Abstract. We find Rodrigues type formula for multi-variate orthogonal. orthogonal polynomial solutions of a second order partial differential

Transcription

1 Bull. Korean Math. Soc. 38 (2001), No. 3, pp RODRIGUES TYPE FORMULA FOR MULTI-VARIATE ORTHOGONAL POLYNOMIALS Yong Ju Kim a, Kil Hyun Kwon, and Jeong Keun Lee Abstract. We find Rodrigues type formula for multi-variate orthogonal polynomial solutions of second order partial differential equations. 1. Introduction and preliminaries Rodrigues formula for classical orthogonal polynomials in one variable is well developed in [1, 2, 6]. In this work, we are concerned with Rodrigues type formula for multivariate orthogonal polynomial solutions of a second order partial differential equation of spectral type in d variables: (1.1) L[u] = i, A ij 2 u + x j B i u = λ n u, n = 0, 1, 2,, where d 2 is an integer. Let N 0 be the set of nonnegative integers and R the set of real numbers. For α = (α 1,, α d ) N d 0 and x = (x 1,, x d ) R d we write x α = x α 1 1 x α d d and α = α 1 + α α d. For any integer n N 0, let Π d n be the space of real polynomials in d variables of (total) degree n and Π d the space of all real polynomials in d variables. Also, let rn d be the Received November 2, Mathematics Subject Classification: 33C50, 35P99. Key words and phrases: multi-variate orthogonal polynomials, partial differential equations, Rodrigues type formula. This work is partially supported by BK-21 project and KOSEF( ).

2 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee number of monomials of degree exactly n. Then ( ) ( ) n + d n + d 1 dim Π d n = and rn d =. d n By a polynomial system(ps), we mean a sequence of polynomials {φ α (x)} α N d 0 such that deg(φ α ) = α and {φ α } α =n are linearly independent modulo Π d n 1 for n N 0 (Π d 1 = {0}). For n N 0, and x R d, and any PS {φ α (x)} α N d 0, we write x n := [x α : α = n] T and Φ n := [φ α : α = n] T which are vectors in R rd n whose elements are arranged according to the lexicographical order of {α N d 0 : α = n} and use also {Φ n } n=0 to denote the PS {φ α } α N d 0. For a matrix Ψ = [ψ ij ] m i=0,j=0 n of polynomials in Π d and a moment functional(i.e., a linear functional) σ on Π d, we let σ, Ψ = [ σ, ψ ij ] m A PS {P n } n=0 is said to be monic if i=0,j=0. n P n (x) = x n modulo Π d n 1, n N 0. For any PS {Φ n } n=0, where Φ n = A n x n modulo Π d n 1, A n, n 0, is an rn d rn d constant non-singular matrix. We then call the monic PS {P n } n=0 the normalization of {Φ n } n=0, where P n := A 1 n Φ n. For any moment functional σ on Π d, we let and σ, φ = σ, φ, i = 1, 2,, d, for any polynomials φ(x) and ψ(x). ψσ, φ = σ, ψφ Definition 1.1. ([4]) A PS {Φ n } n=0 is a weak orthogonal polynomial system (WOPS) if there is a non-zero moment functional σ such that σ, Φ m Φ T n = K n δ mn if m n and m, n N 0 464

3 Rodrigues type formula where K n := σ, Φ n Φ T n, n N 0, is an rn d rn d constant diagonal matrix. If furthermore K n, n N 0, is nonsingular (respectively, positivedefinite) diagonal matrix, we call {Φ n } n=0 an orthogonal polynomial system (OPS) (respectively, a positive-definite OPS). In this case, we say that {Φ n } n=0 is a WOPS or an OPS relative to σ. A PS {Φ n } n=0 is a WOPS relative to σ if and only if σ, Φ n R = 0 for any polynomial R(x) Π d n 1. For any PS {Φ n } n=0, there is a unique moment functional σ, called the canonical moment functional of {Φ n } n=0, defined by the conditions σ, 1 = 1 and σ, Φ n = 0, n 1. Note that if {Φ n } n=0 is a WOPS relative to σ, then σ must be a nonzero constant multiple of the canonical moment functional of {Φ n } n=0. Definition 1.2. A moment functional σ is quasi-definite (respectively, positive-definite) if there is an OPS (respectively, a positive-definite OPS) relative to σ. The following was proved in [4](see also [5]). Proposition 1.1. For a moment functional σ 0, the following statements are all equivalent: (i) σ is quasi-definite (respectively, positive-definite); (ii) There is a unique monic WOPS {P n } n=0 relative to σ; (iii) There is a monic WOPS {P n } n=0 relative to σ such that H n := σ, P n P T n, n N 0, is a nonsingular (respectively, positive-definite) symmetric matrix; (iv) D n is nonsingular (respectively, positive-definite), where D n := [σ α+β ] n n α =0, β =0, n N 0, and σ α = σ, x α, α N d 0, are the moments of σ. Let {P n } n=0 be the monic WOPS relative to a quasi-definite moment functional σ and A n the rn d rn d nonsingular matrix such that A n H n A T n = σ, (A n P n )(A n P n ) T is diagonal. Then {Φ n := A n P n } n=0 is an OPS relative to σ. It is also easy(cf. [8]) to see that σ is positivedefinite if and only if σ, φ 2 > 0 for any polynomial φ(x)

4 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee Lemma 1.2. (see Lemma 2.2 in [3]) Let σ and τ be moment functionals and R(x) a polynomial in Π d. Then (i) σ = 0 if and only if σ = 0 for some i = 1,, d. Assume that σ is quasi-definite and let {Φ n } n=0 be an OPS relative to σ. Then (ii) R(x)σ = 0 if and only if R(x) = 0; (iii) τ, φ α = 0, α > k (k N 0 ) if and only if τ = ψ(x)σ for some polynomial ψ(x) Π d k. Proof. (i) and (ii) are obvious. (iii) : It is trivial from the orthogonality of {Φ n } n=0 relative to σ. (iii) : Consider a moment functional τ = ( k C j Φ j )σ, where C j = [C α ] α =j, 0 j k, are arbitrary constant row vectors. Then j=0 τ, Φ n = k σ, Φ n Φ T j C T j = j=0 { 0, if n > k σ, Φ n Φ T n Cn T, if 0 n k. Hence, if we take C n = τ, Φ n T H 1 n, 0 n k, then τ, Φ n = τ, Φ n, n N 0, so that τ = τ. 2. Differential operator L[ ] We now recall quickly results from [3], which we need to develop Rodrigues type formula. If the differential equation (1.1) has a PS {Φ n } n=0 as solutions, then it must be of the form (2.1) L[u] = = 2 u A ij (x) + x j i, i, ( ax i x j + = λ n u, n N 0, k=1 B i (x) u ) b k 2 u ijx k + c ij + x j 466 (gx i + h i ) u

5 Rodrigues type formula where λ n := an(n 1)+gn. Without the loss of generality, we may assume that the matrix [A ij (x)] d i, is symmetric, that is, A ij (x) = A ji (x), 1 i, j d. We also assume that A ij 0 and a + g = 0 since i,j=0 otherwise the differential equation (2.1) can not have an OPS as solutions. The differential operator L[ ] in (2.1) is called to be admissible if λ m λ n for m n or equivalently an + g 0 for n 0. It is then easy to see (cf. [3, 4]) that the differential equation (2.1) is admissible if and only if the differential equation (2.1) has a unique monic PS as solutions. From now on, we always assume that L[ ] is admissible. Proposition 2.1. (see Theorem 3.7 in [3]) Let σ be the canonical moment functional of a PS {Φ n } n=0 of solutions to the differential equation (2.1). Then the following statements are all equivalent : (i) {Φ n } n=0 is a WOPS relative to σ; (A ij σ) (ii) M i [σ] := B i σ = 0, i = 1,, d; x j (iii) σ, x i Φ n = 0, n 2 and i = 1,, d. We say that the differential operator L[ ] is symmetric if L[ ] = L [ ] where L [ ] is the formal Lagrange adjoint of L[ ] defined by L [u] := i, 2 (A ij u) x j (B i u). We also say that L[ ] is symmetrizable if there is a non-zero C 2 function s(x) in some open subset of R d such that sl[ ] is symmetric. In this case, we call s(x) a symmetry factor of L[ ]. In fact(see [3]), a C 2 function s(x) ( 0) is a symmetry factor of L[ ] if and only if s(x) is a non-zero solution of the following so-called symmetry equations : (2.2) M i [s] = (A ij s) x j B i s = 0, i = 1,, d. 467

6 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee Proposition 2.2. (see Lemma 3.9 and Theorem 3.11 in [3]) If the admissible differential equation (2.1) has an OPS {Φ n } n=0 as solutions, then [A ij ] d i, 0 in any non-empty open subset of R d and L[ ] is symmetrizable. 3. Rodrigues type formula From now on, we may and shall assume (see Proposition 2.2) that [A ij ] d i, 0 and the differential operator L[ ] in (2.1) is symmetrizable. Let s(x)( 0) be a symmetry factor of L[ ]. That is, s(x) is any non-zero solution of the symmetry equations (3.1) M i [s] = (A ij s) xj B i s = 0, i = 1,, d. Solving the equations (3.1) for s xi, i = 1,, d, yields (3.2) α s = β i s, i = 1,, d where α := [A ij ] d i, and β i is the determinant of the matrix obtained from [A ij ] d i, by replacing i-th column of [A ij ] d i, with [ B 1 A 1j x j,, B d A dj x j ] T. Note that deg(α) 2d 1 and deg(β i ) 2d 2, i = 1,, d. Decompose A ij, 1 i, j d, into { A ii = D1D i 2, i i = 1,, d, (3.3) A ij = D1E i ij D1, j 1 i, j d, and i j where D1 i 0, i = 1,, d. Then d α = D1 1 D1α d 0 = α 0 D1, j β i = D 1 1 D i 1 1 D i+1 1 D d 1β i 0 = β i d D1, j j i i = 1,, d

7 where Rodrigues type formula { α 0 = ( ) l ij, lij = D2, i D1E j ij, i = j i j and for i = 1,, d, (A jn ) β0 i = ( ) B j, k = i x m jk, mjk = n=1 n D2, j k = j, k i D1E k jk, k i, j. Then the equations (3.2) become (3.4) p i s = β i 0s, i = 1,, d where p i = D1α i 0, i = 1,, d. Note that for each i = 1,, d, p i 0 and deg(p i ) 2d 1, deg(β0) i 2d 2. Proposition 3.1. Let σ be the canonical moment functional of a PS {Φ n } n=0 satisfying the differential equation (2.1). Assume that (3.5) and (3.6) D i 1 x j = 0, i j and 1 i, j d p i σ = β i 0σ, i = 1,, d. Then for any multi-index γ N d 0 (3.7) γ [( d ) ] p γ i i σ = ψ γ σ where ψ γ (x) is a polynomial of degree (2d 2) γ and (3.8) σ, x γ ψ γ (x) = 0 for any γ N d 0 with 0 γ γ and γ γ. If moreover, σ is quasi-definite and (3.9) deg(p i ) 2 and deg(β i 0) 1, i = 1,, d, 469

8 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee then {Ψ n } n=0 with Ψ n = [ψ γ : γ = n] T satisfies the differential equation (2.1). is a WOPS relative to σ and Proof. Assume that the conditions (3.5) and (3.6) hold. Then for any polynomial π(x) and any multi-index γ = (γ 1,, γ d ) N d 0, we have, for each i = 1,, d with γ i 0 where (πσ d pγ j j ) = σp γ i 1 i π i d j i p γ j j, π i = γ i π p i + ( γ γ i )πd i α p i π xi + β x 0π. i i Since deg(π i ) deg(π) + max{deg(p i ) 1, deg(β i 0)} deg(π) + 2d 2, the first conclusion follows easily by induction on γ N d 0. If moreover, the condition (3.9) holds, we have deg(ψ γ ) γ. Now, for all γ N d 0 with 0 γ γ and γ γ σ, x γ ψ γ = ψ γ σ, x γ = γ [( = ( 1) γ ( d d p γ i i )σ], xγ p γ i i )σ, γ x γ = 0 from which (3.8) follows. Next we further assume that σ is quasi-definite and the condition (3.9) holds. We first claim that deg(ψ γ ) = γ and σ, x γ ψ γ = 0 for γ N d 0. Assume that deg(ψ γ ) γ 1 for some γ N d 0 with γ 1. Then ψ γ 0 since σ, πψ γ = 0 for any π(x) in Π d γ 1 by (3.8) and σ is quasi-definite. Hence γ [( d pγ i i )σ] = 0 so that p i(x) 0 for some i with 1 i d which is a contradiction. Therefore deg(ψ γ ) = γ and σ, x γ ψ γ = 0 for all γ N d 0 by (3.8). We now claim that for each n 0, {ψ γ } γ =n are linearly independent modulo Π d n 1 so that {Ψ n } n=0 is a PS. For any integer n 1, let C γ, γ N d 0 with γ = n, be constants such that φ(x) = γ =n C γ ψ γ is of degree n 1. Then φ(x) 0 since σ, φ(x)π(x) = 0 for any π Π d n 1 by 470

9 Rodrigues type formula (3.8). Then, for all γ N d 0 with γ = n σ, φ(x)x γ = C γ σ, ψ γ x γ = 0 so that C γ = 0, γ N d 0 with γ = n by the first claim. Hence {ψ γ } γ =n are linearly independent modulo Π d n 1 and so {Ψ n } n=0 is a WOPS relative to σ by (3.8). Finally, in order to see that {Ψ n } n=0 satisfy the differential equation (2.1), we let {P n } n=0 and {Q n } n=0 be the normalizations of {Φ n } n=0 and {Ψ n } n=0 respectively. Then {P n } n=0 and {Q n } n=0 are monic WOPS s relative to σ so that P n = Q n, n 0, by Proposition 1.1. Since {Φ n } n=0 satisfy the differential equation (2.1), {Q n } n=0 and so {Ψ n } n=0 also satisfy the differential equation (2.1). In passing, we note: The moment functional σ in Proposition 3.1 satisfies (3.1) and so (3.2) (see Proposition 2.1). However we can not drop the condition (3.6) since the condition (3.2) for σ : p i σ = β i 0σ (i = 1,, d) does not necessarily imply the condition (3.6). From Proposition 3.1, we now have: Theorem 3.2. Assume that the differential equation (2.1) has an OPS {Φ n } n=0 relative to σ as solutions. If the conditions (3.5), (3.6), and (3.9) hold, then the PS {Ψ n } n=0 with Ψ n = [ψ γ : γ = n] T defined by (3.7) is a WOPS relative to σ and satisfies the differential equation (2.1). However, σ, Ψ n Ψ T n, with {Ψ n } n=0 as in Theorem 3.2, is nonsingular but need not be diagonal, that is, {Ψ n } n=0 is a WOPS but need not be an OPS in general(see Example 3.2 below). We may call (3.7) a (functional) Rodrigues type formula for orthogonal polynomial solutions of the differential equations (2.1). If s(x) happens to be an orthogonalizing weight for a PS {Φ n } n=0, then σ in (3.7) can be replaced by s(x), which is an ordinary Rodrigues type formula. Example 3.1. Assume that { A ij 0, A ii (x) = A ii (x i ), i j, 1 i, j d 1 i d 471

10 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee so that the differential equation (2.1) is of the form (3.10) L[u] = A ii (x i ) 2 u 2 + B i (x i ) u = λ n u where A ii (x i ) = d i x i + f i, B i (x i ) = gx i + h i, i = 1,, d, and g 0. Then it is shown in [3] (cf. [7]) that (3.11) the differential equation (3.10) has a unique monic PS {P n } n=0 as solutions; P nei (x) = P nei (x i ), i = 1,, d, and P γ (x) = P γ1 e 1 (x 1 ) P γd e d (x d ), γ N d 0, where e i, i = 1,, d, is the i-th fundamental vector in R d ; {P n } n=0 is a WOPS; For each i = 1,, d, P nei (x i ), n 0, satisfy A ii (x i )P ne i (x i ) + B i (x i )P ne i (x i ) = λ n P nei (x i ). In decomposition (3.3), we take D i 2 = 1, i = 1,, d and E ij = 0, i, j = 1,, d. Then D i 1 = A ii, α 0 = 1, β i 0 = B i A ii, p i = A ii, i = 1,, d so that the conditions (3.5) and (3.9) hold. On the other hand, the canonical moment functional σ of {P n } n=0 is equal to σ = σ (x1) σ (xd), where for each i = 1,, d, σ (xi) is the canonical moment functional of {P nei } n=0. Since A ii (x i ) σ (x i) = (B i (x i ) A ii(x i ))σ (xi) for each i = 1,, d, σ = σ (x1) σ (xd) satisfies the condition (3.6). Hence, by Proposition 3.1, (3.12) γ [( d ) ] A γ i ii σ = ψ γ σ, γ N d 0 where {ψ γ } is a WOPS relative to σ. In fact, we have ψ γ (x) = g γ P γ1 e 1 (x 1 ) P γd e d (x d ), γ N d 0, so that the Rodrigues type formula (3.12) is nothing but the tensor product of one dimensional Rodrigues formulas for {P nei } n=0, i = 1,, d(see [1, 2, 6]). 472

11 Rodrigues type formula We may, of course, replace σ by a symmetry factor s(x) = s 1 (x i ) s d (x i ) of the differential equation (3.10), where for each i = 1,, d, s i (x i ) is symmetry factor of the differential equations (3.11). Example 3.2. Consider the differential equation: (3.13) L[u] = (x i x j δ ij ) 2 u + xj i, gx i u = λ n u. In decomposition (3.3), we take D i 1 = 1, i = 1,, d so that E ij = A ij = x i x j and α = p = q = x 2 i 1, β i = β0 i = (g 3)x i, i = 1,, d. Then if g 1, 0, 1,, then the differential equation (3.13) has an OPS {Φ n } n=0 as solutions and the canonical moment functional σ of {Φ n } n=0 satisfies the condition (3.6) so that by Theorem 3.2, {Ψ n } n=0 with Ψ n = [ψ γ : γ = n] T defined by γ [( ) γ σ ] x 2 i 1 = ψ γ (x)σ, γ N d 0 is a WOPS. But, even in this case, {ψ γ } is not an OPS. For if we let σ be the canonical moment functional of {ψ γ }, then σ satisfies L [σ] = 0, that is, γ ( γ 1 + g)σ γ γ i (γ i 1)σ γ 2ei = 0, γ N d 0 i=0 473

12 Yong Ju Kim, Kil Hyun Kwon, and Jeong Keun Lee so that σ 0,,0 = 1, σ ei = 0, i = 1,, d σ ei +e j = 0, i j and 1 i, j d σ 2ei = 1, i = 1,, d g+1 σ γ = 0, γ N d 0 with γ = 3, σ γ = 0, γ N d 0 with γ = 4 and γ i = 1 for some i 3 σ 4ei = (g+1)(g+3), i = 1,, d 1 σ 2ei +2e j =, i j and 1 i, j d. (g+1)(g+3) Hence, it is easy to show that is nonsingular but not diagonal. H 2 = σ, Ψ 2 Ψ T 2 References [1] W. C. Brenke, On polynomial solutions of a class of linear differential equations of the second order, Bull. Amer. Math. Soc. 36 (1930), [2] C. W. Cryer, Rodrigues formulas and the classical orthogonal polynomials, Boll. Unione. Mat. Ital. 25 (1970), [3] Y. J. Kim, K. H. Kwon, and J. K. Lee, Multi-variate orthogonal polynomials and second order partial differential equation, Comm. Appl. Anal., to appear. [4] H. L. Krall and I. M. Sheffer, Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. seri 4, 76 (1967), [5] L. L. Littlejohn, Orthogonal polynomial solutions to ordinary and partial differential equations, Proc. 2nd Intern. Symp. Orthog. Polyn. and their Appl., M. Alfaro and al. Ed s, Segovia(Spain), (1986), Lect. Notes Math. 1329, Springer-Verlag (1988), [6] F. Marcellán, A. Branquinho, and J. Petronilho, Classical orthogonal polynomials: A functional approach, Acta Appl. Math. 34 (1994), [7] P. K. Suetin, Orthogonal polynomials in two variables, Nauka, Moscow, 1988 (in Russian). [8] Y. Xu, On multivariate orthogonal polynomials, SIAM J. Math. Anal. 4 (1993),

13 Rodrigues type formula Yong Ju Kim and Kil Hyun Kwon, Division of Applied Mathematics, KAIST, Taejon , Korea Jeong Keun Lee, Department of Mathematics, Sunmoon university, Asan , Korea 475