Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra Jérémy Berthomieu, Jean-Charles Faugère To cite this version: Jérémy Berthomieu, Jean-Charles Faugère. Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra. 41st International Symposium on Symbolic and Algebraic Computation, Jul 2016, Waterloo, ON, Canada. pp.95-102, 10.1145/2930889.2930926. hal-01314266 HAL Id: hal-01314266 https://hal.inria.fr/hal-01314266 Submitted on 11 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra Jérémy Berthomieu Jean-Charles Faugère [email protected] [email protected] Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d’Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75252 Paris Cedex 05, France ABSTRACT Thanks to equations (2a) and (2b), we can compute any Given several n-dimensional sequences, we first present an term of b starting with b0;0 = 1. This makes b P-recursive. algorithm for computing the Gr¨obner basis of their module Deciding if a sequence is P-recursive, as defined by Stan- of linear recurrence relations. ley [28] finds applications for instance in Combinatorics and in Computer algebra [27]. We refer the reader to the series n A P-recursive sequence (ui)i2N satisfies linear recurrence relations with polynomial coefficients in i, as defined by of papers on planar [10, 11, 12] and 3D-space walks [7, 8]. P i1 in The generating series ui ;:::;i x ··· xn of a Stanley in 1980. Calling directly the aforementioned algo- i1;:::;in2N 1 n 1 j n rithm on the tuple of sequences (i u ) n for retrieving P-recursive sequence (ui1;:::;in )(i1;:::;in)2N is a D-finite func- i i2N j the relations yields redundant relations. Since the module of tion in x1; : : : ; xn, see [19, Theorem 3.7]. A D-finite function relations of a P-recursive sequence also has an extra struc- is such that the dimension of the vector space spanned by ture of a 0-dimensional right ideal of an Ore algebra, we all its derivatives is finite over K(x1; : : : ; xn), this allows one design a more efficient algorithm that takes advantage of to characterize such functions with initial conditions and this extra structure for computing the relations. differential equations. For instance, the Dynamical Dic- [3] generates dy- Finally, we show how to incorporate Gr¨obner bases com- tionary of Mathematical Functions namically and automatically web pages on elementary and putations in an Ore algebra K ht1; : : : ; tn; x1; : : : ; xni, with commutators x x − x x = t t − t t = t x − x t = 0 special functions thanks to the equations they satisfy. Like- k ` ` k k ` ` k k ` ` k wise, a P-recursive sequence is uniquely defined by the linear for k 6= ` and tk xk −xk tk = xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster recurrence relations with polynomial coefficients it satisfies and a finite number of initial terms. the Gr¨obner basis of the ideal spanned by the first relations, such as in 2D/3D-space walks examples. 1.1 Related work CCS Concepts For one-dimensional sequences, C-relations (linear recur- rence relations with constant coefficients) can be guessed •Comput. method. ! Symbolic calculus algorithms; with Berlekamp { Massey algorithm [4, 20], while for multi- Keywords dimensional sequences, they can be guessed with Berlekamp { Massey { Sakata algorithm [24, 26] or Scalar-FGLM [5, 6]. Gr¨obner bases, P-recursive sequences, FGLM algorithm All these algorithms only guess and check that the relations 1. INTRODUCTION are valid for a finite number of terms of the sequence. Then, usually, one need to prove that they are satisfied for all the Computing linear recurrence relations of multi-dimensional terms of the sequence. In this paper, a table shall denote sequences is a fundamental problem in Computer Science. a finite subset of terms of a sequence: it is one of the in- As this is the core of the paper, we start with an exam- i ! put parameter of the algorithms since one cannot handle an ple. The binomial sequence b = ( 1 =(i2!(i1 − i2)!)) 2 (i1;i2)2N infinite sequence. satisfies Pascal's rule, a linear recurrence relation: Likewise, for one-dimensional sequences, P-relations (lin- 2 8 (i1; i2) 2 N ; bi1+1;i2+1 = bi1;i2+1 + bi1;i2 : (1) ear recurrence relations with polynomial coefficients) can be guessed with Beckermann { Labahn algorithm [2] for com- It has constant coefficients. But this sequence also satisfies puting Hermite-Pad´eapproximants. relations with polynomial coefficients in (i1; i2): When computing parameterized sums of multivariate se- quences, or integrals of multivariate functions, in general, it 8 (i ; i ) 2 2; (i − i + 1) b = (i + 1) b ; (2a) 1 2 N 1 2 i1+1;i2 1 i1;i2 is hopeless to obtain a closed form. Indeed, even for elemen- (i2 + 1) bi1;i2+1 = (i1 − i2) bi1;i2 : (2b) tary univariate function we may not have closed forms when computing their integral on some domain. This motivated Zeilberger in 1990 [29] to introduce the so-called creative telescoping method. It is now one of the most efficient ways to do so and has been a tremendously active research topic. We refer the reader to the up-to-date nice survey on the state ISSAC ’16 July 19–22, 2016, Waterloo, ON, Canada of the art in creative telescoping in [13] and the references ACM ISBN . therein. DOI: http://dx.doi.org/10.1145/2930889.2930926 Although sequences can satisfy some linear recurrence re- lations with constant or polynomial coefficients, they still ity of the linear system solving [9]. We also investigate the may be neither C-recursive nor P-recursive. Given a sub- number of table queries performed by the algorithms. set S of steps in f0; ±1g2, one can define the sequence u = (u ) 3 of the number of planar walks of length n n;i;j (n;i;j)2N ending at (i; j) with steps in S. Determining if u is P- 2. DEFINITIONS AND NOTATION recursive has applications in lattice path enumeration [1, We recall the classical definitions of a C-recursive sequence 11, 12]. Likewise, now 3D-space walks are investigated. and of a P-recursive sequence. We shall also see how to deal In [7], the authors use several different methods to prove with C-relations between terms of multiple sequences and that somes sequences of walks are P-recursive among which: see that from a P-recursive sequence, we can create a tuple the algebraic kernel method, the reduction to smaller dimen- of sequences satisfying C-relations. sions and the guessing and proving method. In all the paper, we shall use standard notation with bolded letters corresponding to vectors and sequences: for ∗ 1.2 Contributions n 2 N , i = (i1; : : : ; in) and x = (x1; : : : ; xn). As usual, i i1 in We start by recalling the definitions of C-recursive (linear x denotes x1 ··· xn and jij = i1 + ··· + in. Finally, n u = (ui)i2N is an n-dimensional sequence over a field K. recursive with constant coefficients) sequences and define C- n i i recursive tuples of sequences. This framework allows us to For i 2 N , we let [x ]u = ui = [x ], if no ambiguity on u transform a P-recursive sequence into a tuple of sequences. shall arise. We extend linearly this [ ] operator to K[x]. C-relations are straightforwardly extended to n-dimensional In Section 3, we extends Scalar-FGLM [5, 6] for guessing n the module of C-relations satisfied by a tuple of tables up to sequences [23, Definition 21]: let S be a finite subset of N , n let α = (αs)s2S be nonzero, then u = (ui)i2N satisfies the a degree bound d. The algorithm returns a (d+1)-truncated n Gr¨obner basis of their module of relations. The main idea of linear relation defined by α if for all i 2 N , this algorithm is the same as Scalar-FGLM's: extracting hX i+si X αs x = αs ui+s = 0: (3) a full-rank submatrix of maximal rank. s2S s2S P-relations with coefficients of degree at most δ satisfied n Operator [ ] allows us to make a one-to-one correspondence by a sequence u = (ui1;:::;in )(i1;:::;in)2N are C-relations sat- j1 jn between C-relations satisfied by u and polynomials in [x]: isfied by the tuple of sequences (i ··· i u ) n , K 1 n i1;:::;in (i1;:::;in)2N relation P α u = 0 is associated to polynomial P α xs. with j ; : : : ; j ≥ 0 and j + ··· + j ≤ δ. Hence, a first idea s2S s s s2S s 1 n 1 n According to Sakata [24] and to Fitzpatrick and Nor- is to compute them with the extended Scalar-FGLM for instance. However, this strategy is not optimal because the ton [17], a C-recursive sequence is defined as follows.