m-fold Hypergeometric Solutions of Linear Recurrence Equations Revisited

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1 -fold Hypergeoetric Solutions of Linear Recurrence Equations Revisited Peter Horn, Wolfra Koepf, Torsten Sprenger Institute of Matheatics, University of Kassel, D Kassel, Gerany Abstract We present two algoriths to copute -fold hypergeoetric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an -fold generalization and q-generalization of the algorith by van Hoeij (1998a), Cluzeau and van Hoeij (2005) for recurrence equations. The second is a cobination of an iproved version of the algoriths by Petkovšek (1992), Abraov et al. (1998) for recurrence and q-recurrence equations and the -fold algorith fro Petkovšek and Salvy (1993) for recurrence equations. We will refer to the classical algoriths as van Hoeij or Petkovšek respectively. To forulate our ideas, we first need to introduce an adapted version of an -fold Newton polygon and its characteristic polynoials for the classical case and q-case, and to prove the iportant properties in this case. Using the data fro the Newton polygon, we are able to present efficient -fold versions of the van Hoeij and Petkovšek algoriths for the classical shift case and for the q-case, respectively. Furtherore, we show how one can use the Newton polygon and our characteristic polynoials to conclude for which N there ight be an - fold hypergeoetric solution at all. Again by using the inforation obtained fro the Newton polygon, the presentation of the q-petkovšek algorith can be siplified and strealined. Finally, we give tiings for the classical q-petkovšek, our q-van Hoeij and our odified q- Petkovšek algorith on soe classes of probles and we present a Maple ipleentation of the -fold algoriths for the q-case. Key words: right factors of -fold hypergeoetric type, linear recurrence operators, -fold hypergeoetric solutions, linear recurrence equations, linear q-recurrence equations, Newton polygon 1. Introduction For the whole paper let F be a coputable field of characteristic zero, x be an indeterinate over F and F(x) the field of rational functions in x over F. We consider Eail addresses: horn@ath.uni-kassel.de (Peter Horn), koepf@ath.uni-kassel.de (Wolfra Koepf), sprenger@ath.uni-kassel.de (Torsten Sprenger). Preprint subitted to Matheatics in Coputer Science 25 January 2012

2 σ : F(x) F(x) to be an autoorphis fixing F. In soe cases we need to specify the considered autoorphis σ. In this paper we deal especially with two autoorphiss, naely the classical shift and the q-shift ε : F(x) F(x), x x + 1 ε q : F(x) F(x), x q x. In the q-case we require that F = K(q), where K is a coputable field of characteristic zero, q an indeterinate over K and K(q) the field of rational functions in q over K. Let furtherore F(x)[σ] denote the non-coutative algebra of recurrence operators with rational function coefficients. Here, we notationally identify the operator σ and the autoorphis σ. We always write occurring operators L F(x)[σ] in expanded noral for n L = a i σ i with a i F(x) and a 0 0, a n 0. Then n is called the order of the operator L. Throughout the paper we oit the arguent x in the notation of polynoials and functions. W.l.o.g. we assue that the coefficients a i are polynoials. In the following we are interested in deterining right factors of -fold hypergeoetric type, i.e. σ r with r F(x), of a given linear recurrence operator L F[x][σ]. The coputation of right factors of -fold hypergeoetric type of L corresponds one-to-one to the coputation of -fold hypergeoetric ter solutions of the recurrence equation L(u) = 0, being solutions u for which cert σ (u) := σ (u) u F(x) holds. We call cert σ (u) the -fold certificate 1 of u w.r.t. σ. Furtherore, we assue that all -fold hypergeoetric ters u considered are priitive, i.e. no rational functions r 0,..., r 2 in x exist with σ 1 (u) + r 2 σ 2 (u) + + r 1 σ(u) + r 0 u = 0. Nonpriitive -fold hypergeoetric solutions are considered in Hendriks and Singer (1998). Exaple 1. Let u be a hypergeoetric ter with r = σ(u) u -fold hypergeoetric ter with 1 since σ (u) u = σ (u) σ 1 (u) σ 1 (u) σ 2 (u) σ2 (u) σ(u) σ(u) u but u is not priitive because of σ(u) ru = 0. F(x). Then u is also an = σ 1 (r)σ 2 (r) σ(r)r F(x), Definition 1. Let γ F and k Z. Then γ σk denotes the root of σ k (x γ). For α, β F define the equivalence relation α σ β : α = β σk for soe k Z. 1 If = 1, then we oit the ter -fold and call a solution just hypergeoetric ter and cert σ (u) its certificate. 2

3 With [α] σ we denote the equivalence class of α in F/ σ. For an -fold hypergeoetric ter u we define the local type at [α] σ F/ σ to be ltype [α] σ (u) = (root order of cert σ (u) in β) (pole order of cert σ (u) in β). β [α] σ Here it is noteworthy that ltype [α] σ (u) is well defined and it is a nonzero integer only for finitely any classes [α] σ. For we define ltype (u) = (c, v ) with cert σ (u) = c x v (1 + O( 1 x )) using the series expansion of cert σ (u) at. In the q-case we have additionally ltype 0 (u) = (c 0, v 0 ) with cert εq (u) = c 0 x v0 (1 + O(x)) as 0 behaves exceptionally like, in the sense that 0 εk q = 0 for all k Z. For this reason we call and 0 (0 only in the q-case) exceptional points. Exaple 2. We consider the shift case with the operator ε and the 2-fold hypergeoetric ter u with cert ε x(x 1) 2(u) = (x + 1) 3 (2x + 1). Then we have ltype [0] ε 2 (u) = 1, ltype [1] ε 2 (u) = 1 3 = 2 and ltype [ 1 (u) = 1. The local 2 ]ε 2 type of u in other finite points is zero. For the local type at we have ltype (u) = ( 1 2, 2) because of cert ε 2(u) = 1 2 x 2 (1+O( 1 x )). Note that if the rational function is the certificate of a hypergeoetric ter u then the equivalence class [0] ε 2 will coincide with [1] ε 2 and ltype [0] ε 1 (u) = = 1. There is in fact a relation between the local types in finite points and the local types in the exceptional points. Lea 3 (Fuchsian relations). The following relations hold for an -fold hypergeoetric ter u with ltype (u) = (c, v ) 0 = v + ltype [α] ε (u). (1) In the q-case we have 0 = v 0 + v + [α] εq F / εq where ltype 0 (u) = (c 0, v 0 ). [α] ε F/ ε ltype [α] εq (u) and c 0 εq c [α] εq F / εq ( α) ltype [α] εq (u), (2) Proof. Let r = s t be the -fold certificate of u with s, t F[x]. We consider the value deg(s) deg(t), which is obviously v. For σ = ε we have deg(s) deg(t) = ltype [α] ε (u), [α] ε F/ ε 3

4 and for σ = ε q deg(s) deg(t) = v 0 + [α] εq F / εq ltype [α] εq (u), hence (1) and the first equation of (2) follows. Applying Vieta s theore to s and t the second relation of (2) follows 2 lcoeff(s) lcoeff(t) } {{ } c [α] εq F / εq ( α) ltype [α] εq (u) εq tcoeff(s) tcoeff(t) = c 0. The representation depends on the choice of the representatives of [α] εq and therefore the ters are only identical odulo q-shifts. We give a definition for the type of an -fold hypergeoetric ter siilar to the classical definition (cf. Cluzeau and van Hoeij (2005)). Definition 2. Two -fold hypergeoetric ters u, w are said to be of the sae type if u w is a rational function or, equivalently, σ (u) w u σ (w) is the -fold certificate of a rational function. The following siple theore, which is crucial for the -fold van Hoeij algorith, connects the two type concepts. Theore 3. Two -fold hypergeoetric ters are of the sae type if and only if their local types in all exceptional points and all finite points coincide. 2. The σ-newton Polygon In this section we consider the σ-newton polygon in detail. This polygon can be easily constructed fro the given recurrence operator L F[x][σ] and it provides useful inforation about -fold hypergeoetric solutions of L Valuations Definition 4. A apping v : F(x) R { } is called a valuation on F(x), if for all f, g F(x) the following properties hold (a) v(f) = f = 0 (b) v(f g) = v(f) + v(g) (c) v(f + g) in{v(f), v(g)} (d) v(f) v(g) v(f + g) = in{v(f), v(g)}. We call a valuation copatible with the autoorphis σ, if v(σ(f)) = v(f) is valid for all f F(x). 2 lcoeff(p) denotes the leading coefficient and tcoeff(p) the trailing coefficient of the polynoial p 4

5 In the case of σ = ε q, there are essentially two copatible valuations 3 on F[x] of our interest v deg : f deg(f) and v ldeg : f ldeg(f) which naturally extend to F(x) by v ( ) f g := v(f) v(g). For σ = ε we only have one iportant copatible valuation, naely v deg. In this case the valuation v ldeg is not copatible, because for n N we have ldeg(x n ) = n, but ldeg(ε(x n )) = ldeg((x + 1) n ) = 0. With these valuations we can easily deterine the local types of an -fold hypergeoetric ter in the exceptional points. Lea 4. For an -fold hypergeoetric ter u with -fold certificate r = f g ltype (u) = (c, v ) the following relations hold In the q-case we have additionally where ltype 0 (u) = (c 0, v 0 ). v = v deg (r) and c = lcoeff(f) lcoeff(g). v 0 = v ldeg (r) and c 0 = tcoeff(f) tcoeff(g), and 2.2. The σ-newton Polygon and its characteristic polynoials In this section, we present the σ-newton polygon and its characteristic polynoials. The σ-newton polygon will give us soe valuable a priori inforation about the structure of the -fold certificates of -fold hypergeoetric solutions of linear recurrence equations. Definition 5. Let L = n a iσ i F[x][σ]. Then n N v (L) := convex hull of {(i, y) y v(a i )} R 2 is the σ-newton polygon of L w.r.t. the valuation v. In the following we consider the edges with finite slope of the σ-newton polygon in detail. We denote the sections of the σ-newton polygon with slope w Q as the edge w. The length of edge w is the length of the projection of that edge onto the x-axis. Definition 6. Let L = n i k=0 α i,kx k σ i F[x][σ] and v a valuation. In the shift case the characteristic polynoial(s) of N v (L) w.r.t. the edge w Q with length l N is (are) given by PL,v,w(T ) := α i+j,k T i i0 F[T ] and in the q-case by P L,v,w(T ) := i,k where (i+j,v(α i+j,k x k )) lies on edge w i,k where (i+j,v(α i+j,k x k )) lies on edge w ( q v(x) i(i 1) ) 2 2 +i j w α i+j,k T i i0 F[T ] 3 deg(p) denotes the degree and ldeg(p) the low degree of the polynoial p 5

6 for all N such that w Z and l, where i 0 N 0 is such that T PL,v,w (T ). In the special case w Z, we have 4 P L,v,w (T ) = α i,k T i i0 F[T ] and P L,v,w (T ) = i(i 1) v(x) q 2 w α i,k T i i0 F[T ] respectively. i,k where (i,v(α i,k x k )) lies on edge w i,k where (i,v(α i,k x k )) lies on edge w In general, the nuber in Definition 6 is not uniquely deterined. For the edge w = 1 of length l N, we have for exaple d(l) characteristic polynoials, where d is the divisor function. Figure 1. The σ-newton polygon N vdeg (L) of Exaple 5 To illustrate the σ-newton polygon, we consider the following exaple. Exaple 5. Let σ = ε q, v = v deg and L = ε 5 q + (q 2 x 2 1)ε 4 q + x ε 3 q (x 4 + x + 1)ε 2 q + q 2 x 4 ε q (x 3 + q). Then the vertices of the ε q -Newton polygon N vdeg (L) are given by {(0, 3), (1, 4), (2, 4), (4, 2), (5, 0)} (see Figure 1). The point (3, 1), which corresponds to the ter x ε 3 q, lies in the interior of the ε q -Newton polygon and therefore it is issing in the above list. The characteristic polynoials of N vdeg (L) are P L,vdeg, 1(T ) = q 2 T 1, P L,vdeg,0(T ) = T + q 2, P L,vdeg,1(T ) = q 8 T 2 q, P L,vdeg,2(T ) = q 20 T + q 14, P 2 L,v deg,1(t ) = q 6 T 1. 4 If = 1, then we oit the nuber in the notation of the characteristic polynoial. 6

7 With the following algorith we can deterine the σ-newton polygon and its characteristic polynoials w.r.t. = 1. Algorith 1 (σ-newton polygon) Deterination of the σ-newton polygon of a linear recurrence operator with associated characteristic polynoials w.r.t. = 1 Input : L = P n aiσi of order n and a valuation v Output : σ-newton polygon as list of vertices and characteristic polynoials w.r.t. the edges as list of triples (w, l, P L,v,w(T )) where w denotes the edge and l its length begin NP, charpols, i 0 while i n do if a i = 0 then next i NP = NP {(i, v(a i))} slope for j i + 1 to n do if a j = 0 then next j s v(a j ) v(a i ) j i if s < slope then slope s charpol a i, v(ai ) end if s = slope then charpol charpol + a j, v(aj ) T j i k j 17 end 18 end 19 charpols = charpols {(slope, deg(charpol), charpol)} 20 i k 21 end 22 NP = NP {(n, v(a n))} 23 return NP, charpols 24 end For the q-case lines 12 and 15 of Algorith 1 have to be adapted by ultiplying k(k 1) v(x) q 2 s to the coefficient a k, v(ak ). In order to deterine all characteristic polynoials we consider one edge w of length l out of the coputed list charpols in detail. Then, for all divisors 1 of l with w Z, we deterine PL,v,w (T ) as in Definition 6. This will be done for every edge w in the list to copute all characteristic polynoials. The ain theore of this paper connects the σ-newton polygon of a recurrence operator with its -fold hypergeoetric solutions. Theore 7. Let u be an -fold hypergeoetric solution of L(u) = 0 with -fold certificate r = c s t, where c F, s, t F[x] onic 5 w.r.t. the given valuation v {v deg, v ldeg } and gcd(s, t) = 1. Then the σ-newton polygon N v (L) has an edge with slope w = v(r) and length k with k N and PL,v,w (c) = 0. 5 For v = v deg we consider lcoeff(s) = lcoeff(t) = 1 and for v = v ldeg we consider tcoeff(s) = tcoeff(t) = 1. 7

8 Proof. The existence of the edge with the corresponding slope can easily (but lengthy) be adopted fro the classical difference or differential case (e.g. Robba (1980), Duval (1983)). Based on the fact that σ (u) = u r holds, we obtain σ i+j (u) = σ j (u) i 1 k=0 σk+j (r) for i N and j = 0,..., 1 by induction. It follows that 1 L(u) = j=0 ( dj i 1 a i+j k=0 ) 1 σ k+j (r) σ j (u) = j=0 ( dj i 1 a i+j k=0 ( σ k+j s ) ) σ j (u) = 0 t where d j is such that a dj+j is the highest nonvanishing polynoial of all a i+j s with i N 0, leading to d j i 1 a i+j c i k=0 d j 1 σ k+j (s) k=i σ k+j (t) = 0 for j = 0,..., 1. Note that this conclusion is correct, since we are dealing with priitive -fold hypergeoetric solutions. Fro the definition d j ( i 1 d j 1 ) P j (T ) := a i+j σ k+j (s) σ k+j (t) T i k=0 k=i } {{ } b j,i:= it follows that P j (c) = 0 for j = 0,..., 1. We consider P j (T ) as polynoial in x and select those polynoials b j,i, whose valuations are inial, because their leading (or trailing resp.) coefficient contributes to the highest (or lowest resp.) coefficient of P j (T ) w.r.t. x. Every coefficient is a polynoial in T vanishing at c. The valuation of b j,i is given by v(b j,i ) = v ( a i+j i 1 k=0 d j 1 σ k+j (s) σ k+j (t) k=0 k=i i 1 d j 1 = v(a i+j ) + v(s) + v(t) k=i = v(a i+j ) + i v(s) + (d j i)v(t) = v(a i+j ) i w + d j v(t), where the second equation follows fro the fact that we only consider valuations which are copatible with σ. The value v(b j,i ) becoes inial, if the corresponding point (i + j, v(a i+j )) lies on the edge w of the σ-newton polygon N v (L), because for all k < i and for all k > i v(a i+j ) v(a k+j ) (i k) < w = v(a i ) i w < v(a k ) k w v(a i+j ) v(a k+j ) > w = v(a (i k) i ) i w < v(a k ) k w, ) 8

9 where i and i denote the sallest and biggest index for which (i + j, v(a i+j )) lies on the edge w. Furtherore, for k with i < k < i, where (k + j, v(a k+j )) does not lie on the edge w, the above estiates are also valid. Hence we obtain for the shift case α i+j,k c i = PL,v,w(c)c i0 = 0 i,k where (i+j,v(α i+j,k x k )) lies on edge w with w = v(r), α i+j,k and i 0 fro Definition 6. For the q-case the characteristic polynoials are ore coplicated since a onic polynoial does not reain onic under the q-shift. We consider the highest coefficient of b j,i in detail (case v = v deg ) and obtain i 1 lcoeff(b j,i ) = lcoeff(a i+j ) k=0 d j 1 q (k+j) deg(s) k=i (k+j) deg(t) q = lcoeff(a i+j )q ji deg(s) q i(i 1) 2 deg(s) q j(dj i) deg(t) q ( dj (d j 1) 2 i(i 1) 2 ) deg(t) = lcoeff(a i+j )q ji(deg(s) deg(t)) q i(i 1) 2 (deg(s) deg(t)) q jdj deg(t) q d j (d j 1) 2 deg(t) = lcoeff(a i+j )q jiw q i(i 1) 2 2w q jdj deg(t) q d j (d j 1) 2 deg(t) with w = deg(s) deg(t) = v deg(r). The factor deg(t) q dj (dj 1) qjdj 2 deg(t), which occurs in every b j,i, is independent of i and therefore can be cancelled. In the case v = v ldeg we analogously consider the trailing coefficient of b j,i and obtain tcoeff(b j,i ) = tcoeff(a i+j )q jiw q i(i 1) 2 2w q jdj ldeg(t) q d j (d j 1) 2 ldeg(t) with w = (ldeg(s) ldeg(t)) = v ldeg(r). Again, we can neglect the occurring constant factor w.r.t. i. Suarizing, we obtain ( q v(x) i(i 1) ) 2 2 +i j w α i+j,k c i = PL,v,w(c)c i0 = 0 i,k where (i+j,v(α i+j,k x k )) lies on edge w with w = v(r), α i+j,k and i 0 fro Definition 6. If we look for all -fold hypergeoetric solutions of a recurrence operator L F[x][σ] with 1, then we deterine all characteristic polynoials and read off all candidates for the nuber. For instance, if there is no characteristic polynoial PL,v,w (T ), then L has no -fold hypergeoetric solutions by Theore 7. Hence, the operator considered in Exaple 5 could only have -fold q-hypergeoetric solutions for {1, 2}. In fact, in this exaple, there are no -fold q-hypergeoetric solutions at all. In the q-case we can actually deterine two Newton polygons and their characteristic polynoials and for each of the we get a set of candidates for. By intersecting both sets we get a finite set of possible s which could correspond to an -fold q-hypergeoetric solution. Once again, in Exaple 5, we have {1, 2} for v = v deg and {1, 5} for v = v ldeg, hence we can even state (in alost no tie), that there are no -fold q-hypergeoetric solutions for 1. Thus, the candidate set for should be coputed before one tries to copute -fold hypergeoetric solutions for a specific. 9

10 2.3. Coputing Candidates for Local Types By the previous theore, the σ-newton polygon gives us relevant inforation about the -fold certificates of possible -fold hypergeoetric solutions, naely the local types in the exceptional points. Therefore we define T0 (L) := {(c 0, v 0 ) N vldeg (L) has edge v0 of length k and P L,v ldeg, v 0 (c 0 ) = 0, where k N and v 0 Z} and additionally for the q-case T (L) := {(c, v ) N vdeg (L) has edge v of length k and (c ) = 0, where k N and v Z}. P L,v deg, v Corollary 6. If u is an -fold hypergeoetric solution of L(u) = 0 then ltype (u) T (L) and in the q-case ltype 0 (u) T 0 (L). Proof. This follows directly fro Theore 7 and the fact that the valuation of a nonzero rational function is always an integer. Exaple 7. We continue Exaple 5 and get the following candidates for the local type of a potential q-hypergeoetric solution of L at { } T (L) = (q 2, 1), (q 2, 0), (q 7 2, 1), ( q 7 2, 1), ( q 6, 2). If we consider 2-fold q-hypergeoetric solutions of L, we obtain T 2 (L) = { (q 6, 2) }. In order to deterine bounds for the local type of an -hypergeoetric solution in a finite point, we consider Petkovšek s noral for. Theore 8. Let L = n a i σ i F[x][σ] and cert σ (u) = z σ (f) f be the -fold certificate of an -fold hypergeoetric solution u in Petkovšek noral for, with z F and f, g, h F[x] onic with g h gcd(g, (σ ) k (h)) = 1 for all k N, gcd(g, f) = 1, gcd(h, σ (f)) = 1, f(0) 0 (only in the q-case), siilar to Lea 1 of Petkovšek and Salvy (1993) for the shift case. Then for the shift case we have (a) ltype (u) = (c, v ) = (z, deg(h) deg(g)) and for the q-case we have (3) 10

11 (b) ltype (u) = (c, v ) = ( zq deg(f), deg(h) deg(g) ), ( (c) ltype 0 (u) = (c 0, v 0 ) = z tcoeff(g) tcoeff(h) ),, ldeg(g) ldeg(h) (d) tcoeff(g) tcoeff(h) c εq c 0. Proof. For (a), (b) and (c) one uses the forulas fro Lea 4 taking into account that all occurring polynoials are onic. Relation (d) follows fro (b) and (c). The core idea of the classical -Petkovšek algorith by Petkovšek and Salvy (1993), which we will odify in the next section, is, that if we use the noral for (3), then one can show that the relations g a 0 and h σ (n ) (a n ) (4) hold (see Petkovšek and Salvy (1993)). The local type of the polynoial part of the -fold hypergeoetric solution is in every finite point 0, because if we consider the corresponding factor σ f f of the -fold certificate, then every root of the denoinator occurs shifted in the nuerator, too. Because of (4), we define for α F { T [α] σ (L) = } ult(σ (n ) (a n ), β),..., ult(a 0, β) β [α] σ β [α] σ where ult(a, β) is the ultiplicity of the root β of the polynoial a. Corollary 8. If u is an -fold hypergeoetric solution of L(u) = 0 then for every α F we have ltype [α] σ (u) T [α] σ (L). Hence to copute these bounds, one first factors the leading and trailing coefficients and collects the factors up to -shift equivalence. Then one adds up the ultiplicities in each class. 3. Right Factors of -fold Hypergeoetric Type We assue that the operators for which right factors of -fold hypergeoetric type are sought do not have polynoial or rational solutions. Otherwise, these should be coputed beforehand with known efficient algoriths by Abraov et al. (1995), Abraov (1995), van Hoeij (1998b), Böing and Koepf (1999) fold Van Hoeij Approach Fro the above sections we get a finite set of possible types of -fold hypergeoetric solutions of a recurrence equation. Each of these candidates aounts to the -fold certificate of such a solution deterined up to -shifts for each irreducible factor of nuerator and denoinator. So we can easily copute a set of possible correct-up-to-shift -fold certificates. In order to reconstruct the real -fold certificate of an -fold hypergeoetric solution, we need the following recurrence operator. 11

12 Definition 9. Let L 1 and L 2 be recurrence operators. The syetric product L 1 SL 2 is defined as the unique onic recurrence operator of inial order such that for all - fold hypergeoetric ters u, w holds: L 1 (u) = 0 L 2 (w) = 0 = (L 1 SL 2 )(u w) = 0. Note that L 1 SL 2 can easily be coputed fro L 1 and L 2 by linear algebra (see e.g. Cluzeau and van Hoeij (2005)). Fro Theore 3 and Definition 2 we know that once we have a candidate -fold certificate r corresponding to the -fold hypergeoetric ter ũ, and copute L := LS(σ 1 r ), then L ust have a rational solution s = ũ u if ũ was of the sae type as an actual solution u with -fold certificate r. Then, this r can be reconstructed fro s and r by σ (s) s r = σ (u) ũ u σ (ũ) σ (ũ) = σ (u) = r. ũ u We obtain the following -version of the van Hoeij-type algorith. Algorith 2 (-fold van Hoeij) Deterination of all right factors of -fold hypergeoetric type of a linear recurrence operator Input : N and a linear recurrence operator L = P n ai εi Output : all right factors of -fold hypergeoetric type of L 1 begin 2 C 3 Copute T (L) and T [α] ε (L) for all α F Q o C n c [α] ε (x F/ ε α)ṽα ( c, ) T (L), ṽ α T [α] ε 4 (L) 5 for all r(x) C do 6 if the Fuchsian relation (1) is not satisfied for r(x) then 7 next 8 end 9 L LS`ε 1 r 10 S rational solutions of L C C ε ε (s) r 11 s S s 12 end 13 return C 14 end Note that in line 3 of Algoriths 2 and 3 the sets T [α] σ (L) are coputed only for those α F that are roots of a 0 σ (n ) (a n ). In line 6 we use the Fuchsian relations to get rid of -fold certificates that certainly do not correspond to proper factors, which reduces the nuber of possible -fold certificates of solutions significantly. 12

13 Algorith 3 (-fold q-van Hoeij) Deterination of all right factors of -fold q-hypergeoetric type of a linear q-recurrence operator Input : N and a linear q-recurrence operator L = P n ai εi q Output : all right factors of -fold q-hypergeoetric type of L 1 begin 2 C 3 Copute T (L), T0 (L) and T εq [α] (L) for all α F n C c Q o 4 xṽ0 [α] εq F / εq (x α)ṽα ( c, ) T (L), (, ṽ 0) T0 (L), ṽ α T εq (L) [α] 5 for all r(x) C do 6 if one of the q-fuchsian relations (2) is not satisfied for r(x) then 7 next 8 end 9 L LS`ε q 1 r 10 S rational solutions of L C C ε q ε q (s) r 11 s s S 12 end 13 return C 14 end We point out that our version of van Hoeij s algorith does not copute solutions over algebraic extensions as described in Cluzeau and van Hoeij (2005). Therefore, our van Hoeij-type algorith consists of only one part of the algorith proposed in Cluzeau and van Hoeij (2005) which for ost purposes is sufficient. We consider another ethod for coputing right factors of -fold hypergeoetric type in the next section fold Petkovšek Revisited In this section, we give a version of the -fold Petkovšek algorith (Petkovšek and Salvy (1993)) that is odified in two ways using the inforation fro the σ-newton polygon. First, we are able to siplify the coputation of the leading coefficient of the certificate. Second, we can significantly reduce the nuber of candidates that have to be taken into consideration, which leads to an efficient algorith to copute -fold hypergeoetric solutions. Once again, we assue that the -fold certificate of an -fold hypergeoetric solution is in Petkovšek noral for. Fro (4) we get a finite nuber of possible choices for g and h. To copute candidates for z, we use the candidates for the local types in the exceptional points 6. If z, g, and h contribute to a proper factor of an -fold certificate of an -fold hypergeoetric solution, f is a polynoial solution of d j ( i 1 z i a i+j k=0 d j 1 σ k+j (g) k=i σ k+j (h) ) σ i (f) = 0. (5) for all j = 0,..., 1. This can be deduced as in the proof of Theore 7 using the noral for (3) for r (instead of c s t ). The odified versions of the -fold Petkovšek algorith which are described in Algoriths 4 and 5 filter out all polynoials g and h according to Corollary 6 and Theore 6 Note that in the classical -Petkovšek algorith z is coputed in a different way. 13

14 Algorith 4 (odified -fold Petkovšek) Deterination of all right factors of -fold hypergeoetric type of a linear recurrence operator Input : N and a linear recurrence operator L = P n ai εi Output : all right factors of -fold hypergeoetric type of L 1 begin 2 C 3 for every onic factor g of a 0 and h of ε (n ) a n do 4 v v deg (g) v deg (h) 5 for all (c, v ) T (L) do 6 z c 7 for j = 0,..., 1 do d P j L j z i Q a i 1 i+j k=0 εk+j (g) Q d j 1 k=i ε k+j (h) ε i 8 9 end 10 S polynoial solutions of L j(f) = 0 for one specific j C C z ε (f) g 11 f h f S and Lj(f) = 0 for all j = 0,..., 1 12 end 13 end 14 return C 15 end 8, which do not contribute to a part of an -fold certificate of an -fold hypergeoetric solution. Thus, the algorith is substantially ore efficient than the classical -fold Petkovšek algorith (especially also for = 1), because in any cases the considered recurrence operator (5) does not need to be constructed and eventually no polynoial solver needs to be used. If we consider linear recurrence operators which have leading and trailing coefficients with any factors, this iproveent is enorous. Notes on Algoriths 4 and 5: (1) Line 3 of Algorith 4 and lines 3 and 6 of Algorith 5 are iterations over the cartesian products. (2) In line 6 of Algorith 5 the values v 0 and v are already coputed and we iterate only over those pairs of T 0 (L) and T (L) with second coponent v 0 and v respectively. In the shift case the sae is true w.r.t. v in line 5 of Algorith 4. (3) Fro line 8 of Algorith 5 we get the possible degrees of the polynoial solutions in line 12 by q deg(f) = c z. (4) In line 10 of Algorith 4 and line 12 of Algorith 5 the nuber j {0,..., 1} can be chosen such that L j is the first nontrivial operator. (5) In line 11 of Algorith 4 and line 13 of Algorith 5 we ust check if f is annihilated by all L j s (j = 0,..., 1). Alternatively, we can perfor a division with reainder of our given operator L and the operator which corresponds to our candidate -fold certificate. If the reainder is zero, the candidate is an -fold certificate of an -fold hypergeoetric solution. Let = 1, then for σ = ε the van Hoeij algorith (van Hoeij (1998a), Cluzeau and van Hoeij (2005)) is the ost efficient algorith. In the q-case the odified q-petkovšek algo- 14

15 Algorith 5 (odified -fold q-petkovšek) Deterination of all right factors of -fold q- hypergeoetric type of a linear q-recurrence operator Input : N and a linear q-recurrence operator L = P n ai εi q Output : all right factors of -fold q-hypergeoetric type of L 1 begin 2 C 3 for every onic factor g of a 0 and h of ε (n ) q a n do 4 v 0 v ldeg (g) v ldeg (h); v v deg (g) v deg (h) 5 c tcoeff(g) tcoeff(h) 6 for all (c 0, v 0) T0 (L) and (c, v ) T (L) do 7 if c c εq c 0 then z c 0 8 c 9 for j = 0,..., 1 do d P j L j z i Q a i 1 i+j k=0 εk+j q (g) Q d j 1 k=i ε k+j q (h) ε i q end 12 S polynoial solutions of L j(f) = 0 for one specific j = 0,..., 1 C C z ε q (f) g 13 f S and L f h j(f) = 0 for all j = 0,..., 1 14 end 15 end 16 end 17 return C 18 end rith (Horn (2008)) is in any cases better than a q-version of the van Hoeij algorith (see next section). For > 1 the odified Petkovšek algorith should be preferred The Special Case = 1 and σ = ε q (q-hypergeoetric Solutions) Now, let = 1 and σ = ε q. We consider operators of order three paraetrized by j N, where the leading coefficient has degree j+1 2 and the trailing coefficient has degree j 2 +1, and both factor into linear factors. To copare the algoriths, we construct two different series of operators, first ( j+1 2 ( x + iq i )) j 2 ε 2 ( + x iq i ) (ε x), (6) i=1 where none of the occurring linear factors is q-shift equivalent to another. Second ( j+1 2 ( x q i )) j 2 ε 2 ( + x iq i ) (ε x), (7) i=1 i=1 i=1 where all linear factors of the leading coefficient are q-shift equivalent whereas the classes of the trailing coefficient are distinct. Obviously, both operators have q-hypergeoetric solutions with q-certificate x. The tiings are in seconds and were recorded on a 3 GHz Intel Xeon with 16GB of RAM using Maple 12. A dash denotes a case in which no result was coputed after three hours. 15

16 Operator (6) Operator (7) Tie Candidates Tie j Pet Hoeij odpet Pet Hoeij odpet Pet Hoeij odpet Table 1. Tiings of different Maple procedures for the deterination of first-order right factors of operators (6) and (7) and the nuber of candidates w.r.t. operator (6). 16

17 The classical q-petkovšek algorith is exponential in j, in fact the nuber of candidates is 2 j for (6) and (7). The odified q-petkovšek algorith is still exponential in j, but has to perfor only tests in ost cases, hence it investigates only few cases in detail. For the operator (6) the nuber of candidates is also shown in Table 1. The q-van Hoeij algorith is exponential in the nuber of occurring shift equivalence classes but uses siilar iproveents as the odified q-petkovšek algorith. For (7) both the q-van Hoeij and the odified Petkovšek algorith reject all but one candidate for all j. With the above knowledge, a hybrid ipleentation would be feasible and reasonable. The decision for one of the algoriths can be ade after the coputation of the q-newton polygon and the factorization of the leading and trailing coefficients Maple Ipleentation A Maple ipleentation of the -fold q-van Hoeij and q-petkovšek algorith can be found at In the following, we give an exaple for the use of the algoriths, where we construct a linear q-recurrence equation of order 5, which has one 2-fold q-hypergeoetric solution and another 3-fold q-hypergeoetric solution. The output of the following function calls of qhypergeosolvere are the -fold q-certificates of the -fold q-hypergeoetric solutions. Exaple 9. > 1:=2; > cert1:=q*(q-1)*x/(x-1); 1 := 2 q (q 1) x cert1 := x 1 > RE1:=deno(cert1)*qshift(f(x),[x$1],q)-nuer(cert1)*f(x)=0; > 2:=3; RE1 := (x 1) ( Sq x,x ) (f (x)) q (q 1) xf (x) = 0 > cert2:=(q-1)*(x-3)^2/(x-q); 2 := 3 cert2 := (x 3)2 (q 1) x q > RE2:=deno(cert2)*qshift(f(x),[x$2],q)-nuer(cert2)*f(x)=0; RE2 := ( x + q) ( Sq x,x,x ) (f (x)) + (q 1) (x 3) 2 f (x) = 0 > RE:=qNoral(qLCM(RE2,RE1,f(x)),f(x)): > qorder(re,f(x)); 5 The coefficients of our q-recurrence equation RE of order 5 are quite huge, hence we surpressed the output. Now, we deterine all -fold hypergeoetric of solutions of RE. > st:=tie(): > qhypergeosolvere(re,f(x),ethod=qvanhoeij,hypersol=2); > tie()-st; 17

18 [ ] q (q 1) x x > st:=tie(): > qhypergeosolvere(re,f(x),ethod=qpetkovsek,hypersol=2); > tie()-st; [ ] q (q 1) x x > st:=tie(): > qhypergeosolvere(re,f(x),ethod=odqpetkovsek,hypersol=2); > tie()-st; [ ] q (q 1) x x > st:=tie(): > qhypergeosolvere(re,f(x),ethod=qvanhoeij,hypersol=3); > tie()-st; [ ] (x 3)2 (q 1) x + q > st:=tie(): > qhypergeosolvere(re,f(x),ethod=qpetkovsek,hypersol=3); > tie()-st; [ ] (x 3)2 (q 1) x + q > st:=tie(): > qhypergeosolvere(re,f(x),ethod=odqpetkovsek,hypersol=3); > tie()-st; [ ] (x 3)2 (q 1) x + q > st:=tie(): > qhypergeosolvere(re,f(x),hypersol=0); > tie()-st; {[ [ ]] [ [ ]]} q (q 1) x 2,, 3, (x 3)2 (q 1) x 1 x + q In the last line we deterine all -fold hypergeoetric solutions in one function call by setting hypersol=0. In that call, the ost efficient algorith, the odified -fold q-petkovšek algorith, is used by default. The -fold q-van Hoeij algorith is rather 18

19 slow in this exaple, because the syetric product has coplex leading and trailing coefficients. We consider another exaple presenting the algorith in practice. Exaple 10. We deterine an -fold q-hypergeoetric series representation with expansion point a for a function F (y) with algoriths described in Sprenger and Koepf (2011), i.e. F (y) = j=0 c j (y a) j q + j=0 c j+1 (y a) j+1 q j=0 c j+( 1) (y a) j+( 1) q, where (y a) j q = (y a) (y aq) (y aq j 1 ). In this exaple we consider the sall q-sine function F (y) = sin q (y). First, we develop a q-differential equation for sin q (y). > qde:=qholonoicde(qsin(y,q),f(y)); qde := F (y) + (q 1) 2 ( Dq y,y ) (F (y)) = 0 Then we convert the q-differential equation into a q-recurrence equation for the q-series coefficients c j with a pattern-atching algorith. > RE:=qDEtoRE(qDE,F(y),c(j),base=qpower,expansionpt=a); ( RE := c (j) + qaq j (1 + q) c (j + 1) + a 2 ( q j) 2 q 4 + ( q j) ) 2 q 3 q j q 2 q j q + 1 c (j + 2) + qaq j (1 + q) ( q j q 3 1 ) ( q j q 2 1 ) c (j + 3) + a 2 ( q j) 2 q 4 ( q j q 4 1 ) ( q j q 3 1 ) c (j + 4) = 0 Note that this q-recurrence equation of c j can be transfored via q j = x and c j = f(x) into a q-recurrence equation of f(x) as in the previous exaple (c j+n = f(q n q j ) = f(q n x) = ε n q f(x)). Finally, we solve this equation by the -fold q-petkovšek algorith. For = 2 we get > qhypergeosolvere(re,c(j),hypersol=2); A 1 ( 1) j qpochhaer (q, q, 2 j), A 2 (q 1) ( 1) j ( ) (q j ) 2 q 1 qpochhaer (q, q, 2 j) Fro the q-taylor theore we obtain the two initial values c 0 = sin q (a) and c 1 = cosq(a) q 1, leading to sin q (y) = ( 1) j sin q(a) (y a) 2j q + ( 1) j cos q (a) (y a) 2j+1 q. (q; q) 2j (q; q) 2j+1 j=0 j=0 Acknowledgent We would like to thank the anonyous referees for their iportant rearks which helped to iprove our paper substantially. 19

20 References Abraov, S., Rational solutions to linear difference and q-difference equations with polynoial coefficients. Prograirovanie (6), Abraov, S., Bronstein, M., Petkovšek, M., On polynoial solutions of linear operator equations. In: ISSAC 1995 Proceedings. pp Abraov, S., Paule, P., Petkovšek, M., q-hypergeoetric solutions of q-difference equations. Discrete Math. 180, Böing, H., Koepf, W., Algoriths for q-hypergeoetric suation in coputer algebra. J. Sybolic Coput. 28, Cluzeau, T., van Hoeij, M., Coputing hypergeoetric solutions of linear recurrence equations. Appl. Algebra Engrg. Co. Coput. 17, Duval, A., Lees de Hensel et Factorisation Forelle pour les Opérateurs aux Différences. Funkcialaj Ekvacioj 26, Hendriks, P. A., Singer, M. F., Solving difference equations in finite ters. J. Sybolic Coput. 27, Horn, P., Faktorisierung in Schief-Polynoringen. Ph.D. thesis, Universität Kassel. Petkovšek, M., Hypergeoetric solutions of linear recurrences with polynoial coefficients. J. Sybolic Coput. 14 (2-3), Petkovšek, M., Salvy, B., Finding all hypergeoetric solutions of linear differential equations. In: ISSAC 1993 Proceedings. pp Robba, P., Lees de Hensel pour les opérateurs différentiels. Application à la réduction forelle des équations différentielles. L Enseigneent Mathéatique 26 (3-4), Sprenger, T., Koepf, W., Algorithic deterination of q-power series for q- holonoic functions. J. Sybolic Coput. doi: /j.jsc van Hoeij, M., 1998a. Finite singularities and hypergeoetric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, van Hoeij, M., 1998b. Rational solutions of linear difference equations. In: ISSAC 1998 Proceedings. pp