Convolutions of Liouvillian Sequences
aa r X i v : . [ c s . S C ] M a r Convolutions of Liouvillian Sequences
Sergei A. Abramov ∗ Marko Petkovˇsek † Helena Zakrajˇsek ‡ Abstract
While Liouvillian sequences are closed under many operations, simpleexamples show that they are not closed under convolution, and the samegoes for d’Alembertian sequences. Nevertheless, we show that d’Alember-tian sequences are closed under convolution with rationally d’Alembertiansequences, and that Liouvillian sequences are closed under convolutionwith rationally
Liouvillian sequences.
Keywords: (rationally) d’Alembertian sequences; (rationally) Liouvilliansequences; closure properties; convolutionMSC (2010) 68W30; 33F10
Let K be an algebraically closed field of characteristic 0, N the set of nonnegativeintegers, and K N the set of all sequences with terms in K . Definition 1
A sequence h a n i ∞ n =0 ∈ K N is: • polynomial if there is p ∈ K [ x ] such that a n = p ( n ) for all n ∈ N , • rational if there is r ∈ K ( x ) such that a n = r ( n ) for all large enough n , • quasi-rational ( cf. [1]) if there are d ∈ N , rational functions r , r , . . . , r d ∈ K ( x ) ∗ , and α , α , . . . , α d ∈ K ∗ such that a n = P di =1 r i ( n ) α ni for all largeenough n . Definition 2
A sequence h a n i ∞ n =0 ∈ K N is P -recursive or holonomic if thereare d ∈ N and polynomials p , p , . . . , p d ∈ K [ n ] , p d = 0 , such that p d ( n ) a n + d + p d − ( n ) a n + d − + · · · + p ( n ) a n = 0 for all n ∈ N . In particular, a holonomic sequence is hypergeometric if ∗ Dorodnicyn Computing Center, Federal Research Center “Computer Science and Con-trol”, Russian Academy of Sciences, Moscow, Russia † Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathemat-ics, Physics and Mechanics, Ljubljana, Slovenia ‡ Faculty of Mechanical Engineering, University of Ljubljana, Slovenia . there are p, q ∈ K [ n ] \ { } such that q ( n ) a n +1 + p ( n ) a n = 0 for all n ≥ ,
2. there is an N ∈ N such that a n = 0 for all n ≥ N .The set of all holonomic sequences in K N will be denoted by P ( K ) , and the setof all hypergeometric sequences in K N by H ( K ) . Example 1
Some hypergeometric sequences: • a n = c n where c ∈ K ∗ (geometric sequences) • a n = p ( n ) where p ∈ K [ n ] \ { } (nonzero polynomial sequences) • a n = r ( n ) for all large enough n where r ∈ K ( n ) ∗ (nonzero rational se-quences) • a n = n ! • a n = (cid:18) nn (cid:19) Our long-term goal is to design algorithms for finding explicit representations of holonomic sequences in terms of • some basic sequences expressible in closed form (such as, e.g., rationalsequences or hypergeometric sequences), • some common operations with sequences which preserve holonomicity.Here we are particularly interested in those explicit representations whose ad-missible operations include convolution, also known as Cauchy product. A short overview of the paper:
In Section 2 we introduce several well-knownholonomicity-preserving operations (see also [8]), and call an operation Ω local if the equivalence relation holding between sequences which eventually agree isa congruence w.r.t. Ω (i.e., if, when given equivalent operands, Ω yields equiv-alent results). Convolution is not local which makes working with it a littleharder. In line with the above-described scheme of defining classes of explicitrepresentations, we describe in Section 3 the rings of d’Alembertian, Liouvil-lian, rationally d’Alembertian (cf. [2]), and rationally Liouvillian sequences. Inthe former two cases, the basis is the set of hypergeometric sequences, and inthe latter two, the ring of rational sequences. While the former two are notclosed under convolution, we show in Sections 4 resp. 5 that the convolutionof a d’Alembertian sequence with a (quasi-)rationally d’Alembertian sequence(see Def. 8) is d’Alembertian (Corollary 3), and the convolution of a Liouvilliansequence with a (quasi-)rationally Liouvillian sequence (see Def. 9) is Liouvil-lian (Corollary 4). We divide the proof (for d’Alembertian sequences) into twoparts: Theorem 5 deals with the “ideal” case where the minimal annihilators of2he hypergeometric resp. rational sequences in the two factors are nonsingular,and Corollary 3 takes care of the rest. In Section 6 we list some open problemsand present an algorithm for finding solutions of a linear recurrence that areconvolutions with a given hyperexponential sequence.
Some further definitions and notation: • For x ∈ K and n ∈ N , we denote by x n := Q n − j =0 ( x − j ) the n -th fallingpower of x . • For n, m ∈ N , m ≥
1, we denote by n div m := ⌊ nm ⌋ the quotient, and by n mod m := n − m ⌊ nm ⌋ the remainder in integer division of n by m . • The shift operator E : K N → K N is defined for all a ∈ K N , n ∈ N by E ( a ) n = a n +1 , and for k ∈ N , its k -fold composition with itself isdenoted by E k . For d ∈ N and p , p , . . . , p d ∈ K [ n ] such that p d = 0, themap L = P dk =0 p k ( n ) E k : K N → K N is a linear recurrence operator oforder ord L = d with polynomial coefficients. We denote the Ore algebraof all such operators (with composition as multiplication) by K [ n ] h E i . Theorem 1 P ( K ) is closed under the following operations: • unary operations a c scalar multiplication: c n = λa n where λ ∈ K shift: c n = E ( a ) n = a n +1 inverse shift: c n = E − λ ( a ) n = (cid:26) a n − , n ≥ λ, n = 0 where λ ∈ K difference: c n = ∆ a n = a n +1 − a n partial summation: c n = P nk =0 a k multisection: c n = a mn + r where m ∈ N \{ } , r ∈ { , , . . . , m − } (the r -th m -section of a ) • binary operations ( a, b ) c addition: c n = a n + b n multiplication: c n = a n b n convolution: c n = ( a ∗ b ) n = P nk =0 a k b n − k • polyadic operations ( a (0) , a (1) , . . . , a ( m − ) c where m ∈ N \ { } interlacing: c n = Λ( a (0) , a (1) , . . . , a ( m − ) n = (cid:0) Λ m − j =0 a ( j ) (cid:1) n = a ( n mod m ) n div m Proof:
Let L = P dk =0 p k ( n ) E k ∈ K [ n ] h E i \ { } be such that L ( a ) = 0.3. L ( λa ) = λL ( a ) = 0, so λa ∈ P ( K ).2. Let L ′ := P dk =0 p k ( n + 1) E k ∈ K [ n ] h E i \ { } . Then L ′ ( E ( a )) = d X k =0 p k ( n + 1) E k +1 ! ( a ) = ( EL )( a ) = E ( L ( a )) = 0 , so E ( a ) ∈ P ( K ).3. Note that EE − λ = id K N , hence( LE )( E − λ ( a )) = ( L ( EE − λ ))( a ) = L ( a ) = 0 , so E − λ ( a ) ∈ P ( K ).4. This follows from items 2, 1, and 7.For proofs in the remaining six cases, see [6, 11, 12]. (cid:3) Note that operations 7 – 9 are associative (and commutative), hence theycan also be considered as polyadic.
Definition 3 [11]
Sequences a, b ∈ K N are equivalent if there is an N ∈ N s.t. a n = b n for all n ≥ N or equivalently, s.t. E N ( a ) = E N ( b ) . We denote this relation by ∼ , and call itsequivalence classes germs (at ∞ of functions N → K ) .We say that a set of sequences C ⊆ K N is closed under equivalence if a ∈ C and a ∼ a ′ implies a ′ ∈ C . Proposition 1
The set H ( K ) is closed under equivalence.Proof: Assume that a ∈ H ( K ) and a ′ ∼ a . Then there are p, q ∈ K [ n ] \ { } and N ∈ N s.t. q ( n ) a n +1 + p ( n ) a n = 0 for all n ∈ N , and a ′ n = a n = 0 for all n ≥ N .Hence n N q ( n ) a ′ n +1 + n N p ( n ) a ′ n = 0for all n ∈ N , so a ′ ∈ H ( K ). (cid:3) Proposition 2
Let
C ⊆ K N be a class of sequences closed under all inverseshifts and addition, and such that ∈ C . Then C is closed under equivalence.Proof: Let a ∈ C and a ′ ∼ a . Then there are k ∈ N and λ , λ , . . . , λ k ∈ K s.t. a ′ − a = h λ , λ , . . . , λ k , , , , . . . i = E − λ E − λ · · · E − λ k (0) , so a ′ = a + E − λ E − λ · · · E − λ k (0) ∈ C . (cid:3) Corollary 1
The holonomic ring P ( K ) is closed under equivalence. efinition 4 An operation ω on K N is local if ∼ is a congruence w.r.t. ω . Proposition 3
The following operations are local: scalar multiplication, shift,inverse shift, difference, multisection, addition, multiplication, interlacing.Proof:
Straightforward. (cid:3)
Example 2
Partial summation is not local: Let, e.g., a = h , , , . . . i ,b = h , , , . . . i . Then a ∼ b but n X k =0 a k = 0 n X k =0 b k = 1 . Since P nk =0 a k = ( a ∗ n , it follows that convolution is not local either. When dealing with local operations, it is customary to work with germsof sequences which simplifies the statements of results and their correspondingproofs. Since here we are especially interested in the non-local operations ofconvolution and partial summation, we have to work with sequences themselves.In this situation, the following auxiliary results are useful.
Lemma 1
Let a, b, ε, η ∈ K N with ε, η ∼ . Then: (i) a ε ∼ , (ii) P nk =0 ε k ∼ C for some C ∈ K , (iii) a ∗ ε = P Nk =0 ε k E − k ( a ) for some N ∈ N , (iv) ε ∗ η ∼ , (v) ( a + η ) ∗ ( b + ε ) ∼ a ∗ b + P N i =0 ε i E − i ( a ) + P N j =0 η j E − j ( b ) for some N , N ∈ N .Proof: (i) This follows from locality of multiplication.(ii) Let N ∈ N be such that ε k = 0 for k > N . Write C = P Nk =0 ε k . For n ≥ N we have P nk =0 ε k = P Nk =0 ε k , so P nk =0 ε k ∼ C .(iii) Let N ∈ N be such that ε k = 0 for k > N . Then for all n ∈ N ,( a ∗ ε ) n = n X k =0 ε k a n − k = min { n,N } X k =0 ε k E − k ( a ) n = (cid:18) N X k =0 ε k E − k ( a ) (cid:19) n (1)where the last equality follows from the fact that E − k ( a ) n = 0 for k > n .5iv) Let N , N ∈ N be such that ε i = 0 for i > N and η j = 0 for j > N .Assume that n > N + N . Then k > N or n − k > N for every k ∈ N ,therefore ( ε ∗ η ) n = n X k =0 ε k η n − k = 0for all such n , so ε ∗ η ∼ a + η ) ∗ ( b + ε ) = a ∗ b + a ∗ ε + b ∗ η + ε ∗ η. The claim now follows from (iii) and (iv). (cid:3)
Here we list some subrings of the holonomic ring ( P ( K ) , + , · ), defined as closuresof a basic set of sequences under a set of holonomicity-preserving operations. Definition 5
The ring of d’Alembertian sequences A ( K ) is the least subring of ( P ( K ) , + , · ) which contains H ( K ) and is closed under • shift, • all inverse shifts, • partial summation. Example 3
Some d’Alembertian sequences: • derangement numbers d n = n ! P nk =0 ( − k k ! • harmonic numbers H n = P nk =1 1 k = P nk =0 1 k +1 − n +1 Definition 6
For d ∈ N \ { } and a (1) , a (2) , . . . , a ( d ) ∈ K N , we shall denote by NS (cid:16) a (1) , a (2) , . . . , a ( d ) (cid:17) = NS di =1 a ( i ) the sequence a ∈ K N defined for all k ∈ N by a k := (cid:16) NS di =1 a ( i ) (cid:17) k = a (1) k k X k =0 a (2) k k X k =0 a (3) k · · · k d − X k d =0 a ( d ) k d (2) and call it the nested sum of sequences a (1) , a (2) , . . . , a ( d ) . We will call thenumber d the nesting depth of this particular representation of a . heorem 2 Let a ∈ K N . Then: (i) a is d’Alembertian iff it can be written as a K -linear combination (possiblyempty) of nested sums of the form (2) where a (1) , a (2) , . . . , a ( d ) ∈ H ( K ) , (ii) a is d’Alembertian iff there are d ∈ N \ { } and L , L , . . . , L d ∈ K [ n ] h E i ,each of order 1, such that L L · · · L d ( a ) = 0 . For a proof, see [3] or [9].
Corollary 2 If y ∈ K N satisfies L ( y ) = a where L is a product of first-orderoperators and a ∈ A ( K ) , then y ∈ A ( K ) .Proof: By Theorem 2.(ii), there are d ∈ N \ { } and L , L , . . . , L d ∈ K [ n ] h E i ,each of order 1, such that L L · · · L d ( a ) = 0. Hence L L · · · L d L ( y ) = L L · · · L d ( a ) = 0 , so, again by Theorem 2.(ii), y ∈ A ( K ). (cid:3) Example 4
It is straightforward to verify that for • derangement numbers: ( E + 1)( E − ( n + 1))( d ) = 0 , • harmonic numbers: (( n + 2) E − ( n + 1))( E − H ) = 0 . Definition 7
The ring of
Liouvillian sequences L ( K ) is the least subring of ( P ( K ) , + , · ) which contains H ( K ) and is closed under • shift, • all inverse shifts, • partial summation, • interlacing. Example 5
Perhaps the simplest element of L ( K ) \ A ( K ) is a n = n !! , definedrecursively by a = a = 1 , a n = na n − for n ≥ . From n !! = ( k k ! , n = 2 k, (2 k +1)!2 k k ! , n = 2 k + 1 we see that n !! is the interlacing of two hypergeometric sequences, hence it isLiouvillian. On the other hand, its annihilating operator L = E − ( n + 2) hasno nonzero d’Alembertian elements in its kernel. Theorem 3
A sequence a ∈ K N is Liouvillian if and only if it is an interlacingof d’Alembertian sequences. For a proof, see [10] or [9]. 7 roposition 4 A ( K ) and L ( K ) are closed under equivalence.Proof: This follows immediately from Proposition 2. (cid:3)
The ring of Liouvillian sequences L ( K ) is, by Definition 7 or by its imme-diate consequences, closed under all the operations listed in Theorem 1, withpossible exception of multisection and convolution. It is easy to see that both A ( K ) and L ( K ) are closed under multisection (cf. [9] and [6]). With convolu-tion, the situation is much more varied already for hypergeometric operands, asdemonstrated by the following three examples. To avoid having to name everysequence that we encounter, we will often use a n ∗ b n to denote either ( a ∗ b ) n or a ∗ b , with the precise meaning determined by the context. Example 6
The convolution of /n ! with itself n ! ∗ n ! = n X k =0 k !( n − k )! = 1 n ! n X k =0 (cid:18) nk (cid:19) = 2 n n ! is hypergeometric. Example 7
Zeilberger’s Creative Telescoping algorithm [13, 14] shows that theconvolution of n ! with itself y n := n ! ∗ n ! = n X k =0 k !( n − k )! satisfies the recurrence y n +1 − ( n + 2) y n = 2( n + 1)! (3) which, together with the initial condition y = 1 , implies that y n = ( n + 1)!2 n n X k =0 k k + 1 . This is a d’Alembertian sequence which is not hypergeometric, as shown byGosper’s summation algorithm [5] , or by algorithm Hyper [7] applied to the ho-mogenization y n +2 − (3 n + 7) y n +1 + ( n + 2) y n = 0 of (3) . Example 8
Zeilberger’s algorithm shows that the convolution of n ! with /n ! y n := n ! ∗ n ! = n X k =0 k !( n − k )! satisfies the recurrence y n +2 − ( n + 2) y n +1 + y n = 1( n + 2)!8 hich can be homogenized (by applying the annihilator ( n + 3) E − of the right-hand side to both sides) to Ly = 0 where L = ( n + 3) E − ( n + 6 n + 10) E + (2 n + 5) E − . (4) This recurrence has no nonzero Liouvillian solutions, as shown by the Hendriks-Singer algorithm [6] . So the convolution of hypergeometric sequences n ! and /n ! is not Liouvillian. As shown by Example 8, neither A ( K ) nor L ( K ) is closed under convolu-tion. To obtain positive results, we define some further subrings of these ringsby replacing hypergeometric sequences with (quasi-)rational sequences as theirbasis. Definition 8
The ring of (quasi-)rationally d’Alembertian sequences A ( q ) rat ( K ) is the least subring of ( P ( K ) , + , · ) which contains all (quasi-)rational sequencesover K and is closed under • shift, • all inverse shifts, • partial summation. Example 9
Harmonic numbers H n = P nk =1 1 k are rationally d’Alembertian. Theorem 4
A sequence a ∈ K N is (quasi-)rationally d’Alembertian iff it can bewritten as a K -linear combination (possibly empty) of nested sums of the form(2) where a (1) , a (2) , . . . , a ( d ) are (quasi-)rational sequences. The proof is analogous to that of Theorem 2(i).
Definition 9
The ring of (quasi-)rationally Liouvillian sequences L ( q ) rat ( K ) isthe least subring of ( P ( K ) , + , · ) which contains all (quasi-)rational sequencesover K and is closed under • shift, • all inverse shifts, • partial summation, • interlacing. Example 10
The interlacing of harmonic numbers H n = P nk =1 1 k with gener-alized harmonic numbers of order 2, H (2) n = P nk =1 1 k , is rationally Liouvillian. D’Alembertian sequences under convolution
Proposition 5
For all k ∈ N and all a, b ∈ A ( K ) , we have a ∗ b ∈ A ( K ) ⇐⇒ E k ( a ) ∗ E k ( b ) ∈ A ( K ) . (5) Proof:
Note that, for all n ∈ N , E ( a ∗ b ) n = n +2 X k =0 a k b n +2 − k = a b n +2 + a n +2 b + n +1 X k =1 a k b n +2 − k = a b n +2 + a n +2 b + n X k =0 a k +1 b n +1 − k = a E ( b ) n + b E ( a ) n + ( E ( a ) ∗ E ( b )) n . As A ( K ) is closed under shift, scalar multiplication and addition, this implies E ( a ∗ b ) ∈ A ( K ) ⇐⇒ E ( a ) ∗ E ( b ) ∈ A ( K ) . By the closure of A ( K ) under shift and all inverse shifts, we have a ∗ b ∈ A ( K ) ⇐⇒ E ( a ∗ b ) ∈ A ( K ) , so a ∗ b ∈ A ( K ) ⇐⇒ E ( a ) ∗ E ( b ) ∈ A ( K ) . (6)As A ( K ) is closed under shift, we can replace a by E k ( a ) and b by E k ( b ) in (6)and obtain E k ( a ) ∗ E k ( b ) ∈ A ( K ) ⇐⇒ E k +1 ( a ) ∗ E k +1 ( b ) ∈ A ( K )for all k ∈ N . Now (5) follows by induction on k . (cid:3) Lemma 2
Let d ∈ N , a (1) , a (2) , . . . , a ( d ) ∈ K N , ε (1) , ε (2) , . . . , ε ( d ) ∈ K N , and ε (1) , ε (2) , . . . , ε ( d ) ∼ . Then there are c , c , . . . , c d ∈ K such that NS di =1 (cid:16) a ( i ) + ε ( i ) (cid:17) ∼ d X i =1 c i NS ij =1 a ( j ) . Proof:
By induction on d .If d = 0 both sides are 0. Now assume that the assertion holds at some d ≥
1, and expand the left-hand side. In line 2 we use the induction hypothesisand compensate for replacing equivalence with equality by adding a sequence η ∼ c : (cid:16) NS d +1 i =1 (cid:16) a ( i ) + ε ( i ) (cid:17)(cid:17) n = (cid:16) a (1) n + ε (1) n (cid:17) n X k =0 (cid:16) NS d +1 i =2 (cid:16) a ( i ) + ε ( i ) (cid:17)(cid:17) k = (cid:16) a (1) n + ε (1) n (cid:17) n X k =0 d +1 X i =2 c i (cid:16) NS ij =2 a ( j ) (cid:17) k + η k ! ∼ a (1) n n X k =0 d +1 X i =2 c i (cid:16) NS ij =2 a ( j ) (cid:17) k + a (1) n n X k =0 η k ∼ d +1 X i =2 c i (cid:16) NS ij =1 a ( j ) (cid:17) n + c a (1) n = d +1 X i =1 c i (cid:16) NS ij =1 a ( j ) (cid:17) n . (cid:3) Lemma 3
Let d ∈ N and a (1) , a (2) , . . . , a ( d ) ∈ K N . If N ∈ N is s.t. a ( i ) n = 0 forall n < N and i ∈ { , , . . . , d } , then E N (cid:16) NS di =1 a ( i ) (cid:17) = NS di =1 E N (cid:16) a ( i ) (cid:17) . Proof:
Write the nested sum on the left as a single sum, shift all summationindices by N , and use the fact that all original summands vanish below N : (cid:16) E N (cid:16) NS di =1 a ( i ) (cid:17)(cid:17) n = a (1) n + N n + N X k =0 a (2) k k X k =0 a (3) k · · · k d − X k d =0 a ( d ) k d = X ≤ k d ≤···≤ k ≤ k ≤ n + N a (1) n + N a (2) k a (3) k · · · a ( d ) k d = X − N ≤ k d ≤···≤ k ≤ k ≤ n a (1) n + N a (2) k + N a (3) k + N · · · a ( d ) k d + N = X ≤ k d ≤···≤ k ≤ k ≤ n a (1) n + N a (2) k + N a (3) k + N · · · a ( d ) k d + N = (cid:16) NS di =1 E N (cid:16) a ( i ) (cid:17)(cid:17) n . (cid:3) Theorem 5
Let d ∈ N , η , η , . . . , η d ∈ N , and h (1) , h (2) , . . . , h ( d ) ∈ H ( K ) . Let p , p , . . . , p d ∈ K [ x ] , q , q , . . . , q d ∈ K [ x ] be such that q i ( n ) h ( i ) n +1 = p i ( n ) h ( i ) n , q i ( n ) = 0 for all n ∈ N and i ∈ { , , . . . , d } . Let e ∈ N , ξ , ξ , . . . , ξ e ∈ N , and ϕ i ( x ) ∈ { α xi x j i , α xi ( x − β i ) − j i } for all i ∈ { , , . . . , e } where j , j , . . . , j e ∈ N , , α , . . . , α e ∈ K ∗ and β , β , . . . , β e ∈ K \ N . Let a ∈ K N be given by a = 0 if d = 0 , and a n = h (1) n n + η X k =0 h (2) k k + η X k =0 h (3) k · · · k d − + η d − X k d =0 h ( d ) k d for all n ∈ N if d ≥ . Let b ∈ K N be given by b = 0 if e = 0 , and b n = ϕ ( n ) n + ξ X k =0 ϕ ( k ) k + ξ X k =0 ϕ ( k ) · · · k e − + ξ e − X k e =0 ϕ e ( k e ) for all n ∈ N if e ≥ . Then a ∗ b is d’Alembertian.Proof: By induction on d + e + P ei =1 j i where d resp. e are the nesting depthsof these representations of a resp. b , and P ei =1 j i is the valuation of b . If d = 0or e = 0, then a ∗ b = 0 ∈ A ( K ). Now let d, e ≥
1. Write a n = h n n + η X k =0 ˜ a k (7)where h = h (1) , η = η and˜ a k = δ k , . . . . . . . . . . . . . . . . . . . . . . . . . , d = 1 ,h (2) k k + η X k =0 h (3) k · · · k d − + η d − X k d =0 h ( d ) k d , d ≥ δ k , = h , , , , . . . i ∈ K N the identity element for convolution. Write b n = ϕ ( n ) n + ξ X k =0 ˜ b k (8)where ϕ = ϕ (with j = j , α = α , β = β ), ξ = ξ and˜ b k = δ k , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , e = 1 ,ϕ ( k ) k + ξ X k =0 ϕ ( k ) · · · k e − + ξ e − X k e =0 ϕ e ( k e ) , e ≥ . We shall prove that the convolution y n := n X k =0 a k b n − k = n X k =0 h k ϕ ( n − k ) k + η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! is d’Alembertian by showing that L ( y ) ∈ A ( K ) for an appropriate operator L ∈ K [ n ] h E i , then invoking Corollary 2. We distinguish three cases:12 ase 1. ϕ ( x ) = α x In this case y n = P nk =0 a k α n − k P n − k + ξk =0 ˜ b k and we take L = E − α . Then( L ( y )) n = y n +1 − αy n = n +1 X k =0 a k α n +1 − k n +1 − k + ξ X k =0 ˜ b k − n X k =0 a k α n +1 − k n − k + ξ X k =0 ˜ b k = a n +1 ξ X k =0 ˜ b k + n X k =0 a k α n +1 − k ˜ b n − k + ξ +1 = E ( a ) n ξ X k =0 ˜ b k + α (cid:16) a n ∗ α n E ξ +1 (˜ b ) n (cid:17) where E ( a ) is d’Alembertian and α n E ξ +1 (˜ b ) n has nesting depth e −
1, hence a n ∗ α n E ξ +1 (˜ b ) n is d’Alembertian by induction hypothesis. Case 2. ϕ ( x ) = α x x j with j ≥ b n = n j c n where c n = α n P n + ξk =0 ˜ b k , and we take L = 1. So y n = n X k =0 a k ( n − k ) j c n − k = j X i =0 ( − i (cid:18) ji (cid:19) n j − i n X k =0 k i a k c n − k = j X i =0 ( − i (cid:18) ji (cid:19) n j − i (cid:0) ( n i a n ) ∗ c n (cid:1) . As the valuation of c is j less than that of b , our induction hypothesis impliesthat y is d’Alembertian. Case 3. ϕ ( x ) = α x ( x − β ) j with j ≥ L = q ( n − β ) E − p ( n − β ) where polynomials p, q ∈ K [ n ] \ { } are such that q ( n ) h n +1 − p ( n ) h n = 0 for all n ∈ N . Then( L ( y )) n == q ( n − β ) n +1 X k =0 h k α n +1 − k ( n + 1 − k − β ) j k + η X k =0 ˜ a k ! n +1 − k + ξ X k =0 ˜ b k ! − p ( n − β ) n X k =0 h k α n − k ( n − k − β ) j k + η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! = q ( n − β ) n X k = − h k +1 α n − k ( n − k − β ) j k +1+ η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! − p ( n − β ) n X k =0 h k α n − k ( n − k − β ) j k + η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! = A n + B n + C n , A n := q ( n − β ) a b n +1 ,B n := q ( n − β ) n X k =0 h k +1 ˜ a k +1+ η α n − k ( n − k − β ) j n − k + ξ X k =0 ˜ b k ! ,C n := n X k =0 q ( n − β ) h k +1 − p ( n − β ) h k ( n − k − β ) j α n − k k + η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! . Clearly A is d’Alembertian. Since B n = q ( n − β ) ( E ( hE η (˜ a )) ∗ b ) n and the nest-ing depth of E ( hE η (˜ a )) is d − B is d’Alembertian by the induction hypothesis.In C n we replace h k +1 with h k p ( k ) /q ( k ) and obtain C n = n X k =0 P ( k ) h k α n − k q ( k )( n − k − β ) j k + η X k =0 ˜ a k ! n − k + ξ X k =0 ˜ b k ! where P ( k ) := q ( n − β ) p ( k ) − p ( n − β ) q ( k ) ∈ K [ n ][ k ]. Since P ( n − β ) = 0, P ( k )is divisible by k − n + β , hence there are s ∈ N and c , c , . . . , c s ∈ K [ x ] suchthat P ( k ) = ( n − k − β ) P si =0 c i ( n ) k i . It follows that C n = s X i =0 c i ( n ) n X k =0 k i a k q ( k ) · α n − k ( n − k − β ) j − n − k + ξ X k =0 ˜ b k ! = s X i =0 c i ( n ) n X k =0 u ( i ) k · α n − k ( n − k − β ) j − n − k + ξ X k =0 ˜ b k ! = s X i =0 c i ( n ) (cid:16) u ( i ) n ∗ ( n − β ) b n (cid:17) where u ( i ) k := k i a k q ( k ) for all k ∈ N and i ∈ { , , . . . , s } . As the valuation of( n − β ) b n is one less than that of b n , our induction hypothesis implies that u ( i ) n ∗ ( n − β ) b n is d’Alembertian, hence so are C and L ( y ) = A + B + C . Since L has order one, Corollary 2 implies that y = a ∗ b is d’Alembertian. (cid:3) Corollary 3 If a ∈ K N is d’Alembertian and b ∈ K N is (quasi-)rationallyd’Alembertian, then their convolution a ∗ b is d’Alembertian.Proof: By Theorem 2.(i), the sequence a can be written as a K -linear combina-tion of sequences of the form NS di =1 h ( i ) where h (1) , h (2) , . . . , h ( d ) ∈ H ( K ). For i = 1 , , . . . , d , let p i , q i ∈ K [ x ] be such that q i ( n ) h ( i ) n +1 = p i ( n ) h ( i ) n for all n ∈ N .By Theorem 4, the sequence b can be written as a K -linear combination ofsequences of the form NS ei =1 r ( i ) where r (1) , r (2) , . . . , r ( e ) are (quasi-)rational se-quences. By the Partial Fraction Decomposition Theorem for rational functions,we can assume that for i = 1 , , . . . , e there are j i ∈ N , α i ∈ K ∗ and β i ∈ K suchthat r ( i ) n = ϕ i ( n ) for all large enough n , where ϕ i ( x ) ∈ { α xi x j i , α xi ( x − β i ) − j i } .14y bilinearity of convolution, it suffices to prove that the convolution of asingle NS di =1 h ( i ) with a single NS ei =1 r ( i ) is d’Alembertian, so henceforth weassume that a ≡ NS di =1 h ( i ) and b ≡ NS ei =1 r ( i ) . Let N ∈ N be such that q i ( n ) = 0 and r ( i ) n = ϕ i ( n ) for all n ≥ N and i ∈ { , , . . . , e } . We shall proveby induction on the sum of nesting depths d + e that a ∗ b is d’Alembertian.If d = 0 or e = 0 then a = 0 or b = 0 and so a ∗ b = 0 ∈ A ( K ).Assume now that d ≥ e ≥
1. Let ˜ a := NS dk =1 ˜ h ( k ) and ˜ b := NS ek =1 ˜ r ( k ) where ˜ h ( k ) = E − N E N (cid:0) h ( k ) (cid:1) and ˜ r ( k ) = E − N E N (cid:0) r ( k ) (cid:1) . Then ˜ h ( k ) n = ˜ r ( k ) n = 0for n < N and ˜ h ( k ) n = h ( k ) n , ˜ r ( k ) n = r ( k ) n for n ≥ N . It follows by Lemma 3 that E N (˜ a ) n = NS dk =1 E N (cid:16) ˜ h ( k ) (cid:17) n = NS dk =1 E N (cid:16) h ( k ) (cid:17) n = NS dk =1 h ( k ) n + N ,E N (˜ b ) n = NS ek =1 E N (cid:16) ˜ r ( k ) (cid:17) n = NS ek =1 E N (cid:16) r ( k ) (cid:17) n = NS ek =1 ϕ k ( n + N ) . Note that by our definition of N , the sequences E N (˜ a ) and E N (˜ b ) satisfy allthe assumptions of Theorem 5, so E N (˜ a ) ∗ E N (˜ b ) ∈ A ( K ). Proposition 5 nowimplies that ˜ a ∗ ˜ b ∈ A ( K ) as well.By Lemma 2, there are c , c , . . . , c d ∈ K and c ′ , c ′ , . . . , c ′ e ∈ K such that a = d X i =1 c i NS ij =1 ˜ h ( j ) + η = c d NS dj =1 ˜ h ( j ) + d − X i =1 c i NS ij =1 ˜ h ( j ) + η,b = e X i =1 c ′ i NS ij =1 ˜ r ( j ) + η ′ = c ′ e NS ej =1 ˜ r ( j ) + e − X i =1 c ′ i NS ij =1 ˜ r ( j ) + η ′ for some sequences η, η ′ ∼
0. Hence a ∗ b = c d c ′ e NS dj =1 ˜ h ( j ) ∗ NS ej =1 ˜ r ( j ) + c d NS dj =1 ˜ h ( j ) ∗ e − X i =1 c ′ i NS ij =1 ˜ r ( j ) + c ′ e NS ej =1 ˜ r ( j ) ∗ d − X i =1 c i NS ij =1 ˜ h ( j ) + d − X i =1 c i NS ij =1 ˜ h ( j ) ∗ e − X i =1 c ′ i NS ij =1 ˜ r ( j ) + η ∗ c ′ e NS ej =1 ˜ r ( j ) + e − X i =1 c ′ i NS ij =1 ˜ r ( j ) + η ′ ! + η ′ ∗ c d NS dj =1 ˜ h ( j ) + d − X i =1 c i NS ij =1 ˜ h ( j ) ! . The first term on the right equals c d c ′ e ˜ a ∗ ˜ b , so it is d’Alembertian as shownin the previous paragraph. The next three terms are linear combinations ofconvolutions of nested sums having nesting depths at most d + e − d + e − d + e −
2, respectively, so they are d’Alembertian by induction hypothesis.By Lemma 1.(iii), the last two terms above are linear combinations of shiftedd’Alembertian sequences, so they are d’Alembertian as well. It follows that a ∗ b is d’Alembertian as claimed. (cid:3) xample 11 By Corollary 3, the convolution of a hypergeometric sequence witha rational sequence, such as y n = (2 n − n !) ∗ n + ! = n X k =0 k − k ! n − k + , is d’Alembertian. By following through the proof of Theorem 5 with a n = 2 n − n ! and b n = 1 / ( n + ) , we will obtain an explicit nested-sum representation of y n .Here the nesting depths of a and b are , j = 1 , β = − / , h n = a n = 2 n − n ! , h n +1 /h n = p ( n ) = 2( n + 1) , q ( n ) = 1 and L = q ( n − β ) E − p ( n − β ) = E − (2 n + 3) . Applying L to y ( n ) we obtain ( L ( y )) n = n +1 X k =0 k − k ! n − k + − (2 n + 3) n X k =0 k − k ! n − k + = n X k = − k ( k + 1)! n − k + − (2 n + 3) n X k =0 k − k ! n − k + = 12 n + 3 + n X k =0 k − k !(2( k + 1) − (2 n + 3)) n − k + = 12 n + 3 + n X k =0 k − k !(2 k − n − n − k + = 12 n + 3 − n X k =0 k k ! = 12 n + 3 − n X k =0 (2 k )!! . (9) By solving this recurrence with initial condition y = 1 , we obtain y n = (2 n + 1)!! n X k =1 k + 1)!! k + 1 − k − X j =0 (2 j )!! . Since (2 n + 1)!! and (2 n )!! are hypergeometric sequences, this shows that y ( n ) isindeed a d’Alembertian sequence.From (9) we can also obtain a fully factored annihilator of y as follows:The right-hand side of (9) is annihilated by the least common left multiple of E − (2 n + 3) / (2 n + 5) and ( E − (2 n + 4))( E − , which is (cid:18) E − (2 n + 3)(2 n + 7) (2 n + 5) (2 n + 9) (cid:19) ( E − (2 n + 4)) ( E − , hence L ( y ) = 0 where L = (cid:18) E − (2 n + 3)(2 n + 7) (2 n + 5) (2 n + 9) (cid:19) ( E − (2 n + 4)) ( E − E − (2 n + 3)) . xample 12 The sequence y n = H n ∗ H n = n X k =0 H k H n − k = n X k =0 k X i =1 i n − k X j =1 j is annihilated by L = (( n + 3) E − ( n + 2)) ( E − . Example 13
The sequence y n = n ! ∗ H n = n X k =0 k ! H n − k = n X k =0 k ! n − k X j =1 j is annihilated by L = (( n + 5) E − ( n + 4)) (( E − ( n + 2))( E − . Here we establish some connections between convolution, interlacing and inverseshift E − which allow us to transfer results about convolutions of d’Alembertiansequences to the corresponding results about Liouvillian sequences. Recall thatthe (ordinary) generating series of a sequence a ∈ K N is defined as the formalpower series g a ( x ) = ∞ X k =0 a k x k , and that for all pairs of sequences a, b ∈ K N , g a + b ( x ) = g a ( x ) + g b ( x ) , g a ∗ b ( x ) = g a ( x ) g b ( x ) . Definition 10 [6]
For m ∈ N \{ } and a ∈ K N , we write Λ m a for Λ( a, m − z }| { , . . . , ,and call it the th m -interlacing of a with zeroes . Lemma 4
Let k ∈ N , m ∈ N \ { } , a, a (0) , a (1) , . . . , a ( m − ∈ K N , and b =Λ m − j =0 a ( j ) . (i) (cid:0) E − k ( a ) (cid:1) n = (cid:26) a n − k , n ≥ k , n < k (ii) (Λ m a ) n = (cid:26) a nm , n ≡ m )0 , n m )(iii) g E − k ( a ) ( x ) = x k g a ( x )(iv) g Λ m a ( x ) = g a ( x m ) 17v) g b ( x ) = P m − j =0 x j g a ( j ) ( x m )(vi) Λ m − j =0 a ( j ) = P m − j =0 E − j (cid:0) Λ m a ( j ) (cid:1) (vii) Λ m E − k = E − km Λ m Proof:
Items (i), (ii) follow immediately from the definitions of E − and Λ m .(iii): g E − k ( a ) ( x ) = ∞ X n =0 E − k ( a ) n x n = ∞ X n = k a n − k x n = ∞ X n =0 a n x n + k = x k g a ( x )(iv): g Λ m a ( x ) = ∞ X n =0 (Λ m a ) n x n = X n ≡ m ) a nm x n = ∞ X k =0 a k x km = g a ( x m )(v): g b ( x ) = P ∞ n =0 a ( n mod m ) n div m x n = P m − j =0 P ∞ k =0 a ( j ) k x km + j = P m − j =0 x j g a ( j ) ( x m )(vi): Using (v), (iv) and (iii) we find that g b ( x ) = m − X j =0 x j g a ( j ) ( x m ) = m − X j =0 x j g Λ m a ( j ) ( x ) = m − X j =0 g E − j ( Λ m a ( j ) )( x )= g P m − j =0 E − j ( Λ m a ( j ) )( x )which implies the assertion.(vii): By applying (iv) and (iii) alternatingly, we obtain g Λ m E − k ( a ) ( x ) = g E − k ( a ) ( x m ) = x km g a ( x m ) = x km g Λ m a ( x ) = g E − km Λ m a ( x )for every a ∈ K N , which implies the assertion. (cid:3) Proposition 6
The convolution of the th m -interlacings of a, b ∈ K N withzeroes is the th m -interlacing of a ∗ b with zeroes: Λ m a ∗ Λ m b = Λ m ( a ∗ b ) . Proof: g Λ m a ∗ Λ m b ( x ) = g Λ m a ( x ) g Λ m b ( x ) = g a ( x m ) g b ( x m )= ∞ X i =0 a i x mi ∞ X j =0 b j x mj = ∞ X i =0 ∞ X j =0 a i b j x m ( i + j ) = ∞ X k =0 x mk k X i =0 a i b k − i = ∞ X k =0 ( a ∗ b ) k ( x m ) k = g a ∗ b ( x m ) = g Λ m ( a ∗ b ) ( x )by using Lemma 4.(iv) three times. (cid:3) roposition 7 Let m ∈ N \ { } , a ( j ) , b ( j ) ∈ K N for all j ∈ { , , . . . , m − } , u = Λ m − j =0 a ( j ) , and v = Λ m − j =0 b ( j ) . Then u ∗ v = m − X k =0 min { k,m − } X j =max { ,k − m +1 } E − k Λ m (cid:16) a ( j ) ∗ b ( k − j ) (cid:17) . (10) Proof:
Using Lemma 4.(v), we obtain g u ∗ v ( x ) = g u ( x ) g v ( x ) = m − X j =0 x j g a ( j ) ( x m ) m − X ℓ =0 x ℓ g b ( ℓ ) ( x m )= m − X j =0 m − X ℓ =0 x j + ℓ g a ( j ) ( x m ) g b ( ℓ ) ( x m )= m − X k =0 k X j =0 + m − X k = m m − X j = k − m +1 x k g a ( j ) ( x m ) g b ( k − j ) ( x m )= m − X k =0 min { k,m − } X j =max { ,k − m +1 } x k g a ( j ) ( x m ) g b ( k − j ) ( x m ) (11)By Lemma 4.(iv), Proposition 6 and Lemma 4.(iii), x k g a ( j ) ( x m ) g b ( k − j ) ( x m ) = x k g Λ m a ( j ) ( x ) g Λ m b ( k − j ) ( x ) = x k g Λ m a ( j ) ∗ Λ m b ( k − j ) ( x )= x k g Λ m ( a ( j ) ∗ b ( k − j ) )( x ) = g E − k Λ m ( a ( j ) ∗ b ( k − j ) )( x )which, together with (11), implies (10). (cid:3) Corollary 4
The convolution of a Liouvillian sequence u with a (quasi-)ratio-nally Liouvillian sequence v is Liouvillian.Proof: Let u = Λ m − i =0 a ( i ) with all a ( i ) d’Alembertian, and v = Λ k − i =0 b ( i ) with all b ( i ) (quasi-)rationally d’Alembertian. Let ℓ = lcm( m, k ). Write u = Λ ℓ − j =0 c ( j ) , v = Λ ℓ − j =0 d ( j ) where c ( j ) and d ( j ) , for j = 0 , , . . . , ℓ −
1, are the j -th l -sections of u and v , respectively. Clearly all c ( j ) , d ( j ) are themselves sections of a ( i ) resp. b ( i ) .Since A ( K ) is closed under multisection [9, Prop. 7], all c ( j ) are d’Alembertian.Similarly one can show that the ring of (quasi-)rationally d’Alembertian se-quences is closed under multisection, hence all d ( j ) are (quasi-)rationally d’Alem-bertian. So by Corollary 3, all convolutions c ( j ) ∗ d ( j ) for j , j ∈ { , , . . . , ℓ − } are d’Alembertian. It follows from Proposition 7 that u ∗ v is a sum of shiftedinterlacings of d’Alembertian sequences, hence it is Liouvillian. (cid:3) Example 14
By Corollary 4, the convolution of a Liouvillian sequence with arational sequence, such as y n := n !! ∗ (cid:18) n + 1 (cid:19) = n X k =0 k !! n − k + 1 , s Liouvillian. By following the proof of Proposition 7 with u n = n !! and v n = n +1 , we will obtain a representation of y n as an interlacing of d’Alembertiansequences. Here m = 2 , u = Λ( a (0) , a (1) ) and v = Λ( b (0) , b (1) ) , where a (0) n = (2 n )!! = 2 n n ! ,a (1) n = (2 n + 1)!! = (2 n + 1)!2 n n ! ,b (0) n = v n = 12 n + 1 ,b (1) n = v n +1 = 12 n + 2 . By Proposition 7 at m = 2 , u ∗ v = Λ (cid:16) a (0) ∗ b (0) (cid:17) + E − Λ (cid:16) a (0) ∗ b (1) + a (1) ∗ b (0) (cid:17) + E − Λ (cid:16) a (1) ∗ b (1) (cid:17) . Denote g (0) := a (0) ∗ b (0) + E − (cid:16) a (1) ∗ b (1) (cid:17) ,g (1) := a (0) ∗ b (1) + a (1) ∗ b (0) . For any a, b, c, d ∈ K N we have Λ( a + b, c + d ) = Λ( a, c ) + Λ( b, d ) , therefore Λ (cid:16) g (0) , g (1) (cid:17) = Λ (cid:16) a (0) ∗ b (0) , a (0) ∗ b (1) (cid:17) + Λ (cid:16) E − (cid:16) a (1) ∗ b (1) (cid:17) , a (1) ∗ b (0) (cid:17) , which by Lemma 4.(vi) at m = 2 equals Λ (cid:16) a (0) ∗ b (0) (cid:17) + E − Λ (cid:16) a (0) ∗ b (1) + a (1) ∗ b (0) (cid:17) + Λ E − (cid:16) a (1) ∗ b (1) (cid:17) . Since Λ E − (cid:0) a (1) ∗ b (1) (cid:1) = E − Λ (cid:0) a (1) ∗ b (1) (cid:1) by Lemma 4.(vii) at m = 2 , itfollows that u ∗ v = Λ (cid:0) g (0) , g (1) (cid:1) . It remains to show that g (0) and g (1) ared’Alembertian. We have ( a (0) ∗ b (0) ) n = n X k =0 k k !2( n − k ) + 1 = n X k =0 k − k ! n − k + , ( a (1) ∗ b (1) ) n − = n X k =0 (2 k + 1)!2 k k !(2( n − k −
1) + 2) = n X k =0 (2 k + 1)!2 k +1 k !( n − k ) , ( a (0) ∗ b (1) ) n = n X k =0 k k !2( n − k ) + 2 = n X k =0 k − k ! n − k + 1 , ( a (1) ∗ b (0) ) n = n X k =0 (2 k + 1)!2 k k !(2( n − k ) + 1) = n X k =0 (2 k + 1)!2 k +1 k !( n − k + ) . In an analogous way as we did it for ( a (0) ∗ b (0) ) n in Example 11, we can computeexplicit d’Alembertian representations for ( a (1) ∗ b (1) ) n − , ( a (0) ∗ b (1) ) n , and a (1) ∗ b (0) ) n . After some additional simplification we obtain g (0) n = (2 n + 1)!! n X k =1 k + 1)!! k + 12 k (2 k + 1) − k − X j =0 j !! ,g (1) n = (2 n + 2)!!
34 + n X k =1 k + 2)!! k + 3(2 k + 1)(2 k + 2) − k − X j =0 j !! . Note that k − X j =0 j !! = k − X j =0 (2 j )!! + k − X j =0 (2 j + 1)!! , k − X j =0 j !! = k − X j =0 (2 j )!! + k − X j =0 (2 j + 1)!! , hence both g (0) and g (1) are d’Alembertian sequences, and u ∗ v , as their inter-lacing, is Liouvillian. As we have seen, the ring of Liouvillian sequences is not closed under convolu-tion. Several questions now arise naturally, such as:
Question:
Given a homogeneous linear recurrence equation with polynomial co-efficients, how can we find all its solutions having the form of: • a convolution of hypergeometric sequences, • a convolution of d’Alembertian sequences, • a convolution of Liouvillian sequences, • an expression built from hypergeometric sequences using the ten operationslisted in Theorem 1? We do not have the answers to these questions (listed by increasing degree ofdifficulty), so we consider their relaxed versions where one of the factors of theconvolutional solution is part of the input data for the problem. For example,the relaxed version of the simplest question above is the following:
Question:
Given a homogeneous linear recurrence equation with polynomial co-efficients and a hypergeometric sequence a , how can we find all hypergeometricsequences b such that a ∗ b is a solution of the given equation? Even to this simple-looking question we do not have a complete answer.We do have a partial answer in the special case when the given sequence a is21 yperexponential , i.e., its generating series g a ( x ) = P ∞ n =0 a n x n ∈ K [[ x ]] satisfiesthe first-order linear differential equation g ′ a ( x ) = r ( x ) g a ( x )for some rational function r ∈ K ( x ). Here we consider the ring of two-waysequences K Z on which the shift operator E is an automorphism, and the actionof operators from K [ n ] h E, E − i on it. To each a ∈ K N we assign its paddingwith zeroes ζ ( a ) ∈ K Z defined by ζ ( a ) n = (cid:26) a n , n ≥ , , n < . We present here an algorithm which, given L ∈ K [ n ] h E i and a fixed hyperexpo-nential sequence a , returns an operator L ′ ∈ K [ n ] h E, E − i such that L ′ ( ζ ( b )) =0 for all b ∈ K N with L ( ζ ( a ∗ b )) = 0. This algorithm is based on the well-known fact that a ∈ K N is P-recursive iff g a ( x ) is D-finite (see [11, Thm.1.4]). Constructively, there exists a skew-Laurent-polynomial algebra isomor-phism R : K [ n ] h E, E − i → K [ x, x − ] h D i (where D = ∂∂x is the derivativeoperator , Dg a ( x ) = g ′ a ( x )) which fixes each element of K and satisfies R : n xD, R : E x − , R : E − x, (12) R − : x E − , R − : x − E, R − : D ( n + 1) E (13)(cf. [4, Sec. 5]), such that for any L ∈ K [ n ] h E, E − i and any a ∈ K N we have L ( ζ ( a )) = 0 ⇐⇒ R ( L )( g a ( x )) = 0 . (14)The algorithm is as follows: Algorithm HyperExpFactorInput: L ∈ K [ n ] h E i with ord L ≥ r ∈ K ( x ) ∗ Output: L ′ ∈ K [ n ] h E, E − i
1. Compute M = R ( L ) ∈ K [ x, x − ] h D i using (12).2. Compute M ′ ∈ K [ x, x − ] h D i such that M ′ ( v ) = 0 ⇐⇒ M ( uv ) = 0whenever u ′ ( x ) = r ( x ) u ( x ).3. Compute and return L ′ = R − ( M ′ ) ∈ K [ n ] h E, E − i using (13).22 roposition 8 Let the output of algorithm
HyperExpFactor with input
L, r be L ′ , and let a ∈ K N be s.t. g ′ a ( x ) = r ( x ) g a ( x ) . Then for every b ∈ K N , we have L ( ζ ( a ∗ b )) = 0 ⇐⇒ L ′ ( ζ ( b )) = 0 . Proof:
Going backwards through algorithm
HyperExpFactor , we obtain L ′ ( ζ ( b )) = 0 ⇐⇒ R − ( M ′ )( ζ ( b )) = 0 by ( ) with L := R − ( M ′ ) ⇐⇒ M ′ ( g b ( x )) = 0 ⇐⇒ M ( g a ( x ) g b ( x )) = 0 ⇐⇒ R ( L )( g a ∗ b ( x )) = 0 by ( ) ⇐⇒ L ( ζ ( a ∗ b )) = 0 (cid:3) Example 15
Let L be the operator from Example 8 and let a n = 1 /n ! , so that r ( x ) = g ′ a ( x ) /g a ( x ) = e x /e x = 1 . Running the above algorithm with input L = ( n + 3) E − ( n + 6 n + 10) E + (2 n + 5) E − and r ( x ) = 1 we obtain1. M = − x − ( x D − ( x − x − D + ( x − x − ,2. M ′ = − x − ( x D + (3 x − D + 1) ,3. L ′ = E (( n + 1) E − ( n + 1) ) = ( n + 3) E − ( n + 3) E .Denote y = ζ ( b ) . Then ( L ′ y ) n = ( n + 3) y n +3 − ( n + 3) y n +2 = 0 holds for all n ≤ − since y n = 0 for n < . For n ≥ − we can cancel thecommon factor n + 3 and obtain y n +3 − ( n + 3) y n +2 = 0 , or equivalently, y n +1 − ( n + 1) y n = b n +1 − ( n + 1) b n = 0 for all n ≥ . It follows that b n = Cn ! for C ∈ K , in agreement with the factthat we constructed L as an annihilator of n ! ∗ n ! . Acknowledgements
S.A.A. acknowledges financial support from the Russian Foundation for BasicResearch (project No. 16-01-001174). M.P. acknowledges financial support fromthe Slovenian Research Agency (research core funding No. P1-0294). The paperwas completed while M.P. was attending the thematic programme “Algorithmicand Enumerative Combinatorics” at the Erwin Schr¨odinger International Insti-tute for Mathematics and Physics in Vienna, Austria, to which he thanks forsupport and hospitality. 23 eferences [1] Abramov, S.A., 1991. An algorithm for finding quasi-rational solutions ofdifferential and difference equations with polynomial coefficients (Russian).
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