Liouvillian Solutions of Irreducible Second Order Linear Difference - PowerPoint PPT Presentation

Preliminaries Tools Liouvillian Liouvillian Solutions of Irreducible Second Order Linear Difference Equations Mark van Hoeij and Giles Levy July 28, 2010 Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Preliminaries Definition τ will refer to the shift operator acting on C ( n ) by τ : n �→ n + 1. i a i τ i acts as Lu ( n ) = � An operator L = � i a i u ( n + i ). Definition C ( n )[ τ ] is the ring of linear difference operators where ring multiplication is composition of operators L 1 L 2 = L 1 ◦ L 2 . Definition Let S = C N / ∼ where s 1 ∼ s 2 if there exists N ∈ N such that, for all n > N , s 1 ( n ) = s 2 ( n ). Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Definition V ( L ) refers to the solution space of the operator L , i.e. V ( L ) := { u ∈ S | Lu = 0 } . If L = � k i =0 a i τ i , a 0 , a k � = 0, then dim ( V ( L )) = k (‘A=B’ Theorem 8.2.1). Definition A function or sequence v ( n ) such that v ( n + 1) / v ( n ) = r ( n ) is a rational function of n will be called a hypergeometric term . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Gauge Transformation Properties Tools Let D = C ( n )[ τ ]. If L ∈ D with L � = 0 then D / DL is a D − module. Definition L 1 is gauge equivalent to L 2 when D / DL 1 and D / DL 2 are isomorphic as D − modules. Lemma L 1 is gauge equivalent to L 2 if and only if ∃ G ∈ D such that G ( V ( L 1 )) = V ( L 2 ) and L 1 , L 2 have the same order. Thus G defines a bijection V ( L 1 ) → V ( L 2 ) . Definition The bijection defined by G in the preceding lemma will be called a gauge transformation . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Gauge Transformation Properties Definition The companion matrix of a monic difference operator L = τ k + a k − 1 τ k − 1 + · · · + a 0 , a i ∈ C ( n ) will refer to the matrix:   0 1 . . . 0 0 . . . . ... . . . .   . . . .   M = .   0 0 . . . 1 0     0 0 . . . 0 1   − a 0 − a 1 . . . − a k − 2 − a k − 1 Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Gauge Transformation Properties The equation Lu = 0 is equivalent to the system τ ( Y ) = MY where   u ( n ) . . Y =  .   .  u ( n + k − 1) Definition Let L = a k τ k + a k − 1 τ k − 1 + · · · + a 0 , a i ∈ C ( n ). The determinant of L , det( L ) := ( − 1) k a 0 / a k , i.e. the determinant of its companion matrix. Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Gauge Transformation Properties Definition Two rational functions will be called shift equivalent , denoted SE r 1 ≡ r 2 , if τ − r 1 / r 2 has a rational solution or, equivalently, the difference modules for τ − r 1 and τ − r 2 are isomorphic. Lemma If there exists a gauge transformation G : V ( L 1 ) → V ( L 2 ) then SE det( L 1 ) ≡ det( L 2 ) . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Liouvillian Liouvillian solutions are defined in Hendriks-Singer 1999 Section 3.2. For irreducible operators they are characterized by the following theorem: Theorem (Propositions 31-32 in Feng-Singer-Wu 2009 or Lemma 4.1 in Hendriks-Singer 1999) An irreducible k’th order operator L has Liouvillian solutions if and only if L is gauge equivalent to τ k + α, α ∈ C ( n ) . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Finding a gauge equivalence to τ k + α is desirable because it is easily solved with interlaced hypergeometric terms, e.g. τ 2 − 4( n + 2) / ( n + 7) has solutions: � Γ( n k 1 , if n even 2 + 1) 2 ) · 2 n · Γ( n 2 + 7 k 2 , if n odd where k 1 , k 2 are arbitrary constants. Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Definition Let L 1 , L 2 ∈ C ( n )[ τ ]. The symmetric product of L 1 and L 2 is defined as the monic operator L ∈ C ( n )[ τ ] of smallest order such that L ( u 1 u 2 ) = 0 for all u 1 , u 2 ∈ S with L 1 u 1 = 0 and L 2 u 2 = 0. Definition The symmetric square of L , denoted L � 2 , will refer to the symmetric product of L and L (i.e. with itself). Lemma Let L = a 2 τ 2 + a 1 τ + a 0 , a 0 , a 2 � = 0 . � 2 , if a 1 = 0 L � 2 has order: 3 , if a 1 � = 0 Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Commutative Diagram L = a 2 τ 2 + a 1 τ + a 0 , a 1 � = 0 L = τ 2 + α ˜ G = τ + g G V (˜ α, g ∈ C ( n ), unknown V ( L ) − − − − → L ) − − − − → 0   u v   ↓ ↓   u 2 v 2 � � G 2 → V (˜ 0 → V ( GCRD ( G 2 , L � 2 )) → V ( L � 2 ) L � 2 ) − − − − − − − − → 0 dim 1 dim 3 dim 2 Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Algorithm Algorithm Find Liouvillian: Input: L ∈ C [ n ][ τ ] a second order, irreducible, homogeneous difference operator. Let L = a 2 ( n ) τ 2 + a 1 ( n ) τ + a 0 ( n ) and let L � 2 = c 3 τ 3 + c 2 τ 2 + c 1 τ + c 0 . Output: A two-term difference operator, ˆ L , with a gauge transformation from ˆ L to L , if it exists. 1 If a 1 = 0 then return ˆ L = L and stop. 2 Let u ( n ) be an indeterminate function. Impose the relation Lu ( n ) = 0, i.e. 1 u ( n + 2) = − a 2 ( n )( a 0 ( n ) u ( n ) + a 1 ( n ) u ( n + 1)) . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Algorithm (continued) 3 Let d = det( L ) = a 0 / a 2 . Let R be a non-zero rational solution of L T := L � 2 ⊗ ( τ + 1 / d ) , if such a solution exists, else return NULL and stop. 4 Let g be an indeterminate and let → V (ˆ G := τ + g : V ( L ) − L ) Compute corresponding G 2 : V ( L � 2 ) → V (ˆ L � 2 ). 5 From R (solution of L T ) take the corresponding solution of L � 2 , plug this corresponding solution into G 2 , and equate to 0. 6 The equation computed above is quadratic in g . Solve the equation for g and choose one solution. Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Example Let L = n τ 2 − τ − ( n 2 − 1)(2 n − 1) , Lu ( n ) = 0: d = − ( n 2 − 1)(2 n − 1) / n L T = n ( n + 3) (2 n + 3) ( n + 1) 2 τ 3 − 2 n 3 + 3 n 2 − n + 1 � � τ 2 − n ( n + 2) 2 n 3 + 3 n 2 − n + 1 � � ( n + 2) ( n + 1) τ + n ( n + 2) ( n − 1) ( n + 1) (2 n − 1) R = 1 A = 1 n · ( g 2 + (3 n − 2) g + (2 n − 1)( n − 1)) n , δ = 1 − n 2 g = 1 − n , Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations

Preliminaries Tools Liouvillian Commutative Diagram Algorithm Example Example (continued) leading to the output: ˆ Lv ( n ) = v ( n + 2) − (2 n − 1)( n + 2) v ( n ) , u ( n ) = 1 1 nv ( n ) + n 2 − 1 v ( n + 1) . Mark van Hoeij and Giles Levy Liouvillian Solutions of Irreducible Second Order Linear Difference Equations