D-finiteness: algorithms and applications

• Authors: Bruno Salvy
• Venue: Proceedings of the 2005 international symposium on Symbolic and algebraic computation
• Year: 2005

Differentially finite series are solutions of lineardifferential equations with polynomial coefficients. P-recursivesequences are solutions of linear recurrences with polynomialcoefficients. Corresponding notions are obtained by replacingclassical differentiation or difference operators by theirq-analogues. All these objects share numerous propertiesthat are described in the framework of "D-finiteness". Our aim inthis area is to enable computer algebra systems to deal in analgorithmic way with a large number of special functions andsequences. Indeed, it can be estimated that approximately 60% ofthe functions described in Abramowitz & Stegun's handbook [1]fall into this category, as well as 25% of the sequences inSloane's encyclopedia [20,21].In a way, D-finite sequences or series are non-commutativeanalogues of algebraic numbers: the role of the minimal polynomialis played by a linear operator.Ore [14] described a non-commutativeversion of Euclidean division and extended Euclid algorithm forthese linear operators (known as Ore polynomials). In the same wayas in the commutative case, these algorithms make several closureproperties effective (see[22]). It follows that identities betweenthese functions or sequences can be proved or computedautomatically. Part of the success of the gfunpackage [17] comes from an implementation of these operations.Another part comes from the possibility ofdiscovering such identities empirically, withPadé-Hermite approximants on power series [2] taking theplace of the LLL algorithm on floating-point numbers.The discovery that a series is D-finite is also important fromthe complexity point of view: several operations can be performedon D-finite series at a lower cost than on arbitrary power series.This includes multiplication, but also evaluation at rationalpoints by binary splitting [4]. A typical application is thenumerical evaluation of π in computer algebra systems; we giveanother one in these proceedings [3].Also, the local behaviour of solutions of linear differentialequations in the neighbourhood of their singularities is wellunderstood [9] and implementations of algorithms computing thecorresponding expansions are available [24, 13]. This gives accessto the asymptotics of numerous sequences or to analytic proofs thatsequences or functions cannot satisfy such equations [10]Results ofa more algebraic nature are obtained by differential Galois theory[18, 19], which naturally shares many subroutines with algorithmsfor D-finite series.The truly spectacular applications of D-finiteness come from themultivariate case: instead of series or sequences, one works withmultivariate series or sequences, or with sequences of series orpolynomials,.... They obey systems of linear operators that may beof differential, difference, q-difference ormixed types, with the extra constraint that a finite number ofinitial conditions are sufficient to specify the solution. This isa non-commutative analogue of polynomial systems with a finitenumber of solutions. It turns out that, as in the polynomial case,Gröbner bases give algorithmic answers to many decisionquestions, by providing normal forms in a finite dimensional vectorspace. This has been observed first in the differential case [11,23] and then extended to the more general multivariate Ore case[8].A crucial insight of Zeilberger [27, 15] is that elimination inthis non-commutative setting computes definite integrals or sums.This is known as creative telescoping. Inthehypergeometric setting (when the quotient isa vector space of dimension1), a fast algorithm for this operationis known as Zeilberger's fast algorithm [26]. In the more generalcase, Gröbner bases are of help in this elimination. This istrue in the differential case [16, 25] and to a large extent in themore general multivariate case [8]. Also, Zeilberger's fastalgorithm has been generalized to the multivariate Ore case byChyzak [5, 6]. Still, various efficiency issues remain andphenomena of non-minimality of the eliminated operators are notcompletely understood.A further generalization of D-finite series is due to Gessel[12] who developed a theory of symmetric series. These series aresuch than when all but a finite number of their variables (in acertain basis) are specialized to0, the resulting series isD-finite in the previous sense. Closure properties under scalarproduct lead to proofs of D-finiteness (in the classical sense) forvarious combinatorial sequences. Again, algorithms based onGröbner bases make these operations effective [7].The talk will survey the nicest of these algorithms and theirapplications. I will also indicate where current work is inprogress, or where more work is needed.