Journal of Computational and Applied Mathematics 233 2010) 1554 1561 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: wwwelseviercom/locate/cam Two-variable orthogonal polynomials of big -Jacobi type Stanisław Lewanowicz, Paweł Woźny Institute of Computer Science, University of Wrocław, ul Joliot-Curie 15, 50-383 Wrocław, Poland a r t i c l e i n f o a b s t r a c t Article history: Received 29 September 2007 Dedicated to Professor Jesús S Dehesa on the occasion of his 60th birthday MSC: 33D50 33C50 A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type The polynomials, which are expressed in terms of univariate big -Jacobi polynomials, form an extension of Dunkl s bivariate little) -Jacobi polynomials [CF Dunkl, Orthogonal polynomials in two variables of -Hahn and -Jacobi type, SIAM J Algebr Discrete Methods 1 1980) 137 151] We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial -difference euation 2009 Elsevier BV All rights reserved Keywords: Bivariate big -Jacobi polynomial Orthogonality weight Three-term relation Partial -difference euation 1 Introduction In this paper we introduce bivariate orthogonal big -Jacobi polynomials, P n,k x, y; a, b, c, d; ) := P n k y; a, bc 2k+1, k ; ) y k /y; ) k P k x/y; c, b, d/y; ) 0, 1), 0 < a, b, c < 1, d < 0, and n N; k = 0, 1,, n), 11) P m t; A, B, C; ) := 3 φ 2 m, AB m+1, t; A, C ; ) m 0) are univariate big -Jacobi polynomials see, eg, [1, Section 73], or [2, Section 35]) The notation used in the paper is explained in the last paragraphs of this section) General properties of discrete orthogonal polynomials of several variables were discussed in [3] Notice that there exist some multivariate extensions of discrete -classical orthogonal polynomials Dunkl [4] introduced bivariate -Hahn, and little) -Jacobi polynomials Gasper and Rahman [5] proposed multivariate -Racah polynomials, from which systems of multivariate -Hahn, -Krawtchouk, -Meixner, and -Charlier polynomials follow as special or limit cases Bivariate big -Jacobi polynomials 11) occupy an important place in a hierarchy of bivariate extensions of -classical orthogonal polynomials, ie, in a two-variable extension of the -Askey tableau [2] Note that limiting forms of these polynomials are the above-mentioned bivariate little -Jacobi polynomials, as well as triangle Jacobi polynomials [6]; see also [7, p 86]) In Section 2, we give some basic properties of polynomials 11) More specifically, in Section 21, we show that they form an orthogonal system with respect to the weight function Wx, y; a, b, c, d; ) := /y, x/cy), x/d, y/a, y/d; ) y d/cy), cy/d, x/y, bx/d, y; ) Corresponding author Tel: +48 71 375 7812; fax: +48 71 375 7801 E-mail addresses: StanislawLewanowicz@iiuniwrocpl S Lewanowicz), PawelWozny@iiuniwrocpl P Woźny) 0377-0427/$ see front matter 2009 Elsevier BV All rights reserved doi:101016/jcam200902070
S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 1555 Let P n := [P n,0, P n,1,, P n,n ] T, P n,k := P n,k x, y; a, b, c, d; ) In Section 22, we show that the following three-term relation holds: z P n = A n,z P n+1 + B n,z P n + C n,z P n 1 z = x, y; n 0), A n,z, B n,z and C n,z are matrices of appropriate dimensions, and P 1 := 0 In Section 23, we show that for any n 0, and 0 k n the polynomial P n,k x, y; a, b, c, d; ) satisfies a linear second-order partial -difference euation In the Appendix, we give some auxilliary results on the univariate big -Jacobi polynomials, needed in the paper We end this section with a list of notation and terminology used in the paper For more details the reader is referred to the monographs [1] or [8] or the report [2] The -shifted factorial is defined for any c C by c; ) k := k 1 j=0 1 c j ) k 0) Assuming that 0 < < 1, we also put c; ) := j=0 1 c j ) In what follows, we make use of the convention c 1, c 2,, c m ; ) k := c 1 ; ) k c m ; ) k k = 0, 1, or ) For c C, we define the -number [ c ] by [ c ] := c 1)/ 1) The generalized -binomial coefficient is given by [ ] n ; ) := n 0 k + l n) k, l ; ) k ; ) l ; ) n k l Partial -derivative operator with respect to x is defined for C \ {1} by f x, y) f x, y) D,x f x, y) := 1)x The -integral is defined by b a f x) d x := b1 ) f b k ) k a1 ) f a k ) k 12) k=0 The basic hypergeometric series is defined by see, eg, [1, Section 109]) a 1,, a r ; ) k )) 1+s r rφ s a 1,, a r ; b 1,, b s ; z) := k 1) k 2 z k,, b 1,, b s ; ) k k=0 r, s Z + and a 1,, a r, b 1,, b s, z C 2 Fundamental properties of the bivariate big -Jacobi polynomials We define the bivariate big -Jacobi polynomials by P n,k x, y; a, b, c, d; ) := P n k y; a, bc 2k+1, k ; ) y k /y; ) k P k x/y; c, b, d/y; ) 0, 1), 0 < a, b, c < 1, d < 0, and k=0 n N; k = 0, 1,, n), 21) P m t; A, B, C; ) := 3 φ 2 m, AB m+1, t; A, C ; ) m 0) are univariate big -Jacobi polynomials see, eg, [1, Section 73], or [2, Section 35]; in the Appendix, we recall basic data of these polynomials, and give some of their properties which are used in this section) Notice that P n,k x, y; a, b, c, 0; ) = γ n,k p n,k x, y; a, b, c ) with γ n,k := 1) n a n k c k k 2 ) + n k 2 ) +n b; ) k bc 2k+2 ; ) n k, the polynomials c; ) k a; ) n k p x, y; a, b, n,k c ) := p y; n k bc2k+1, a ) y k p k x/y; b, c ) are closely related to Dunkl s bivariate little) -Jacobi polynomials [4] Here we use the notation p m x; a, b ) := p m x/b); a, b ), p m x; a, b ) := 2 φ 1 m, ab m+1 ; a ; ) x are little -Jacobi polynomials of one variable see, eg, [8, p 182], or [2, Section 312]) Notice that lim 1 p n,k x, y; α, β, γ ) = const n,k α, β, γ ) P α+1/2,β+1/2,γ +1/2) n,k 1 y, x), P A,B,C) n,k u, v) are the triangle Jacobi polynomials see [7, p 86], or [6]) In [9,10], we gave connections of triangle and Dunkl s polynomials with two-variable classical) Bernstein and -Bernstein polynomials, respectively
1556 S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 21 Orthogonality property Theorem 21 Polynomials P n,k x, y; ) P n,k x, y; a, b, c, d; ), 0 < a, b, c < 1, d < 0, satisfy the orthogonality relation a cy Wx, y; a, b, c, d; ) P n,k x, y; ) P m,l x, y; ) d x d y = Λ n,k a, b, c, d; ) δ n,m δ k,l, 22) /y, c 1 x/y, x/d, y/a, y/d; ) Wx, y; a, b, c, d; ) := y c 1 d/y, cy/d, x/y, bx/d, y; ), 23) is a positive weight function, with Λ n,k a, b, c, d; ) := π 1 k b, c, d; ) λ n k a, bc 2k+1, k ; ) π k b, c, d; ) := k+1)2 1 bc 2k+1 ) c, bc; ) k b, c; ) c k+1 d 2k 1 )1 bc), b; ) k, bc2 ; ), and the notation used being that of A4) Proof Notice that by 12), the functional Lf ) := cy f x, y) Wx, y; a, b, c, d; ) d x d y, may be expressed as Lf ) = εx, y) f x, y) Wx, y; a, b, c, d; ), x,y) S S := S 1 S 2 S 3, and S 1 := {ac i+1, a j+1 ) j = 0, 1, ; i j + 1}, S 2 := {a i+1, j+1 ) i, j = 0, 1, }, S 3 := { i+1, j+1 ) j = 0, 1, ; i j}, εx, y) being a rather simple positive function Now, an easy analysis shows that the function Wx, y; a, b, c, d; ) is positive on S It can be checked that Wx, y; a, b, c, d; ) = π 1 k b, c, d; )Vx, y; a, b, c, d; ), Vx, y; a, b, c, d; ) := wy; a, bc2k+1, k ; ) wx/y; c, b, d/y; ) y 2k+1 /y; ) 2 λ, k kc, b, d/y; ) and we use the notation of A3) and A4) We have cy = Vx, y; a, b, c, d; ) P n,k x, y; ) P m,l x, y; ) d x d y wy; a, bc 2k+1, k ; ) y l /y; ) l y k+1 /y; ) k λ k c, b, d/y; ) P n k y; a, bc 2k+1, k ; )P m l y; a, bc 2l+1, l ; ) cy wx/y; c, b, d/y; )P k x/y; c, b, d/y; ) P l x/y; c, b, d/y; ) d x d y By A2), the inner integral euals y λ k c, b, d/y; ) δ k,l, hence cy I := Vx, y; a, b, c, d; ) P n,k x, y; ) P m,l x, y; ) d x d y = δ k,l I, wy; a, bc 2k+1, k ; )P n k y; a, bc 2k+1, k ; )P m k y; a, bc 2k+1, k ; ) d y
S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 1557 Observe that the function wy; a, bc 2k+1, k ; ) contains the factor assumptions on the function f, we have y/ k ); ) f y) d y = y/ k ); ) f y) d y, y/k ); ) cf A3)) Now, under certain k+1 so that, using again A2), we obtain I = δ n,m λ n k a, bc 2k+1, k ; ) Hence follows E 22) 22 The three-term relation According to the general theory of orthogonal polynomials in several variables see [7, Section 32]), a three-term relation holds in a vector-matrix form We show that for the polynomials 21) this property has the following formulation Theorem 22 Denote by P n the column polynomial vector P n := [P n,0, P n,1,, P n,n ] T, 24) P n,k := P n,k x, y; a, b, c, d; ) For n 0, the following relation holds: z P n = A n,z P n+1 + B n,z P n + C n,z P n 1 z = x, y), 25) we define P 1 := 0, and A n,z, B n,z and C n,z are matrices of the size n + 1) n + 2), n + 1) n + 1) and n + 1) n, respectively, given by the formulas a n,0 b n,0 a n,1 A n,y :=, b n,1 B n,y :=, 26) an,n 0 bn,n c n,0 c n,1 C n,y :=, cn,n 1 0 27) a n,k := 1 a n k+1 )1 abc n+k+2 )1 n+1 )/ abc 2n+2 ; ) 2 0 k n), 28) b n,k := 1 a n,k c n,k 0 k n; c nn := 0), 29) c n,k := a n+1 n k 1)1 bc n+k+1 )1 abcd 1 n+1 )/ abc 2n+1 ; ) 2 0 k n 1); 210) and f n,0 g n,0 e n,1 f n,1 g n,1 A n,x :=, 211) e n,n 1 f n,n 1 g n,n 1 0 e n,n f n,n g n,n e n,k := τ k bc k k 1)1 n+1 )a n k+1 ; ) 2 /abc 2n+2 ; ) 2 f n,k := a n,k bc k τ k σ k + 1) 0 k n), g n,k := σ k 1 n+1 )abc n+k+2 ; ) 2 /[1 k+1 )abc 2n+2 ; ) 2 ] 1 k n), 0 k n); s n,0 t n,0 r n,1 s n,1 t n,1 B n,x :=, 212) r n,n 1 s n,n 1 t n,n 1 r n,n s n,n
1558 S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 r n,k := τ k z n k 1)1 a n k+1 )1 bc n+k+1 ) s n,k := b n,k bc k τ k σ k + 1) + k+1 σ k τ k ) 1 k n), 0 k n), t n,k := k+1 σ k z n 1 n k )1 abc n+k+2 )/1 k+1 ) 0 k n 1) with z n := { abc n+1 1 + n+1 ) } d /[1 abc 2n+1 )1 abc 2n+3 )]; and, finally, v n,0 w n,0 u n,1 v n,1 w n,1 C n,x := u n,n 1 v, 213) n,n 1 u n,n Here u n,k := τ k a n k+1 k 1)abc n+1 d)bc n+k ; ) 2 /abc 2n+1 ; ) 2 v n,k := c n,k bc k τ k σ k + 1) 0 k n 1), 1 k n), w n,k := abcσ k n+2k+3 abc n+1 d) n k 1 ; ) 2 /[1 k+1 )abc 2n+1 ; ) 2 ] 0 k n 2) σ k := 1 ck+1 )1 bc k+1 ), τ bc 2k+1 k := c k+1 1 k )1 b k ) 214) ; ) 2 bc 2k ; ) 2 In Es 26), 27) and 211) 213), the elements of matrices, which are not shown, are eual to zero Proof i) Using 21) and the three-term recurrence satisfied by the univariate big -Jacobi polynomials cf A5), in the Appendix), we obtain y P n,k x, y; a, b, c, d; ) = y P n k y; a, bc 2k+1, k ; ) y k /y; ) k P k x/y; c, b, d/y; ) = { a n,k P n k+1 y; a, bc 2k+1, k ; ) + b n,k P n k y; a, bc 2k+1, k ; ) + c n,k P n k y; a, bc 2k+1, k ; ) } y k /y; ) k P k x/y; c, b, d/y; ) = a n,k P n+1,k x, y; a, b, c, d; ) + b n,k P n,k x, y; a, b, c, d; ) + c n,k P n 1,k x, y; a, b, c, d; ), 215) the notation used is that of 28) 210) The obtained result justifies 25) with z = y ii) Again, using 21) and A5), we obtain x P n,k x, y; a, b, c, d; ) = y P n k y; a, bc 2k+1, k ; ) y k /y; ) k x/y) P k x/y; c, b, d/y; ) = σ k P n k y; a, bc 2k+1, k ; ) y k+1 /y; ) k+1 P k+1 x/y; c, b, d/y; ) [σ k y k+1 ) + τ k d bc k y) y] P n,k x, y; a, b, c, d; ) + τ k y k )d bc k y) P n k y; a, bc 2k+1, k ; )y k 1 /y; ) k 1 P k 1 x/y; c, b, d/y; ) with the notation used being that of 214) By Propositions A1 and A2 see Appendix), we have P n k y; a, bc 2k+1, k ; ) = n k j=n k 2 y k )d bc k y) P n k y; a, bc 2k+1, k ; ) = C j P j y; a, bc 2k+3, k+1 ; ), n k+2 j=n k D j P j y; a, bc 2k 1, k 1 ; ), C j := C n k,j a, bc 2k+1, k ), D j := k 1 D n k,j a, bc 2k+1, k ), notation used being that of A7) and A9) Using these results as well as E 215), being the scalar form of 25) with z = y, we obtain x P n,k x, y; a, b, c, d; ) = n+1 k+1 m=n 1 l=k 1 ξ m,l P m,l x, y; a, b, c, d; ) with ξ m,k 1 := τ k D m k+1, ξ m,k := η m,k bc k τ k σ k + 1) + δ m,n k+1 σ k dτ k ), ξ m,k+1 := σ k C m k+1, η m,k := δ m,n+1 a n,k + δ m,n b n,k + δ m,n 1 c n,k m = n 1, n, n + 1) Hence, we arrived to the scalar form of 25) with z = x
S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 1559 23 Second-order partial -difference euation The main result of this subsection is given in the following theorem Theorem 23 For n 0 and 0 k n, polynomial P n,k x, y; ) P n,k x, y; a, b, c, d; ) satisfies the partial -difference euation L n P n,k x, y; ) { l 11 x)d,x D 1,x + l 22 y)d,y D 1,y + l 12 x, y)d 1,x D 1,y with I denoting the identity operator, and +l + 12 x, y)d,xd,y + m 1 x)d,x + m 2 y)d,y + µ n I}P n,k x, y; ) = 0 216) l 11 x) := x )x ac 2 ), l 22 y) := y a)y ), l x, y) := 12 1 x )y a), l + x, y) := 12 ac3 bx d)y 1), m 1 x) := { abc 3 1)x 1) ac 2 1) 1) } / 1), m 2 y) := { abc 3 1)y 1) a 1) 1) } / 1), µ n := [ n ] 1 n abc n+2 1)/ 1) In the proof of the theorem, which will be preceded by three lemmas, we shall use the bivariate generalized Bernstein polynomials of total degree n, defined by [10] ] B n k,l x, y; ω ) := ω; ) 1 n [ n k, l x k ω/x; ) k y l x/y; ) l y; ) n k l, 217) 0 k + l n Here ω is a real parameter, ω 1, 1,, 1 n In what follows, we adopt the convention that B n k,l x, y; ω ) = 0, if k < 0, or l < 0, or k + l > n Notice that when 1, polynomials 217), being a two-variable analogue of univariate generalized Bernstein polynomials introduced in [11] see also [12]), reduce to classical Bernstein polynomials in two variables see, eg, [13, Section 184]) Lemma 24 For n 0 and 0 k + l n, the following identity holds: L n B n x, y; ) = [ n ] k,l 1 n { ac n k+2 b k 1) B n 1 k 1,l x, y; ) 1 + a n k l 1) B n 1 k,l x, y; ) + a n k l+1 c l 1) B n 1 x, y; )} k,l 1 Lemma 25 [10]) For any k, n N such that 0 k n, we have n k n i P n,k x, y; a, b, c, d; ) = A n, k) i,j Bn n i j,j x, y; ), 218) i=0 j=0 with A i,j n, k) := abcn+2 ) i, i n ; ) k n k 1 /bc); ) i f n i,j 219) ; ) k a; ) i f i,j := 3 φ 2 j, k, bc k+1 ; i n, c ; ) 220) Lemma 26 Quantity 220) satisfies the following recurrence relation: c j+k+1 i n 1)b n i j 1) f i,j i+k n 1)bc n i+k+1 1) f i+1,j + k i n 1)c j+1 1) f i,j+1 = 0 0 i n k; 0 j n i 1; f n k+1,j := 0) Proof of Theorem 23 Apply the operator L n to both members of E 218) see Lemma 25) to obtain n k n i L n P n,k x, y; a, b, c, d; ) = A n, k) i,j L n B n n i j,j x, y; ) 221) i=0 j=0
1560 S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 By Lemma 24, we have L n B n x, y; ) = [ n ] n i j,j 1 n { ac i+j+2 b n i j 1) B n 1 n i j 1,j x, y; ) 1 + a i 1) B n 1 n i j,j x, y; ) + ai+1 c j 1) B n 1 n i j,j 1 x, y; )} Hence, the second member of 221) can be written, after a rearrangement, as 1 n [ n ] 1 n k i=0 n i 1 j=0 { ac i+j+2 b n i j 1) A n, k) + i,j ai+1 1) A i+1,j n, k) + a i+1 c j+1 1) A n, k) } i,j+1 B n 1 n i j 1,j x, y; ) Now, the expression in parantheses { } can be written, in view of 219), as, i n ; ) a i k+1 a i+1 1)abc n+2 ) i k n k 1 /bc); ) i 1 i n ) n ; ) k a; ) i+1 { } c j+k+1 i n 1)b n i j 1) f i,j i+k n 1)bc n+k i+1 1) f i+1,j + k i n 1)c j+1 1) f i,j+1, which, by Lemma 26, euals zero Hence the thesis Acknowledgments The authors thank the referees for their helpful suggestions Appendix A Auxilliary results on the univariate big -Jacobi polynomials The big -Jacobi polynomials of a single variable see, e g, [8, Section 73], or [2, Section 35]) P k x) P k x; a, b, c; ) := 3 φ 2 k, ab k+1, x; a, c ; ), a, b and c being parameters, 0 < a, b < 1, c < 0, satisfy the orthogonality relation and c wx; a, b, c; )P k x) P l x) d x = λ k a, b, c; )δ kl, wx; a, b, c; ) := x/a, x/c; ) /x, bx/c; ), ) 1 ab), b, ab/c; ) k k λ k a, b, c; ) := M ac 2 ) k 2 1 ab 2k+1 ) a, ab, c; ) k with M := a1 ), ab 2, c/a, a/c; ) /a, b, c, ab/c; ) Also, they satisfy the three-term recurrence relation x P k x) = A k P k+1 A k + B k 1) P k x) + B k P k 1 x) k 0), A5) P 0 x) 1, P 1 x) 0, and A k := 1 a k+1 )1 ab k+1 )1 c k+1 )/ ab 2k+1 ; ) 2, B k := ac k+1 1 k )1 b k )1 abc 1 k )/ ab 2k ; ) 2 A1) A2) A3) A4) Proposition A1 The formula P n x; a, b, c; ) = n k=n 2 C n,k a, b, c; ) P k x; a, b 2, c; ) holds with C n,n a, b, c; ) := 1 c n+1 ) ab n+1 ; ) /[1 c) ab 2n+1 ; ) ], 2 2 C n,n 1 a, b, c; ) := {abn 1 + c n+1 ) c}1 n )1 ab n+1 ), 1 c)1 ab 2n )1 ab 2n+2 ) C n,n 2 a, b, c; ) := ab n+2 ab n c) n 1 ; ) /[1 c) ab 2n ; ) ] 2 2 A6) A7)
S Lewanowicz, P Woźny / Journal of Computational and Applied Mathematics 233 2010) 1554 1561 1561 Proposition A2 The formula x c)c bx) P n x; a, b, c; ) = n+2 k=n D n,k a, b, c; ) P k x; a, b/ 2, c/; ) holds with D n,n+2 a, b, c; ) := bc 1)1 c n+1 ) a n+1 ; ) / ab 2n+1 ; ), 2 2 D n,n+1 a, b, c; ) := {ab n 1 + c n+1 ) c} c 1)1 an+1 )1 b n ), 1 ab 2n )1 ab 2n+2 ) D n,n a, b, c; ) := a n+2 c 1)ab n c) b n 1 ; ) / ab 2n ; ) 2 2 A8) A9) References [1] GE Andrews, R Askey, R Roy, Special Functions, Cambridge Univ Press, Cambridge, 1999 [2] R Koekoek, RF Swarttouw, The Askey scheme of hypergeometric orthogonal polynomials and its -analogue, Rep 98 17, Fac Techn Math Informatics, Delft Univ of Technology, 1998 [3] Y Xu, On discrete orthogonal polynomials of several variables, Adv Appl Math 33 2004) 615 632 [4] CF Dunkl, Orthogonal polynomials in two variables of -Hahn and -Jacobi type, SIAM J Algebr Discrete Methods 1 1980) 137 151 [5] G Gasper, M Rahman, Some systems of multivariable orthogonal -Racah polynomials, Ramanujan J 13 2007) 389 405 [6] TH Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: RA Askey Ed), Theory and Application of Special Functions, Academic Press, New York, 1975, pp 435 495 [7] CF Dunkl, Y Xu, Orthogonal Polynomials of Several Variables, Cambridge Univ Press, Cambridge, 2001 [8] G Gasper, M Rahman, Basic Hypergeometric Series, 2nd ed, Cambridge University Press, Cambridge, 2004 [9] S Lewanowicz, P Woźny, Connections between two-variable Bernstein and Jacobi polynomials on the triangle, J Comput Appl Math 197 2006) 520 533 [10] S Lewanowicz, P Woźny, I Area, E Godoy, Multivariate generalized Bernstein polynomials: Identities for orthogonal polynomials of two variables, Numer Algor 49 2008) 199 220 [11] S Lewanowicz, P Woźny, Generalized Bernstein polynomials, BIT Numer Math 44 2004) 63 78 [12] S Lewanowicz, P Woźny, Dual generalized Bernstein basis, J Approx Theory 138 2006) 129 150 [13] GE Farin, Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide, 3rd ed, Academic Press, Boston, 1996