first order linear difference systems

We propose an algorithm to compute rational function solutions for a first order system of linear difference equations with rational coefficients. This algorithm does not require preliminary uncoupling of the given system.

On solutions of linear functional systems

We describe a new direct algorithm for transforming a linear system of recurrences into an equivalent one with nonsingular leading or trailing matrix. Our algorithm, which is an improvement to the EG elimination method [2], uses only elementary linear algebra operations (ranks, kernels and determinants) to produce an equation satisfied by the degrees of the solutions with finite support. As a consequence, we can bound and compute the polynomial and rational solutions of very general linear functional systems such as systems of differential or (q—) difference equations.

q -hypergeometric solutions of q -difference equations

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.

When does Zeilberger's algorithm succeed?

A terminating condition of the well-known Zeilbergers algorithm for a given hypergeometric term T(n,k) is presented. It is shown that the only information on T(n,k) that one needs in order to determine in advance whether this algorithm will succeed is the rational function T(n,k+1)/T(n,k).

Integration of solutions of linear functional equations

We introduce the notion of the adjoint Ore ring and give a definition of an adjoint polynomial, operator and equation. We apply this for integrating solutions of Ore equations. ∗

Indefinite sums of rational functions

We propose a new algorithm for indefinite rational summa-tion which, given a rational function F(z), extracts a rational part R(z) from the indefinite sum of F(z): If H(x) is not equal to O then the denominator of this rational function has the lowest possible degree. We then solve the same probleme in the g-difference case. 1 The decomposition problem We discuss here the problem of indefinite summation of rational functions. This problem is equivalent to the problem of solving the difference equation y(z + 1) – y(z) = F(z) (1) where F(z) is a rational function over a field K of characteristic O. The decomposition problem is to find whether (1) has a rational solution, and if it does not, then to extract an additive rational part l?(x) from the solution s.t. the remaining part satisfies a simpler difference equation, where the denominator of the new right-hand side has the lowest possible degree. This gives an equality ~F(z) = R(Lz)+ ~H(z) (2) values then we can use some integer bounds s < t for our sums: t t It is probable that the publication [Abr75] was the first in which the rational and nonrational parts were computed in an algorithmic way. In [Pau93], P. Paule introduced the concept of shift-saturated extension which allows one to give some useful explicit formulas, and presented two new algorithms to construct (2). One works iteratively and is similar to Hermites algorithm. The other is an analogue of the algorithm of Ostrogradsky. Neither of them requires full factorization of polynomials. Our old algorithm [Abr75] works iteratively. In the next paragraph we discuss a new algorithm which is an analogue of the algorithm of Ostrogradsky. We will compare it with Paules algorithm. 2 A new algorithm to solve the decomposition problem We can assume that F(x) is a proper rational function (the degree of its numerator is lower than the degree of the denominator). If the degree of the numerator of F(x) is not lower than the degree of its denominator, then one can extract the polynomial part p(x) from F(z): F(z) = p(z) + F*(z), where H(z) is a rational function whose denominator has the lowest possible degree. Similar algorithms are well known where F* (x) is a proper rational function. A polynomial in integration theory: we have in mind especially the al-q(z) s.t. q(z + 1) – q(z) = p(z) can be found, so …

On the structure of multivariate hypergeometric terms

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.

Minimal decomposition of indefinite hypergeometric sums

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>).