ANALOG OF FAVARD S THEOREM FOR POLYNOMIALS CONNECTED WITH DIFFERENCE EQUATION OF 4-TH ORDER. S. M. Zagorodniuk

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2 Serdica Math J 27 (2001, ANALOG OF FAVARD S THEOREM FOR POLYNOMIALS CONNECTED WITH DIFFERENCE EQUATION OF 4-TH ORDER S M Zagorodniuk Communicated by E I Horozov Abstract Orthonormal polynomials on the real line {p n (λ} n=0 satisfy the recurrent relation of the form: λ n 1 p n 1 (λ + α n p n (λ + λ n p n+1 (λ = λp n (λ, n = 0, 1, 2,, where λ n > 0, α n R, n = 0, 1, ; λ 1 = p 1 = 0, λ C In this paper we study systems of polynomials {p n (λ} n=0 which satisfy the equation: α n 2 p n 2 (λ + β n 1 p n 1 (λ + γ n p n (λ + β n p n+1 (λ + α n p n+2 (λ = λ 2 p n (λ, n = 0, 1, 2,, where α n > 0, β n C, γ n R, n = 0, 1, 2,, α 1 = α 2 = β 1 = 0, p 1 = p 2 = 0, p 0 (λ = 1, p 1 (λ = cλ + b, c > 0, b C, λ C It is shown that they are orthonormal on the real and the imaginary axes pm (λ in the complex plane: (p n (λ, p n ( λdσ(λ = δ R T p m ( λ n,m, n, ( m = 0, ; T = (, with respect to some matrix measure σ(λ = σ1 (λ σ 2 (λ σ 3 (λ σ 4 (λ Also the Green formula for difference equation of 4-th order is built 2000 Mathematics Subject Classification: 42C05, 39A05 Key words: orthogonal polynomials, difference equation

3 194 S M Zagorodniuk Let us consider the recurrent relation for the system of polynomials {p n (λ} n=0 of the form: (1 α n 2p n 2 (λ + β n 1 p n 1 (λ + γ n p n (λ + β n p n+1 (λ + α n p n+2 (λ = λ 2 p n (λ, n = 0,1,2,, where α n > 0, β n C, γ n R, n = 0,1,2,, α 1 = α 2 = β 1 = 0, p 1 = p 2 = 0, p 0 (λ = 1, p 1 (λ = cλ + b, c > 0, b C, λ C This relation can be written in the matrix form: or γ 0 β 0 α β 0 γ 1 β 1 α α 0 β 1 γ 2 β 2 α α 1 β 2 γ 3 β 3 α 3 J 5 p = λ 2 p, p 0 p 1 p 2 p 3 = λ2 where J 5 is five-diagonal, symmetric, semi-infinite matrix and p is vector of polynomials Let us study the properties of these systems of polynomials Note, that orthonormal polynomials on the real line belong to this class of polynomials For them the following equation is fulfilled: J 3 p = λp with three-diagonal, symmetric, semi-infinite matrix and from this immediately it follows, that J3 2p = λ2 p and J3 2 is five-diagonal, symmetric, semi-infinite matrix (Question: how larger is the considered class than the class of real polynomials? For the answer see [1, Theorem 5, p 272] ( σ1 (λ σ Definition ([1, p ] Find matrix measure σ(λ= 2 (λ σ 3 (λ σ 4 (λ λ C; σ i (λ : C C is piecewise continuous mapping on the real and the imaginary axis, i = 1,4, such that 1 σ(λ is symmetric, monotonically increasing matrix function: σ 1 (λ = σ 1 (λ, σ 4 (λ = σ 4 (λ, σ 2 (λ = σ 3 (λ; σ(λ 2 σ(λ 1, λ 2 λ 1, λ 1,λ 2 R p 0 p 1 p 2 p 3, 2 R T λ 2 σ(λ 2 σ(λ 1, λ 1 i i, λ 1,λ 2 ( i,i (λ k,( λ k 1 d σ(λ = s 1 k, k = 0,,

4 Analog of Favard s theorem 195 (λ k 1,( λ k 1 λ d σ(λ = m R T λ k, k = 1,, where {s k } k=0, {m k} k=1 are fixed sequences of complex numbers; We ll call this problem generalized symmetric moments problem Definition ([1, p 266] We call a pair of sequences {s k,m k+1 } k=0, s k C,m k+1 C, k = 0, symmetric if s 2k+1 = m 2k+1 ; s 2k = s 2k,m 2k+2 = m 2k+2, k = 0, We call a pair of sequences {s k,m k+1 } k=0, s k C, m k+1 C, k = 0, positive one if s 0 s 1 s k s 0 s 1 s k m 1 m 2 m k+1 m 1 m 2 m k+1 s 2 s 3 s k+2 > 0, k = 2l + 1; s 2 s 3 s k+2 > 0,k = 2l m k m k+1 m 2k s k s k+1 s 2k l = 0, The following theorem holds true: Theorem 1 ([1, Theorem 6, p 274] Let moments problem in general form be given For the existance of a problem s solution σ(λ (with an infinite number of increasing points it is necessary and sufficient the pair of sequences {s k,m k+1 } k=0 to be symmetric and positive Zolotarev suggested to investigate systems of polynomials, which satisfy (1 He also gave the usefull notes The next theorem is an analog of Favard s theorem [2, Theorem 15, p 60] It gives the orthonormality properties for the systems from (1 Theorem 2 Let the system of polynomials {p n (λ} n=0 satisfy (1 Then these polynomials are orthonormal on the real and the imaginary axes in the complex plane: R T pm (λ (p n (λ,p n ( λdσ(λ = δ p m ( λ n,m, n,m = 0, ;T = (,,

5 196 S M Zagorodniuk σ1 (λ σ where σ(λ = 2 (λ is symmetric, non-decreasing matrix-function σ 3 (λ σ 4 (λ with infinite number of points of increasing: σ 1 (λ = σ 1 (λ, σ 4 (λ = σ 4 (λ, σ 2 (λ = σ 3 (λ; σ(λ 2 σ(λ 1, λ 2 λ 1, λ 1,λ 2 R, σ(λ 2 σ(λ 1, λ 2 i λ 1,λ 2 T, T = ( i,i λ 1 i, Proof Let {p n (λ} n=0 be given system of polynomials We define a functional σ(u,v, where u,v P, P is a space of all polynomials If u(λ,v(λ P, than we can write: u(λ = n a i p i (λ, v(λ = m b j p j (λ, a i C, i = 0,n, b j C,j = 0,m, (a n 0, b m 0, n,m N Note, that for arbitrary polynomials u(λ, v(λ such decomposition is always possible because the polynomials p n (λ have positive leading coefficients Also, for arbitrary u(λ,v(λ such representation is unique By definition σ(u,v = k a i b i, k = min{n,m} This functional is bilinear Also, it satisfies the relation: σ(u,v = σ(v,u Note, that for the polynomials {p n (λ} n=0 we have: (2 σ(p n (λ,p m (λ = δ n,m, n,m = 0,1, Let us show that this functional has the property: (3 σ(λ 2 u,v = σ(u,λ 2 v, u,v P Really, if u(λ = n a i p i (λ, v(λ = m b j p j (λ, a i C, i = 0,n, b j C, j = 0,m, (a n 0, b m 0, n,m N, then ( ( σ(λ 2 u,v = σ λ 2 n a i p i (λ, m n b j p j (λ = σ λ 2 a i p i (λ, m b j p j (λ = If we prove that = n m a i b j σ(λ 2 p i (λ,p j (λ (4 σ(λ 2 p i (λ,p j (λ = σ(p i (λ,λ 2 p j (λ, i,j = 0,1,,

6 Analog of Favard s theorem 197 then σ(λ 2 u,v = n and relation (3 will be proved But ( m n a i b j σ(p i (λ,λ 2 p j (λ = σ a i p i (λ, m λ 2 b j p j (λ = = σ(u,λ 2 v σ(λ 2 p i (λ,p j (λ = σ(α i 2 p i 2 (λ+β i 1 p i 1 (λ+γ i p i (λ+β i p i+1 (λ+α i p i+2 (λ, p j (λ = α i 2 δ i 2,j + β i 1 δ i 1,j + γ i δ i,j + β i δ i+1,j + α i δ i+2,j, i,j = 0,1,2, σ(p i (λ,λ 2 p j (λ = σ(p i (λ,α j 2 p j 2 (λ + β j 1 p j 1 (λ + γ j p j (λ + β j p j+1 (λ+ +α j p j+2 (λ = α j 2 δ i,j 2 + β j 1 δ i,j 1 + γ j δ i,j + β j δ i,j+1 + α j δ i,j+2 = = α i δ i,j 2 + β i δ i,j 1 + γ i δ i,j + β i 1 δ i,j+1 + α i 2 δ i,j+2 = α i δ i+2,j + β i δ i+1,j + +γ i δ i,j + β i 1 δ i 1,j + α i 2 δ i 2,j, i,j = 0,1,2, This implies the correctness of (3 Let s k = σ(λ k,1, k = 0,1, m k = σ(λ k 1,λ, k = 1,2, Then s 2k = σ(λ 2k,1 = σ(1,λ 2k = σ(λ 2k,1 = s 2k, s 2k+1 = σ(λ 2k+1,1 = σ(1,λ 2k+1 = σ(λ 2k,λ = m 2k+1, k = 0,1, ; m 2k = σ(λ 2k 1,λ = σ(λ,λ 2k 1 = σ(λ 2k 1,λ = m 2k, k = 1,2, Hence, the pair of sequences {s k,m k+1 } k=0, s k C, m k+1 C, k = 0, is symmetric Note that σ(u,u > 0, u P, u 0 Using this we have ( n n 0 < σ a k λ k, a i λ i = n a k a i σ(λ k,λ i = k=0 k,

7 198 S M Zagorodniuk = n k=0 n k=0 [ n/2 a k a 2j s k+2j + n/2 1 [ (n 1/2 a k a 2j s k+2j + a k a 2j+1 m k+2j+1 ], if n is even (n 1/2 a i C,i = 0,n,a n 0, n N a k a 2j+1 m k+2j+1 ], if n is odd It follows that the pair of sequences {s k,m k+1 } k=0 is positive σ1 (λ σ Let σ(λ = 2 (λ be a solution of corresponding generalized σ 3 (λ σ 4 (λ symmetric problem of moments Let us consider a functional: ˆσ(u,v = R T v(λ (u(λ, u( λdσ(λ, u,v P v( λ Let u(λ = n a k λ k, v(λ = m b i λ i, a k C,k = 0,n, b i C, i = 0,m Then k=0 ( n ˆσ(u,v = ˆσ a k λ k, m b i λ i = n k=0 The following equality holds true: Actually, k=0 ˆσ(λ k,λ i = σ(λ k,λ i, k,i = 0,1, m a k b iˆσ(λ k,λ i ˆσ(λ k,λ 2j = ˆσ(λ k+2j,1 = s k+2j = σ(λ k+2j,1 = σ(λ k,λ 2j, ˆσ(λ k,λ 2j+1 = ˆσ(λ k+2j,λ = m k+2j+1 = σ(λ k+2j,λ = σ(λ k,λ 2j+1, This implies that ˆσ(u,v = n k=0 k,j = 0,1,2, m a k b i σ(λ k,λ i = σ(u,v, u,v P Using the orthonormality relation (2 we obtain that polynomials {p n (λ} n=0 satisfy the required property of orthonormality and this completes the proof

8 Analog of Favard s theorem 199 We now turn to the difference equation of the type: (5 α n 2 y n 2 + β n 1 y n 1 + γ n y n + β n y n+1 + α n y n+2 = λ 2 y n (λ, n = 2,3,, y 0 y 1 where α n > 0, β n C, γ n R, n = 0,1,2,; λ C is a parameter, y = y ṇ is solution vector Let {p n (λ} n=0 be the system of polynomials such that: p 0(λ = 1, p 1 (λ = cλ + b, c > 0, b C, λ C, the polynomials p 2 (λ, p 3 (λ can be found from equalities: γ 0 p 0 (λ + β 0 p 1 (λ + α 0 p 2 (λ = λ 2 p 0 (λ, β 0 p 0 (λ + γ 1 p 1 (λ + β 1 p 2 (λ + α 1 p 3 (λ = λ 2 p 1 (λ, γ 0,γ 1 R,β 0 C, and {p n (λ} n=0 is solution of (5 This system of polynomials satisfy (1 Let σ(u,v,u,v P be a functional defined as in the proof of Theorem 2 Let us write the first 4 polynomials in the sequence {p n (λ} n=0 : p 0 (λ = 1,p 1 (λ = λ+ν 1,p 2 (λ = µ 2 λ 2 +ν 2 λ+η 2,p 3 (λ = µ 3 λ 3 +ν 3 λ 2 +η 3 λ+ξ 3, where µ k > 0,ν k C,k = 1,3, η 2,η 3,ξ 3 C Consider the following systems of polynomials: p + n (λ = p n(λ + p n ( λ 2, p n (λ = p n(λ p n ( λ, 2λ, f n (λ = σ u ( pn (u up n (λ p+ n (λ u 2 λ 2,1 ( pn (u up n g k (λ = σ (λ p+ n (λ u u 2 λ 2,p 1 (u (6 n = 0,1, The f n (λ,g n (λ can also be written in the form: ( p + f n (λ = σ n (u p + n (λ ( u u 2 λ 2,1 + σ u u p n (u p n (λ u 2 λ 2,1,,

9 200 S M Zagorodniuk ( p + g n (λ = σ n (u p + ( n (λ u u 2 λ 2,p 1 (u + σ u u p n (u p n (λ u 2 λ 2,p 1 (u, n = 0, Now we can construct the fundamental system of solutions of (5 (see [3]: Theorem 3 Let us consider the difference equation (5 The systems of polynomials {p + n (λ} n=0, {p n (λ} n=0, {f n(λ} n=0, {g n(λ} n=0 form the fundamental system of solutions of (5 The initial conditions are as follows: (7 p + 0 (λ = 1 p + 1 (λ = ν 1 p + 2 (λ = µ 2λ 2 ; + η 2 p + 3 (λ = ν 3λ 2 + ξ 3 f 0 (λ = 0 f 1 (λ = 0 f 2 (λ = µ 2 f 3 (λ = µ 3ν 1 + ν 3 ; p 0 (λ = 0 p 1 (λ = p 2 (λ = ν ; 2 p 3 (λ = µ 3λ 2 + η 3 g 0 (λ = 0 g 1 (λ = 0 g 2 (λ = 0 g 3 (λ = µ 3 Proof The systems {p ± n (λ} n=0 are solutions of (5 because they are linear combination of solutions of this equation The conditions (7 are easily verified Substituting f n (λ in the left-hand side of (5 we have: ( u 2 p + n σ (u λ2 p + n (λ ( u u 2 λ 2,1 + σ u u u2 p n (u λ2 p n (λ u 2 λ 2,1 = σ u (p + n (u,1+ ( p +λ 2 + σ n (u p + n (λ ( u u 2 λ 2,1 + σ u (up n (u,1 + λ 2 σ u u p n (u p n (λ u 2 λ 2,1 = = λ 2 f n (λ + σ u (p + n (u + up n (u,1 = λ 2 f n (λ + σ u (p n (u,1 = λ 2 f n (λ, Analogously for polynomials g k (λ we have: n = 1,2, ( u 2 p + n (u λ 2 p + ( n (λ σ u u 2 λ 2,p 1 (u + σ u u u2 p n (u λ 2 p n (λ u 2 λ 2,p 1 (u = λ 2 g n (λ + σ u (p n (u,p 1 (u = λ 2 g n (λ, n = 2,3, =

10 Analog of Favard s theorem 201 It follows that the systems {f n (λ} n=0, {g n(λ} n=0 are solutions of (5 Using formula (6 and conditions (7 for p ± n (λ,n = 0,3 we obtain: f 0 (λ = g 0 (λ = 0, f 1 (λ = g 1 (λ = 0, f 2 (λ = σ u (µ 2,1 = µ 2, g 2 (λ = σ u (µ 2,p 1 (u = µ 2 σ u (1,p 1 (u = 0, p1 (u ν 1 f 3 (λ = σ u (ν 3,1 + σ u (uµ 3,1 = ν 3 + µ 3 σ u (u,1 = ν 3 + µ 3 σ u,1 = ν 3 + µ 3 (σ u (p 1 (u,1 σ u (ν 1,1 = ν 3 µ 3ν 1, g 3 (λ = σ u (ν 3,p 1 (u + σ u (uµ 3,p 1 (u = µ 3 σ u ( p1 (u ν 1,p 1 (u = µ 3 The linear independence of solutions {p ± n (λ} n=0, {f n(λ} n=0, {g n(λ} n=0 is evident from (7 The proof is complete Let {y n } n=0, {v n} n=0 be solutions of equation (5 corresponding to parameter values λ and ξ respectively: λ 2 y n (λ = α n 2 y n 2 (λ + β n 1 y n 1 (λ + γ n y n (λ + β n y n+1 (λ + α n y n+2 (λ, Then ξ 2 v n (ξ = α n 2 v n 2 (ξ + β n 1 v n 1 (ξ + γ n v n (ξ + β n v n+1 (ξ + α n v n+2 (ξ, n = 2,3, λ 2 y n (λv n (ξ = α n 2 y n 2 (λv n (ξ + β n 1 y n 1 (λv n (ξ + γ n y n (λv n (ξ+ +β n y n+1 (λv n (ξ + α n y n+2 (λv n (ξ, ξ 2 y n (λv n (ξ = α n 2 y n (λv n 2 (ξ + β n 1 y n (λv n 1 (ξ + γ n y n (λv n (ξ+ +β n y n (λv n+1 (ξ + α n y n (λv n+2 (ξ, n = 2,3, Subtracting the second equality from the first one we have: (λ 2 ξ 2 y n (λv n (ξ = α n 2 (y n 2 (λv n (ξ y n (λv n 2 (ξ + β n 1 y n 1 (λv n (ξ β n 1 y n (λv n 1 (ξ + β n y n+1 (λv n (ξ β n y n (λv n+1 (ξ + α n (y n+2 (λv n (ξ y n (λv n+2 (ξ = A n 2 (λ,ξ B n 1 (λ,ξ + B n (λ,ξ + A n (λ,ξ, =

11 202 S M Zagorodniuk where A n (λ,ξ = α n (y n+2 (λv n (ξ y n (λv n+2 (ξ, B n (λ,ξ = β n y n+1 (λv n (ξ β n y n (λv n+1 (ξ Summing up over n we obtain: (λ 2 ξ 2 m y n (λv n (ξ = B m (λ,ξ + A m (λ,ξ + A m 1 (λ,ξ A k 2 (λ,ξ n=k So, we have proved the following theorem: B k 1 (λ,ξ A k 1 (λ,ξ, 2 k m Theorem 4 Let {y n } n=0, {v n} n=0 be solutions of equation (5 corresponding to parameters λ and ξ respectively Then the following formula holds: (λ 2 ξ 2 m y n (λv n (ξ = α m (y m+2 (λv m (ξ y m (λv m+2 (ξ+ n=k +α m 1 (y m+1 (λv m 1 (ξ y m 1 (λv m+1 (ξ+β m y m+1 (λv m (ξ β m y m (λv m+1 (ξ α k 2 (y k (λv k 2 (ξ y k 2 (λv k (ξ α k 1 (y k+1 (λv k 1 (ξ y k 1 (λv k+1 (ξ (β k 1 y k (λv k 1 (ξ β k 1 y k 1 (λv k (ξ, 2 k m REFERENCES [1] S Zagorodniuk On a five-diagonal Jacobi matrices and orthogonal polynomials on rays in the complex plane Serdica Math J 24 (1998, [2] G Freud Orthogonal polynomials Academia Kiado, Budapest, 1971 [3] D I Martyniuk Lectures on qualitative theory of difference equations Kiev, Naukova dumka, 1972 (in Russian Faculty of Mathematics and Informatics Kharkov University Kharkov 77 Ukraine Received August 25, 2000 Revised July 30, 2001