Walks in the Quarter Plane with Multiple Steps
aa r X i v : . [ m a t h . C O ] N ov Discrete Mathematics and Theoretical Computer Science
DMTCS vol. (subm.) , by the authors, 1–1
Walks in the Quarter Plane withMultiple Steps
Manuel Kauers † and Rika Yatchak ‡ RISC, Johannes Kepler University, Linz, Austria received tba , revised tba , accepted tba . We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities areattached to each direction in the step set. Our study leads to a small number of infinite families that completelycharacterize all the models whose associated group is D4, D6, or D8. These families cover all the models withmultiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite — with three noteworthy exceptions.
Keywords:
Lattice Walks, D-finiteness, Computer Algebra
We consider quadrant walk models where step sets may contain several distinguishable steps pointing intothe same direction. For example, the step sets {← , ↓ , ր} and {← , ← ′ , ↓ , ր} are considered different, asthe latter contains two different ways of going to the left. The objects being counted are then walks inthe quarter plane starting at the origin, consisting of n consecutive steps taken from the step set in sucha way that the walk never leaves the first quadrant, ending at a point ( i, j ) ∈ N , and one of k differentcolors is attached to each step in the walk whose multiplicity in the step set is k . For each model (viz., foreach multiset of admissible directions), we want to know whether the corresponding generating function f ( x, y, t ) = P ∞ n =0 P i,j f i,j,n x i y j t n which counts the number f i,j,n of walks of length n ending at ( i, j ) is D-finite. As usual, a power series in t is D-finite if it satisfies an ordinary linear differential equationwith polynomial coefficients.If we let a u,v denote the multiplicity of the direction ( u, v ) ∈ {− , , } \ { (0 , } , then the number f i,j,n of walks of length n ending at ( i, j ) is uniquely determined by the recurrence equation f i,j,n +1 = X u,v a u,v f i − u,j − v,n ( n ∈ N , i, j ∈ N ) † Email: [email protected] . Partially supported by the Austria FWF grants Y464-N18 and F50-04. ‡ Email: [email protected] . Partially supported by the Austria FWF grant F50-04subm. to DMTCS c (cid:13) by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
Manuel Kauers and Rika Yatchak together with the initial values f , , = 1 , f i,j, = 0 for ( i, j ) = (0 , , and the boundary conditions f − ,j,n = f i, − ,n = 0 for all i, j, n . Equivalently, we can say that the generating function f ( x, y, t ) = P ∞ n =0 P i,j f i,j,n x i y j t n ∈ Q [ x, y ][[ t ]] satisfies the functional equation (cid:16) − t X u,v a u,v x u y v (cid:17) f ( x, y, t )= 1 − ty (cid:16)X u a u, − x u (cid:17) f ( x, , t ) − tx (cid:16)X v a − ,v y v (cid:17) f (0 , y, t ) + a − , − txy f (0 , , t ) . (1)Its first terms are f ( x, y, t ) = 1 + (cid:0) a , xy + a , x + a , y (cid:1) t + (cid:0) a , x y + 2 a , a , x y + ( a , + a , − a , ) x + 2 a , a , xy + 2 a , a , xy + ( a , a , − + a , − a , ) x + ( a , + a − , a , ) y + ( a − , a , + a − , a , ) y + ( a , − a , + a − , a , + a − , − a , ) (cid:1) t + · · · . This means, for example, that there are a − , a , + a − , a , many walks of length n = 2 ending at ( i, j ) = (0 , .For the models where all multiplicities a u,v are in { , } , a complete classification is available: amongthe = 256 different models, Bousquet-M´elou and Mishna (2010) identified 79 nontrivial cases. For 22of them they prove that the generating function is D-finite using certain symmetry groups G associated toeach of the models. For a 23rd model, the notorious Gessel model {← , → , ր , ւ} , their techniques donot apply but a proof by a different method based on computer algebra was found by Bostan and Kauers(2010). A computer-free proof was later found by Bostan et al. (2013). The remaining 56 models arenot D-finite: Mishna and Rechnitzer (2009) and Melczer and Mishna (2013) showed that the generatingfunctions of five of these models have infinitely many singularities and therefore are not D-finite. For theremaining models, Bostan et al. (2014b) proved that the counting sequences f , ,n for walks returning tothe origin have asymptotic behaviour for n → ∞ that D-finite functions cannot possibly have.The need for a classification of quarter plane models with multiplicities arose in the classificationproject for octant models in 3D (Bostan et al., 2014a), as it turns out that some models in 3D can bereduced by projection to 2D models with multiplicities. For example, it is easy to see that the generatingfunction for the octant model with step set { (cid:16) − (cid:17) , (cid:16) − (cid:17) , (cid:16) − (cid:17) , (cid:16) (cid:17) } is D-finite if and only if the quadrantmodel with step set {← , ← ′ , ↓ , ր} is. Bostan et al. (2014a) have classified only the 527 models that theyneeded for their study, and point out that the classification problem for models with multiplicities is ofinterest in its own right.For the present paper we carried out a systematic search over all the = 65536 models where each ofthe eight directions may have any of the four multiplicities 0, 1, 2, 3. Of these, 30307 are nontrivial andessentially different, and of these, 1457 turn out to be D-finite, and of these, 79 are even algebraic. Goingone step further, we have identified families of D-finite models in which some or all of the “multiplicities”are arbitrary complex numbers. Rather than asking for a fixed model what the corresponding group is,we ask for a fixed group what all the models leading to this group are. In this way we obtain a smallnumber of families that completely characterize all the models which lead to groups with at most eightelements. This characterization covers 1454 of the 1457 D-finite cases we discovered for multiplicities in { , , , } , the remaining three models have a group of order 10, which was too hard for us to analyze alks in the Quarter Plane with Multiple Steps a u,v ∈ { , } it is noteworthy that the generating function for a model isD-finite if and only if the associated group is finite, and it is algebraic if and only if the so-called orbit-sum(cf. Section 4 below) is zero. It seems that these equivalences remain true for models with multiplicities. Our reasoning largely follows that of Bousquet-M´elou and Mishna (2010). Their first step is to identifythe interesting models. By a model, we understand here a particular choice of multiplicities a u,v ∈ C (not necessarily integers). For each such model there is a corresponding generating function f ( x, y, t ) ∈ C [ x, y ][[ t ]] , and we want to identify the models whose generating functions are D-finite.A model is uninteresting if a , − = a , = a , = 0 or a − , = a , = a , = 0 or a − , − = a − , = a − , = 0 or a − , − = a , − = a , − = 0 , because in either of these cases the corresponding generatingfunction is algebraic and it is well-understood why (Flajolet and Sedgewick, 2009, Section VII.8). Sec-ondly, if two models can be obtained from one another by reflecting the step set about the diagonal x = y ,then the corresponding generating functions can be obtained from one another by exchanging the vari-ables x ↔ y , and therefore either both are D-finite or neither is. Similarly, if one model can be obtainedfrom another by multiplying all multiplicities by a nonzero constant λ , then its generating function can beobtained from the generating function of the other by sending t to λt , and therefore again either both areD-finite or neither is.Applying all these filters to the = 65536 models with possible multiplicities a u,v ∈ { , , , } leaves us with 30307 nontrivial models (including, for the sake of completeness, the 79 interesting modelswith a u,v ∈ { , } that have already been completely classified). For a fixed model, i.e., for a fixed choice of multiplicities a u,v ∈ C , consider the functional equation (1).The group associated to the model acts on this equation. Its elements map the variables x and y to certainrational functions in x and y , which are chosen in such a way that all the group elements leave the kernelpolynomial K ( x, y, t ) := 1 − t X u,v a u,v x u y v fixed. It is easy to check that the two particular transformations Φ , Ψ : C ( x, y ) → C ( x, y ) defined by Φ : ( x, y ) (cid:16) x P v a − ,v y v P v a ,v y v , y (cid:17) , Ψ : ( x, y ) (cid:16) x, y P u a u, − x u P u a u, x u (cid:17) have this property. It is also easy to check that Φ and Ψ are involutions, i.e., Φ = Ψ = id .The group G is defined as the group generated by Φ and Ψ under composition.Note that we do not need to worry that one of the denominators P u a u, x u or P v a ,v y v is identicallyzero, because this only happens for models that are uninteresting in the sense of the previous section. For Manuel Kauers and Rika Yatchak the same reason, we may also assume that the numerators P u a u, − x u and P v a − ,v y v , respectively, arenonzero polynomials. In order to argue that the composition of rational functions into the power seriesof equation (1) is algebraically meaningful, recall that the series in question belong to Q [ x, y ][[ t ]] , so theresult of the composition can be naturally interpreted as an element of C ( x, y )[[ t ]] .The group G is finite if and only if (ΦΨ) n = id for some n ∈ N , and this is the case if and only if G = (cid:8) id , ΦΨ , (ΦΨ) , · · · , (ΦΨ) n − , Φ , (ΦΨ)Φ , (ΦΨ) Φ , · · · , (ΦΨ) n − Φ (cid:9) , where all the listed elements are distinct. In other words, G is either the dihedral group with n elements,or infinite. The sign sgn( g ) of an element g ∈ G is defined to be if g = (ΦΨ) k for some k , and − otherwise. As there is obviously no way to choose a u,v such that ΦΨ = ( x r ( y ) , y s ( x )) = ( x, y ) = id , the smallestpossible n ∈ N with (ΦΨ) n = id is . The group with (ΦΨ) = id is the dihedral group D4 with fourelements. In order to determine the models which lead to this group, regard the a u,v as variables andcompute ( p, q ) := Φ(Ψ( x, y )) − Ψ(Φ( x, y )) . This is a pair of rational functions in x, y whose coefficientsare rational functions in the a u,v over the rational numbers. Write p, q as quotients of polynomials in x, y whose coefficients are polynomials in a u,v with integer coefficients. We want to know the possiblechoices of a u,v for which p and q become zero. (Note that ΦΨ = ΨΦ ⇐⇒ (ΦΨ) = id because Φ and Ψ are involutions.) In order to find these a u,v , consider the ideal in Q [ a − , − , . . . , a , ] generated bythe coefficients of all monomials x i y j in the numerator of p and the coefficients of all monomials x i y j in the numerator of q . This ideal basis consists of 36 homogeneous polynomials of degree 4, which wedon’t reproduce here because of its length. Using Gr¨obner basis techniques (Becker et al., 1993), we candetermine the irreducible components of the radical of this ideal. We have used the commands facstd and minAssGTZ of the software package Singular (Greuel and Pfister, 2002) for this step. It turns outthat the two irreducible components are generated by { a , a , − − a , − a , , a − , a , − − a − , − a , , a − , a , − − a − , − a , } , and { a , a − , − a − , a , , a , − a − , − a − , − a , , a , − a − , − a − , − a , } . As the latter is obtained from the former by replacing all a u,v by a v,u , it suffices to consider one of thetwo components, say the first. The equations in this component are equivalent to saying that the vectors ( a − , − , a , − , a , − ) and ( a − , , a , , a , ) are linearly dependent. Since the models where one or bothof these vectors are zero are uninteresting, the interesting models leading to the group D4 are preciselythose for which there exists a constant λ = 0 such that a − ,v = λa ,v for v = − , , . We then have Φ( x, y ) = (cid:16) λx , y (cid:17) and Ψ( x, y ) = (cid:16) x, y λa , − x − + a , − + a , − xλa , x − + a , + a , x (cid:17) . At this point, we can proceed analogously to Bousquet-M´elou and Mishna (cf. their Proposition 5): mul-tiplying (1) on both sides by xy/K ( x, y, t ) and forming the orbit sum gives the general relation X g ∈ G sgn( g ) g ( xy f ( x, y, t )) = 1 K ( x, y, t ) X g ∈ G sgn( g ) g ( xy ) , alks in the Quarter Plane with Multiple Steps K ( x, y, t ) (cid:16) xy − yλx − xy λa , − x − + a , − + a , − xλa , x − + a , + a , x + λxy λa , − x − + a , − + a , − xλa , x − + a , + a , x (cid:17) = ( x − λ )( a , xy − a , − x − ( λ + x )( a , − − a , y )) xy ( a , ( λ + x ) + a , x ) K ( x, y, t ) . For the left hand side, we have xy f ( x, y, t ) − λyx f (cid:16) λx , y, t (cid:17) − xy s ( x ) f (cid:16) x, y s ( x ) , t (cid:17) + λxy s ( x ) f (cid:16) λx , y s ( x ) , t (cid:17) , where we abbreviate s ( x ) = λa , − x − + a , − + a , − xλa , x − + a , + a , x . The identity holds in Q ( x, y )[[ t ]] , but it can be seenthat all quantities actually belong to Q ( x )[ y, y − ][[ t ]] . The last two terms of the equation involve onlynegative exponents with respect to y , so taking the positive part [ y > ] will kill them. The remaining termshappen to belong to Q [ x, x − ][[ t ]] , and since the second term only has negative exponents with respectto x , taking the positive part [ x > ] will eliminate it and only leave the first. It follows that f ( x, y, t ) = 1 xy [ x > ][ y > ] ( x − λ )( a , xy − a , − x − ( λ + x )( a , − − a , y )) xy ( a , ( λ + x ) + a , x ) K ( x, y, t ) . Alternatively, we could interpret the elements of Q ( x, y )[[ t ]] as elements of multivariate formal Laurentseries field Q ≤ (( x, y, t )) for a term order ≤ with x, y ≤ ≤ t and do the positive part extraction withrespect to x and y simultaneously. See Aparicio Monforte and Kauers (2013) for a discussion of formalLaurent series in several variables. In any case, we can summarize the result of this section as follows. Theorem 1
The interesting quarter plane models whose group is D4 are precisely those where a − ,v = λa ,v for v = − , , and some λ = 0 . All these models are D-finite. Family 0Defining equations: a , a , − = a , − a , , a − , a , − = a − , − a , , a − , a , − = a − , − a , Example: − −
713 6 − We now determine all the choices for a u,v such that (ΦΨ) = id . As before, we compute ( p, q ) :=Ψ(Φ(Ψ( x, y ))) − Φ(Ψ(Φ( x, y ))) and consider the ideal generated by the coefficients of the numeratorswith respect to x, y . The basis consists of 210 homogeneous polynomials of degree 9. The ideal has 34irreducible components, 18 of which turn out to contain only uninteresting models. Of the remaining 16components, 6 can be discarded because their solution sets are properly contained in the solution set ofothers. Of the remaining 10 components, 4 can be discarded because they are reflections of others. Thisleaves us with the following 6 families:
Manuel Kauers and Rika Yatchak
Family 1aDefining equations: a , = a − , − = 0 , a − , a , − = a − , a , = a , a , − Example: / / / Family 1bDefining equations: a , − = a − , = 0 , a − , a , = a − , − a , = a , − a , Example: / / / Family 2aDefining equations: a , = a , = 0 , a , − a − , = 2 a , a − , − , a , − = 4 a , − a − , − , a , − a , = 2 a − , a , − Example:
Family 2bDefining equations: a , = a , − = 0 , a , a − , − = 2 a , − a − , , a , = 4 a , a − , , a , a , − = 2 a − , − a , Example:
Family 3aDefining equations: a − , = a − , − = 0 , a , a , − = 2 a , − a , , a , = 4 a − , a , , a , a , − = 2 a , − a − , Example:
77 5121
Family 3bDefining equations: a − , = a − , = 0 , a , − a , = 2 a , a , − , a , − = 4 a − , − a , − , a , − a , = 2 a , a − , − Example:
77 5121
Note that the families on the right can be obtained from those on the left by reflection about the hori-zontal axis and the families in the third row can be obtained from those in the second row by reversing allarrows. The families in the first row are closed under reversing arrows.
Theorem 2
The interesting quarter plane models whose group is D6 are precisely those that belong toone or more of the families described in the table above. All these models are D-finite.
The remainder of this section is devoted to the D-finiteness claim of this theorem.
These families can be handled very much like the family in Section 4 above. Without going into furtherdetails, we just report the resulting formulas for the generating functions.For family 1a, let λ = a − , a , − = a − , a , = a , a , − . If λ = 0 , then all the a u,v are nonzero(except a , and a − , − of course). In this case, the resulting formula for the generating function is f ( x, y, t ) = 1 xy [ x > y > ] ( a − , − a , − xy − )( a , − − a − , yx − )( λxy − a − , a , − ) λ K ( x, y, t ) . Otherwise, if λ = 0 and a − , = 0 , then a − , = 0 and a , = 0 (otherwise the model is not interesting),but then a , = 0 and a , − = 0 (by the defining equations), and then a , − = 0 (otherwise again themodel is not interesting). In this case, the resulting formula for the generating function is f ( x, y, t ) = 1 xy [ x > y > ] ( a , − a , − xy − )( a , − − a − , x − y )( a , xy − a − , ) a , a , − K ( x, y, t ) . alks in the Quarter Plane with Multiple Steps λ = 0 and a − , = 0 , then a , − = 0 and the only interesting cases have a , = 0 , a , − = 0 ,and a − , = a , = 0 . This case is symmetric to the previous case and therefore not interesting.For family 2a, we may assume that a , − = 0 , because otherwise the model is uninteresting. Then wecan also assume a , = 0 , because if a , = 0 , then the last defining equation would imply a − , = 0 ,which together with a , = 0 would also render the model not interesting. Under the assumption a , − =0 , a , = 0 , the generating function can be expressed as f ( x, y, t ) = 1 xy [ x > y > ] P ( x, y ) (cid:0) a − , − − a , − x + a − , y + a − , y (cid:1)(cid:0) a , y − a , − x − a , − (cid:1) x y a , a , − K ( x, y, t ) where P ( x, y ) = 2 a − , − − a , xy + a , − x + 2 a − , y + 2 a − , y .For family 3a, models are interesting only when a − , = 0 and a , − = 0 and ( a , − , a , , a , ) =(0 , , . Under these assumptions, we obtain the following expression for the generating function: f ( x, y, t ) = 1 xy [ x > y > ] Q ( x, y ) (cid:0) a , y − a , − +2 a − , y /x (cid:1)(cid:0) a , x y + a , x + a , − x /y − a − , y (cid:1)(cid:0) a , − + a , y + a , y (cid:1)(cid:0) a − , a , − + (2 a − , + a , x (cid:1) Q ( x, y )) K ( x, y, t ) where Q ( x, y ) = 2 a , xy + 2 a , xy + 2 a , − x + a , y − a , − . Note that in this case the denominatorcontains nontrivial factors involving both x and y , so the ad-hoc reasoning used in Section 4, whichalso works for the families 1a and 2a, does not work here. However, there is no problem if we take theviewpoint of multivariate Laurent series (Aparicio Monforte and Kauers, 2013), because all that is neededfor the argument to go through is the property that there exists a term order ≤ so that for all g ∈ G \ { id } and all positive integers i, j the expansion of g ( x ) i g ( y ) j ∈ C ( x, y ) in the multivariate Laurent series field C ≤ (( x, y )) contains no terms x k y ℓ where both k and ℓ are positive. This turns out to be the case. For the family 1b there are three cases to distinguish. First, when a − , − = a , = a , = 0 , then a , , a − , , a , − all must be nonzero in order for the model to be interesting. In this case, the generatingfunction is f ( x, y, t ) = k (cid:16) q a , − a , /a − , x, q a − , a , /a , − y, q a − , a , − a , t (cid:17) , where k ( x, y, t ) is the generating function for classical Kreweras walks (i.e., a , = a − , = a , − = 1 ),which is well-known to be algebraic (Kreweras, 1965; Bousquet-M´elou, 2005). Secondly, when a , = a − , = a , − = 0 , algebraicity of the generating function can be established by a similar argument. Thethird case is when a , , a − , − , a , , a − , , a , , a , − are all nonzero. In this case it is impossible toexpress the generating function in terms of the generating function for the corresponding model withoutmultiplicities, known as the double Kreweras model. However, if we let f λ ( x, y, t ) be the generatingfunction for the family where a − , − = a − , = a , − = 1 and a , = a , = a , = λ = 0 , then f ( x, y, t ) = f a , a , / ( a , − a , ) (cid:16) a , a , x, a , a , y, a − , − a , a , a , t (cid:17) is the generating function of an arbitrary model of family 1b with a , a − , − = 0 . It therefore sufficesto show that f λ ( x, y, t ) is D-finite. We will show that it is in fact algebraic, following the treatment in Manuel Kauers and Rika Yatchak
Section 6.3 of Bousquet-M´elou and Mishna (2010) step by step with the added parameter λ . The orbit sumargument does not work here because the orbit sum turns out to be zero. We therefore sum (1) over onlyhalf the orbit to obtain a nonzero expression on both sides. This new expression will be more complicatedthan in the orbit sum case: in general, it will involve the unknown series f λ ( x, y, t ) , f λ ( x, , t ) and f λ (0 , , t ) . However, by careful coefficient extraction, the algebraicity result is still attainable.Writing A v = P u a u,v x u for v = − , , , the half-orbit sum equation reads xyf λ ( x, y, t ) − λx f λ (cid:16) λxy , y (cid:17) + 1 λy f λ (cid:16) λxy , x (cid:17) = xy − λx + λy − txA − f λ ( x, , t ) + tf λ (0 , , t ) K ( x, y, t ) . Next we extract the coefficient of y . In order to do so, we use Lemma 7 from Bousquet-M´elou and Mishna(2010). That is, we solve K ( x, y, t ) = 0 for y in terms of x and t : writing ∆( x ) := t x − − t +2 λt ) x − + (1 − λt ) − λt (1 + 2 t ) x + λ t x for the discriminant of K ( x, y, t ) , the two solutionsare Y = (cid:0) − tA − p ∆( x ) (cid:1) / (2 tA ) and Y = 1 / ( λxY ) . The coefficient of y n in /K ( x, y, t ) can beexpressed in terms of Y , Y , and ∆( x ) via [ y n ] 1 K ( x, y, t ) = 1 p ∆( x ) × (cid:26) Y − n if n ≤ Y − n if n ≥ . Using these facts, extracting the coefficient of y on both sides of the half-orbit sum equation leads to − λx d λ (cid:16) λx , t (cid:17) = 1 p ∆( x ) (cid:16) xY − λx + 1 λY − txA − f λ ( x, , t ) + tf λ (0 , , t ) (cid:17) , (2)where d λ ( x, t ) := P i,n ( f λ ) i,i,n x i t n is the generating function for walks ending on the diagonal.Now we write ∆( x ) = t Z ∆ − ( x )∆ + ( x ) , where ∆ + ( x ) = 1 − λZ (1 + 2 Z + 2 λZ + 2 λ Z + λ Z )(1 − λZ ) x + λ Z x , ∆ − ( x ) = ∆ + (cid:16) x (cid:17) , and where Z ∈ Q [ λ ][[ t ]] is defined through Z = t (1+3 λZ +4 λ (1+ λ ) Z +3 λ Z + λ Z )(1 − λZ ) and Z (0) = 0 .Multiplying (2) by A p ∆ − ( x ) and using the explicit expressions for Y and Y given above, we obtain p ∆ − ( x ) (cid:16) xt − λx A d λ (cid:16) λx , t (cid:17)(cid:17) = ZA t p ∆ + ( x ) (cid:16) x (1 − tA ) tA − xλ − tA − f λ ( x, , t ) + tf λ (0 , , t ) (cid:17) . From this equation, we extract the coefficient of x . Using [ x ] d λ ( λx , t ) = [ x ] f λ ( x, , t ) = f λ (0 , , t ) ,we find f λ (0 , , t ) = Z − λZ − λZ − λ Z − λ Z t (1 − λZ ) . With this knowledge, we can now extract the positive part in x on both sides of the same equation toobtain f λ ( x, , t ) = x ( λZ −
1) + 2 xZ ( λZ + 1) − λZ + Z λtx ( x + 1) Z (1 − λZ ) p ∆ + ( x ) − Z t (1 + x ) (cid:18) λtx + 2 tx + t − x λtx ( x + 1) Z + 2( λ Z + λ ( Z + 3) Z − − λZ ) + 1 (cid:19) . Noting that f λ (0 , y, t ) = f λ ( y, , t ) , we conclude from equation (1) that f λ ( x, y, t ) is algebraic. alks in the Quarter Plane with Multiple Steps The orbit sum argument also fails for these families. For the models in family 2b the orbit sum is zero,while in family 3b the orbit sum is nonzero but the desired term f ( x, y, t ) cannot be isolated by taking thepositive part because there are group elements g = id for which f ( g ( x ) , g ( y ) , t ) also contributes termswith positive exponents to the orbit sum. Because of the lack of symmetry, the half orbit sum argumentused for family 1b does not seem to apply either.One model from each of these two families were already encountered by Bostan et al. (2014a), andcomputer proofs have been given there that the generating function for the model belonging to family 2bis algebraic and the model belonging to family 3b is (transcendental) D-finite. The models consideredby Bostan et al. (2014a) are a , = a , − = a − , = 0 , a − , = a , = a , = a − , − = a , − = 1 (case 2b), and its reverse a − , = a − , = a , = 0 , a , − = a , − = a − , − = a , = a , = 1 (case 3b).We were able to extend these computer proofs to the more general cases where a − , = λ (case 2b), and a , = λ (case 3b), respectively, are formal parameters. From here, every other model of the respectivefamily can be reached by an appropriate algebraic substitution: if f λ ( x, y, t ) is the generating function forthe model a , = a , − = a − , = 0 , a − , = a , = a , = a − , − = a , − = 1 , a − , = λ , then f ( x, y, t ) = f a − , / √ a − , − a − , (cid:16) a , − a − , − x, r a − , a − , − y, r a − , − a − , t (cid:17) is the generating function for an arbitrary model of family 2b, and likewise for family 3b. (Models wherethe a u,v ’s appearing in the denominators are zero are not interesting.)The computational techniques we used were introduced by Kauers and Zeilberger (2008); Kauers et al.(2009); Bostan and Kauers (2010), and they have been described for the cases λ = 0 in the paperof Bostan et al. (2014a). We do not repeat these explanations again but only remark that the additionalsymbolic parameter λ has made the calculations considerably more expensive. The computations weredone using software of Kauers (2009) and Koutschan (2010). The bottleneck was the construction of acertified recurrence for ( f λ ) , ,n . The (nonminimal) recurrence we found has order 14 and degrees 30, 26in n , λ , respectively; the certificate for this recurrence is 16 gigabytes long! From this recurrence it canbe deduced that f λ (0 , , t ) is the unique formal power series T ∈ Q [ λ ][[ t ]] with T (0) = 1 and t T + (2 tλ + 1) t T + t (4 t + 1) − (3 t ( λ −
4) + 3 t + 1) Z + t (6 t + 1)( λ + 2) Z = 0 , where Z ∈ Q [ λ ][[ t ]] is the unique formal power series with Z (0) = 0 and t = Z (4 Z +1)1+6 Z +12 Z +4(2+ λ ) Z .Using this equation and the functional equation (1) (with f λ in place of f ), we could then prove thecorrectness of guessed polynomial equations P ( x, t, λ, f λ ( x, , t )) = Q ( y, t, λ, f λ (0 , y, t )) = 0 , whichin turn can be used to deduce that f λ ( x, , t ) is the unique formal power series U ∈ Q [ x, λ ][[ t ]] with U (0) = 1 and ( x + 1) t U + (2 tλ − x + 1) t U + t ( t ( x + 4) + 1) − (3 t ( λ −
4) + 3 t + 1) Z + t (6 t + 1)( λ + 2) Z = 0 and that f λ (0 , y, t ) is (cid:0) − p tyV (cid:1) / ( ty ) where V the unique formal power series V ∈ Q [ y, λ ][[ t ]] Manuel Kauers and Rika Yatchak with V (0) = 1 and ( λy + y + 1) t V + (cid:0) t (6 t + 1)( λ + 2) yZ − ty (3 t λ − t + 3 t + 1) Z + t (6 t λy + 4 t λ + 2 t y + 18 t y − tλy + 2 ty + 4 ty + 2 t − y ) (cid:1) V + t (4 tλy + ty + 16 t + 2 y + 4) − t λ − t + 3 t + 1) Z + 4(6 t + 1) t ( λ + 2) Z = 0 . Together with the functional equation, it finally follows that f λ ( x, y, t ) is algebraic.For the generating function ¯ f λ ( x, y, t ) of the model with a − , = a − , = a , = 0 , a − , − = a , − = a , − = a , = a , = 1 , a , = λ from family 3b, we have that ¯ f λ (0 , , t ) = f λ (0 , , t ) (for combinatorial reasons), and we can use this and the functional equation to certify guessed systemsof partial linear differential equations for ¯ f λ ( x, , t ) and ¯ f λ (0 , y, t ) which then together with the functionequation (1) (now with ¯ f λ in place of f ) imply that ¯ f λ ( x, y, t ) is D-finite. The equations are somewhattoo large to be included here: ¯ f λ ( x, , t ) satisfies a differential equation of order 11 with respect to t with polynomial coefficients of respective degrees 82, 90, 110 in x , λ and t , while ¯ f λ (0 , y, t ) satisfiesa differential equation of order 11 with respect to t with polynomial coefficients of respective degrees70, 58, 90 in y , λ and t . For the possible values of a u,v such that (ΦΨ) = id , we obtain, after discarding components that onlycontain uninteresting models or are redundant or are reflections of others, three essentially different primeideals. One of them is the ideal from Section 4, which appears again because (ΦΨ) = id implies (ΦΨ) = id . The other two define the following families:Family 4aDefining equations: a , − a − , = a , a − , , a , = a , = a , − = a − , − = 0 Example:
63 24
Family 4bDefining equations: a , a − , − = a , a − , , a , − = a , = a − , = a − , = 0 Example:
In family 4a, we must have a , − = 0 and a − , = 0 for a model to be interesting. But then a , − a − , = 0 implies also a , = 0 and a − , = 0 through the first defining equation. Similarly,we can assume for the models in family 4b that a , , a − , , a , a − , − all are nonzero.For family 4a, the orbit sum argument applies and yields f ( x, y, t ) = 1 xy [ x > y > ] ( a , − x/y − a − , y/x )( a , − /y − a , )( a , x − a − , /x )( a , x − a − , y/x ) a − , a , K ( x, y, t ) as expression for the generating function.For family 4b the orbit sum is zero, but it was pointed out by Bostan et al. (2014a) in their Section 6.2that its D-finiteness can be deduced from the D-finiteness of the corresponding model without multiplic-ities. Indeed, if g ( x, y, t ) denotes the generating function for the Gessel model {ւ , ← , → , ր} withoutmultiplicities, then we have f ( x, y, t ) = g (cid:16)r a , a − , x, a , a , y, q a , a − , t (cid:17) alks in the Quarter Plane with Multiple Steps g ( x, y, t ) is known to be algebraic(Bostan and Kauers, 2010; Bostan et al., 2013), it follows that all the models of family 4b are algebraic. Theorem 3
The interesting quarter plane models whose group is D8 are precisely those that belong toone of the families described in the table above. All these models are D-finite.
For n ≥ we failed to compute the prime decomposition of the ideal of relations among the a u,v thatensures (ΦΨ) n = id . The required calculations become too expensive. However, in our search over all the30307 quarter plane models with multiplicites in { , , , } we did encounter, very much to our surprise,the following three models that do not belong to any of the families discussed so far. Their group is D10.
11 2 1211 121 1 112 121 2 111
The orbit sum is zero, and guessing suggests that for all three models the generating function is algebraic.The models on the left and in the middle can be obtained from one another by reversing arrows,therefore these two models have the same number of walks returning to the origin. If Z ∈ Q [[ t ]] is the unique formal power series with Z (0) = 0 satisfying Z = t (4 Z + 8 Z + 2 Z + 1) , so that Z = t + 2 t + 12 t + 60 t + · · · , then we believe that f (0 , , t ) = Zt (1 − Z + 2 Z ) . More generally,for the model on the left we seem to have f ( x, , t ) = f (0 , x, t ) = P ( x, Z ) − ( x − Z )(2 xZ + x − p − xZ ( Z + 1)2 tx ( x + 1) Z with P ( x, Z ) = 2 Z + ( x − x (4 Z + 8 Z − x − Z + 1) , and then, using equation (1), the fullgenerating function f ( x, y, t ) can be expressed in terms of all these algebraic series. For the model in themiddle, we find a slightly messier expression for f ( x, , t ) = f (0 , x, t ) , also quadratic over t, Z, x , whichagain implies (if correct) that f ( x, y, t ) is algebraic. Since the first two models are symmetric about thediagonal, we expect that these formulas can be proven in a similar way as the models of family 1b inSection 5.2 above, but we have not gone through the details of the required calculations.The model on the right also seems to have an algebraic generating function. We found that f ( x, , t ) seems to satisfy an algebraic equation P ( x, t, f ( x, , t )) = 0 for some irreducible polynomial P ∈ Z [ x, t, T ] of respective degrees 40, 45, 24 in x, t, T , and f (0 , y, t ) seems to satisfy an algebraic equa-tion Q ( y, t, f (0 , y, t )) = 0 for some irreducible polynomial Q ∈ Z [ y, t, T ] of respective degrees 64, 45,24 in y, t, T . We expect that these algebraic equations can be proven by computer algebra in a similarway as the models of family 2b in Section 5.3 above, but this would require immense calculations whichwe have not carried out.The model obtained from the model on the right by reversing arrows is just its reflection and thereforealso algebraic but not of interest.Using substitutions like in earlier sections, the three models can be used to generate three families ofmodels. The corresponding ideals of defining relations for the a u,v have dimension three. We do notknow whether these families completely characterize all the interesting models whose group is D10, nordo we know anything about models for even larger groups. Does there exist for every n ≥ a quarterplane model with multiplicities whose group is D n ?2 Manuel Kauers and Rika Yatchak
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