Jacobi-Sobolev-type orthogonal polynomials: second-order differential equation and zeros
Abstract
We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Qn} associated with the inner product 〈p,q〉=∫−11 p(x)q(x)p(x)dx + A1p(1)q(1) + B1p(−1)q(−1) + A2p′(1)q′(1) + B2p′(−1)q′(−1), where p(x) = (1 − x)α(1 + x)β is the Jacobi weight function, α,β> − 1, A1,B1,A2,B2⩾0 and p, q ∈ P, the linear space of polynomials with real coefficients. The hypergeometric representation (6F5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Qn(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.
Full text
JACOBI-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS: SECOND ORDER DIFFERENTIAL EQUATION AND ZEROS. J. Arves u, R. Alvarez-No darse, F. Marcellan, and K. Pan Preprint MA/UC3M/7/1997 Dedicated to Professor Mario Rosario Occorsio on his 65-th birthday. Key words and phrases: Orthogonal p olynomials, Jacobi p olynomials, hyp ergeometric function, Sob olev-typ e orthogonal p olynomials, WKB metho d. AMS (MOS) sub ject classication: 33C45, 33A65, 42C05. Abstract We obtain an explicit expression for the Sob olev-typ e orthogonal p olynomials Q n ( x ) asso ciated with the inner pro duct < p; q > = Z 1 ? 1 p ( x ) q ( x ) ( x ) dx + A 1 p (1) q (1) + B 1 p ( ? 1) q ( ? 1) + A 2 p 0 (1) q 0 (1) + B 2 p 0 ( ? 1) q 0 ( ? 1) ; where ( x ) = (1 ? x ) (1 + x ) is the Jacobi weight function, ; > ? 1, A 1 ; B 1 ; A 2 ; B 2 0 and p , q 2 IP , the linear space of p olynomials with real co ecients. The hypergeometric representation ( 6 F 5 ) and the second order linear dierential equation that such polynomials satisfy are also obtained. The asymptotic b ehaviour of such polynomials in [-1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q n ( x ). Such a zero is lo cated outside the interval [-1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented. 1 Intro duction. The study of some particular cases of orthogonal p olynomials in Sob olev spaces has attracted the interest of several authors [1], [9], [15], [20], [21] and [25 ]. Particular emphasis was given to the so-called classical Sob olev p olynomials of discrete typ e, i.e., p olynomials orthogonal with resp ect to an inner pro duct < p; q > = Z p ( x ) q ( x ) d ( x ) + N X k =0 Z p ( k ) ( x ) q ( k ) ( x ) d k ( x ) ; where d ( x ) is a classical measure (Jacobi [1], Gegenbauer [8 ], Laguerre [16], Bessel [21 ]) and d k ( x ) are Dirac measures. Decemb er 14, 1997 1
p p p y g p p < p; q > = Z 1 ? 1 p ( x ) q ( x ) ( x ) dx + A 1 p (1) q (1) + B 1 p ( ? 1) q ( ? 1) + A 2 p 0 (1) q 0 (1) + B 2 p 0 ( ? 1) q 0 ( ? 1) ; where ( x ) = (1 ? x ) (1 + x ) is the Jacobi weight function, ; > ? 1, A 1 ; B 1 ; A 2 ; B 2 0 and p , q 2 IP , the linear space of p olynomials with real co ecients. Some estimates concerning to this kind of p olynomials have b een obtained in [3]. However, the explicit form of these p olynomials in the general case remains as an op en question as well as the study of their zeros. We are trying in this pap er to cover this lack. Moreover, some of the usual prop erties of classical orthogonal p olynomials { symmetry prop erty, their representation as hyp ergeometric series and the second order linear dierential equation { are translated to the context of Sob olev-typ e ortogonality. The structure of the pap er is the following. In Section 2 we give some results concerning to classical Jacobi p olynomials. Using these results, in Section 3 we obtain an explicit formula for the JacobiSob olev-typ e orthogonal p olynomials in terms of the classical ones and their rst and second derivatives which allows us to deduce a symmetry prop erty. In Section 4 we establish the recurrence relation that the Jacobi-Sob olev-typ e orthogonal p olynomials satisfy, when the masses A 2 and B 2 are b oth dierent from zero. In Section 5 a representation of our p olynomials as a 6 F 5 hyp ergeometric function is deduced. Finally, in Section 6 a general algorithm in order to generate the second order linear dierential equations that such p olynomials satisfy is given. This result is basic for the development of the Section 8, more precisely for the WKB metho d, in order to obtain the distribution of their zeros. In Section 7, some asymptotic formulas, useful in the study of the zeros, are presented. Finally, in Section 8 we obtain the sp eed of convergence of those zeros lo cated outside [-1, 1]. On the other hand, we show some graphics concerning the WKB density as well as the analytic b ehaviour of the distribution of zeros for Jacobi-Sob olev-typ e orthogonal p olynomials. 2 Classical Jacobi p olynomials. In this section we have enclosed some formulas for the classical Jacobi p olynomials which will b e useful to obtain some prop erties of the Sob olev-typ e orthogonal p olynomials. All the formulas as well as some sp ecial prop erties for the classical Jacobi p olynomials can b e found in the literature [23, Chapter 1-2], [27]. In this work we will use monic p olynomials, i.e., p olynomials with leading co ecient equal to 1. The classical Jacobi p olynomials P ; n ( x ) satisfy the orthogonality relation Z 1 ? 1 P ; n ( x ) P ; m ( x )(1 ? x ) (1 + x ) dx = nm d 2 n ; (1) where d 2 n = jj P ; n ( x ) jj 2 = 2 2 n + + +1 n !?( n + + 1)?( n + + 1)?( n + + + 1) ?(2 n + + + 1)?(2 n + + + 2) : They are the p olynomial solution of the second order linear dierential equation of hyp ergeometric typ e ( x ) y 00 ( x ) + ( x ) y 0 ( x ) + n y ( x ) = 0 ; (2) where ( x ) = (1 ? x 2 ) ; ( x ) = ? ? ( + + 2) x; n = n ( n + + + 1) ; resp ectively. Notice that deg =2 and deg =1. Also they verify the symmetry prop erty P ; n ( x ) = ( ? 1) n P ; n ( ? x ) ; (3) 2
d d x P ; n ( x ) ( P ; n ( x )) ( ) = n ! ( n ? )! P + ; + n ? ( x ) ; with n and n = 0 ; 1 ; 2 ; :::; (4) as well as the three-term recurrence relation xP n ( x ) = P ; n +1 ( x ) + ; n P ; n ( x ) + ; n P ; n ? 1 ( x ) ; (5) where ; n = 2 ? 2 (2 n + + )(2 n + 2 + + ) ; ; n = 4 n ( n + )( n + )( n + + ) (2 n + + ? 1)(2 n + + ) 2 (2 n + + + 1) : (6) They are represented as the hyp ergeometric series P ; n ( x ) = 2 n ( + 1) n ( n + + + 1) n 2 F 1 ? n; n + + + 1 + 1 1 ? x 2 ! ; (7) where p F q a 1 ; a 2 ; :::; a p b 1 ; b 2 ; :::; b q x ! = 1 X k =0 ( a 1 ) k ( a 2 ) k ( a p ) k ( b 1 ) k ( b 2 ) k ( b q ) k x k k ! ; (8) and ( a ) k is the Po chhammer symb ol or shifted factorial ( a ) 0 := 1, ( a ) k := a ( a + 1)( a + 2) ( a + k ? 1) = = ?( a + k ) ?( a ) , k = 1 ; 2 ; 3 ; ::: . As a consequence of this representation we get P ; n (1) = 2 n ( + 1) n ( n + + + 1) n ; P ; n ( ? 1) = ( ? 1) n 2 n ( + 1) n ( n + + + 1) n : (9) The Christoel-Darb oux formula is n ? 1 X m =0 P ; m ( x ) P ; m ( y ) d 2 m = 1 x ? y P ; n ( x ) P ; n ? 1 ( y ) ? P ; n ? 1 ( x ) P ; n ( y ) d 2 n ? 1 ; n = 1 ; 2 ; 3 ; ::: (10) Throughout the work we will denote K ; ( p;q ) n ( x; y ) = n X m =0 ( P ; m ) ( p ) ( x )( P ; m ) ( q ) ( y ) d 2 m = @ p + q @ x p @ y q K ; n ( x; y ) ; (11) the kernels of the Jacobi p olynomials, as well as their derivatives with resp ect to x and y , resp ectively. By using the symmetry prop erty (3) and (11) it is straightforward to prove that the following symmetry prop erties for the Jacobi kernels K ; n ( x; y ) = K ; n ( ? x; ? y ) ; K ; (0 ; 1) n ( x; y ) = ? K ; (0 ; 1) n ( ? x; ? y ) ; K ; (1 ; 1) n ( x; y ) = K ; (1 ; 1) n ( ? x; ? y ) ; (12) hold. In our work we need the explicit expressions of the kernels K ; n ? 1 ( x; 1), K ; (0 ; 1) n ? 1 ( x; 1), K ; n ? 1 ( x; ? 1) and K ; (0 ; 1) n ? 1 ( x; ? 1), resp ectively. To obtain these kernels we can use the Christoel-Darb oux formula, the structure relation, the three-term recurrence relation and the dierentiation formula for classical monic Jacobi p olynomials, resp ectively. The detailed computation can b e found in [5]. We will provide 3
g p p notation ; n = (2 n + + + 1) ; n K ; n ? 1 ( x; 1) = P ; n (1) d 2 n ? 1 ; n h (1 + x )( P ; n ) 0 ( x ) ? nP ; n ( x ) i ; (13) K ; (0 ; 1) n ? 1 ( x; 1) = ( P ; n ) 0 (1) d 2 n ? 1 ; n h (1 + x )( P ; n ) 0 ( x ) ? nP ; n ( x ) i ? ? P ; n (1) d 2 n ? 1 ; n ( + 1) h (1 + )( P ; n ) 0 ( x ) + ( x + 1)( P ; n ) 00 ( x ) i : (14) From the two previous formulas and using the symmetry prop erties (12), we nd K ; n ? 1 ( x; ? 1) = ? P ; n ( ? 1) h (1 ? x )( P ; n ) 0 ( x ) + nP ; n ( x ) i d 2 n ? 1 ; n ; K ; (0 ; 1) n ? 1 ( x; ? 1) = ? ( P ; n ) 0 ( ? 1) h (1 ? x )( P ; n ) 0 ( x ) + nP ; n ( x ) i d 2 n ? 1 ; n + + P ; n ( ? 1) h (1 ? x )( P ; n ) 00 ( x ) ? ( + 1)( P ; n ) 0 ( x ) i d 2 n ? 1 ; n ( + 1) : (15) Also the following values are needed [5 ] K ; n ? 1 (1 ; 1) = ( P ; n (1)) 2 n ( n + ) d 2 n ? 1 ; n ( + 1) ; K ; (0 ; 1) n ? 1 (1 ; 1) = ( P ; n ) 0 (1) P ; n (1)( n + ) d 2 n ? 1 ; n ( + 2)( n ? 1) ? 1 ; K ; n ? 1 (1 ; ? 1) = ? nP ; n ( ? 1) P ; n (1) d 2 n ? 1 ; n ; K ; (0 ; 1) n ? 1 (1 ; ? 1) = ( P ; n ) 0 ( ? 1) P ; n (1)(1 ? n ) d 2 n ? 1 ; n ; K ; (1 ; 1) n ? 1 (1 ; 1) = P ; n (1)( P ; n ) 0 (1)( n + ) ( + 2)( n 2 + n + n ) ? ( + 1)( + + 2) 2 d 2 n ? 1 ; n ( + 1)( + 2)( + 3)( n ? 1) ? 1 ; K ; (1 ; 1) n ? 1 (1 ; ? 1) = ( P ; n ) 0 ( ? 1) P ; n (1)(1 ? n ) n 2 + n + n ? ? ? 2 2 d 2 n ? 1 ; n ( + 1) : (16) 3 Jacobi-Sob olev-typ e orthogonal p olynomials. Consider the inner pro duct in the linear space of p olynomials with real co ecients < p; q > = < p; q > c + A 1 p (1) q (1) + B 1 p ( ? 1) q ( ? 1) + A 2 p 0 (1) q 0 (1) + B 2 p 0 ( ? 1) q 0 ( ? 1) ; (17) where < p; q > c is the Jacobi inner pro duct < p; q > c = Z 1 ? 1 p ( x ) q ( x )(1 ? x ) (1 + x ) dx; > ? 1 ; > ? 1 ; (18) 4
1 2 1 2 g We will denote f Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) g n the monic orthogonal p olynomial sequence with resp ect to the inner pro duct (17). They will b e called Jacobi-Sobolev-type orthogonal polynomials . Let us now to nd an explicit representation of the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) in terms of the classical ones. To obtain this we write the Fourier expansion of the Jacobi-Sob olev-typ e p olynomials in terms of the Jacobi p olynomials ~ Q n ( x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = P ; n ( x ) + n ? 1 X k =0 a n;k P ; k ( x ) ; (19) where P ; n ( x ) is the classical Jacobi monic p olynomial of degree n . To nd the co ecients a n;k we can use the orthogonality of the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) with resp ect to <; > , i.e., < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; P ; k ( x ) > = 0 0 k < n: (20) Thus, according to (17) we nd < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; P ; k ( x ) > = < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; P ; k ( x ) > c + + A 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) P ; k (1) + B 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) P k ( ? 1)+ + A 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1)( P ; k ) 0 (1) + B 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1)( P ; k ) 0 ( ? 1) ; (21) If we use the decomp osition (19) and taking into account (20) we nd the following expression for the co ecients a n;k a n;k = ? A 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) P ; k (1) + B 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) P ; k ( ? 1) d 2 k ? k < n ? A 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1)( P ; k ) 0 (1) + B 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1)( P ; k ) 0 ( ? 1) d 2 k ; (22) where d 2 k denotes the square norm of the classical Jacobi p olynomials (1). Finally, the equation (19) b ecomes Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = P ; n ( x ) ? A 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) K ; n ? 1 ( x; 1) ? ? B 1 Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) K ; n ? 1 ( x; ? 1) ? A 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1) K ; (0 ; 1) n ? 1 ( x; 1) ? ? B 2 ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1) K ; (0 ; 1) n ? 1 ( x; ? 1) : (23) In order to nd the unknowns Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1), Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1), ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1) and ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1) we can take derivatives in (23) and evaluate the resulting equation, as well as (23), at x = 1 and x = ? 1. This leads to a linear system of equations IK ~ Q n = Q n ; (24) 5
y y 1 2 3 4 k 1 = 0 B B B B @ 1 + A 1 K ; n ? 1 (1 ; 1) A 1 K ; n ? 1 (1 ; ? 1) A 1 K ; (0 ; 1) n ? 1 (1 ; 1) A 1 K ; (0 ; 1) n ? 1 (1 ; ? 1) 1 C C C C A ; k 2 = 0 B B B B @ B 1 K ; n ? 1 (1 ; ? 1) 1 + B 1 K ; n ? 1 ( ? 1 ; ? 1) B 1 K ; (0 ; 1) n ? 1 ( ? 1 ; 1) B 1 K ; (0 ; 1) n ? 1 ( ? 1 ; ? 1) 1 C C C C A ; k 3 = 0 B B B B @ A 2 K ; (0 ; 1) n ? 1 (1 ; 1) A 2 K ; (0 ; 1) n ? 1 ( ? 1 ; 1) 1 + A 2 K ; (1 ; 1) n ? 1 (1 ; 1) A 2 K ; (1 ; 1) n ? 1 (1 ; ? 1) 1 C C C C A ; k 4 = 0 B B B B @ B 2 K ; (0 ; 1) n ? 1 (1 ; ? 1) B 2 K ; (0 ; 1) n ? 1 ( ? 1 ; ? 1) B 2 K ; (1 ; 1) n ? 1 (1 ; ? 1) 1 + B 2 K ; (1 ; 1) n ? 1 ( ? 1 ; ? 1) 1 C C C C A ; and ~ Q n and Q n are the column vectors ~ Q n = 0 B B B @ Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1) ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1) 1 C C C A ; Q n = 0 B B B @ P ; n (1) P ; n ( ? 1) ( P ; n ) 0 (1) ( P ; n ) 0 ( ? 1) 1 C C C A ; resp ectively. Let us denote IK j ( Q n ) the matrix obtained substituting the j column in IK by Q n . Then, from the Cramer's, rule the system (24) has a unique solution if and only if the determinant of IK do es not vanish. Moreover, the solution is given by Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) = det IK 1 ( Q n ) det IK ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) = det IK 2 ( Q n ) det IK ; ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1) = det IK 3 ( Q n ) det IK ; ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1) = det IK 4 ( Q n ) det IK : (25) Here we want to remark that, since our p olynomials are orthogonal with resp ect to (17), then the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) exist for all values of the nonnegative masses A 1 , B 1 , A 2 and B 2 . In particular this implies that det IK 6 = 0. This situation is very dierent from one studied in [5] where the p olynomials are orthogonal with resp ect to a linear functional which is not p ositive denite (in general it is not a quasi-denite linear fuctional). Prop osition 1 The fol lowing symmetry property for the Jacobi-Sobolev polynomials holds Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? x ) = ( ? 1) n Q ;;B 1 ;A 1 ;B 2 ;A 2 n ( x ) : (26) Pro of : Let us denote the determinant of IK by 4 ; ;A 1 ;B 1 ;A 2 ;B 2 n and the determinant of IK j ( Q n ) by 4 ; ;A 1 ;B 1 ;A 2 ;B 2 n;j ( Q n ). If we interchange in IK 2 ( Q n ) the rst and second columns and the rst and second rows, the third and fourth columns and the third and fourth rows, resp ectively, and then we use the symmetry prop erty of Jacobi p olynomials (9) and their kernels (12) we nd the following relation for the determinants 4 ;;B 1 ;A 1 ;B 2 ;A 2 n; 2 ( Q n ) = ( ? 1) n 4 ; ;A 1 ;B 1 ;A 2 ;B 2 n; 1 ( Q n ) : If we handle with the same rows and columns but in IK we get 4 ; ;A 1 ;B 1 ;A 2 ;B 2 n = 4 ;;B 1 ;A 1 ;B 2 ;A 2 n : Then, from (25) we obtain Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( ? 1) = ( ? 1) n Q ;;B 1 ;A 1 ;B 2 ;A 2 n (1) : (27) 6
y ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 ( ? 1) = ( ? 1) n ? 1 ( Q ;;B 1 ;A 1 ;B 2 ;A 2 n ) 0 (1) : (28) Now, if we provide the change of parameters $ , A 1 $ B 1 and A 2 $ B 2 in (23) and then use the symmetry prop erties for the Jacobi kernels (12) and (27)-(28) the prop osition holds. Let us now to obtain an explicit formula of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) in terms of the classical Jacobi p olynomials and their rst and second derivatives. We start from formula (23) where we substitute the kernels by their explicit expressions (13)-(15) and use the formulas (25). This leads to the following. Prop osition 2 The Jacobi-Sobolev orthogonal polynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) can be given in terms of the classical Jacobi polynomials and their rst and second derivatives Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = (1 + n n + n n ) P ; n ( x ) + [ n (1 ? x ) ? n (1 + x )+ +( + 1) n + ( + 1) ! n ]( P ; n ( x )) 0 + [ n (1 + x ) ? ! n (1 ? x )] ( P ; n ( x )) 00 ; (29) where n = B 1 C ;;B 1 ;A 1 ;B 2 ;A 2 n + B 2 D ;;B 1 ;A 1 ;B 2 ;A 2 n ; n = A 1 C ; ;A 1 ;B 1 ;A 2 ;B 2 n + A 2 D ; ;A 1 ;B 1 ;A 2 ;B 2 n ; (30) n = A 2 E ; ;A 1 ;B 1 ;A 2 ;B 2 n ; ! n = B 2 E ;;B 1 ;A 1 ;B 2 ;A 2 n ; (31) and C ; ;A 1 ;B 1 ;A 2 ;B 2 n = Q ; ;A 1 ;B 1 ;A 2 ;B 2 n (1) P ; n (1) d 2 n ? 1 ; n ; D ; ;A 1 ;B 1 ;A 2 ;B 2 n = ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1)( P ; n ) 0 (1) d 2 n ? 1 ; n ; E ; ;A 1 ;B 1 ;A 2 ;B 2 n = ( Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ) 0 (1) P ; n (1) d 2 n ? 1 ; n (1 + ) : (32) Notice that the constants n ; n ; n and ! n depend on n; ; and the masses A 1 ; B 1 ; A 2 and B 2 . In the next Section we will establish the recurrence relation that the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) satisfy. Notice that, since the matrix of the moments of the inner pro duct dened by (17) is not of Hankel typ e b ecause < x ; x > 6 = < 1 ; x 2 > , then the Sob olev-typ e orthogonal p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) don't satisfy a three-term recurrence relation. In fact they will satisfy a seventerm recurrence relation (see [15 ]). 4 The seven-term recurrence relation for Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) . Here we will prove that the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) satisfy a seven-term recurrence relation. In fact, it's straightforward to prove that the multiplication op erator by ( x 2 ? 1) 2 is symmetric with resp ect to (17). The problem is to nd a p olynomial op erator of the lowest degree which b e symmetric with resp ect to the Sob olev inner pro duct (17). Cases: 1. If A 2 = B 2 = 0 we have a standard inner pro duct, hence the multiplication op erator by x is symmetric. 7
2 2 6 p p y ( y obtain a ve-term recurrence relation. 3. If A 2 6 = 0 and B 2 = 0 then the multiplication op erator b y ( x ? 1) 2 is symmetric. Hence we obtain a ve-term recurrence relation. There is another interesting case, when the masses A 2 and B 2 are b oth dierent from zero. This situation will b e considered b elow. We assume that A 2 6 = 0 and B 2 6 = 0. In particular, from (17) we get < hp; q > = < p; hq > p; q 2 IP ; (33) for some p olynomial h ( x ) of degree less than or equal to four. This implies that A 2 ( hp ) 0 (1) q 0 (1) + B 2 ( hp ) 0 ( ? 1) q 0 ( ? 1) = A 2 ( hq ) 0 (1) p 0 (1) + B 2 ( hq ) 0 ( ? 1) p 0 ( ? 1) ; 8 p; q 2 IP : (34) Therefore A 2 h 0 (1) p (1) q 0 (1) + B 2 h 0 ( ? 1) p ( ? 1) q 0 ( ? 1) = A 2 h 0 (1) q (1) p 0 (1) + B 2 h 0 ( ? 1) q ( ? 1) p 0 ( ? 1) ; (35) or, equivalently, A 2 h 0 (1) p (1) q 0 (1) ? p 0 (1) q (1) + B 2 h 0 ( ? 1) p ( ? 1) q 0 ( ? 1) ? q ( ? 1) p 0 ( ? 1) = 0 ; 8 p; q 2 IP : (36) If p ( x ) = 1 and q ( x ) = x the equation (36) yields A 2 h 0 (1) + B 2 h 0 ( ? 1) = 0 : (37) If p ( x ) = 1 and q ( x ) = x 2 the equation (36) leads 2 A 2 h 0 (1) ? 2 B 2 h 0 ( ? 1) = 0 : (38) Thus, from (37)-(38) we get ( A 2 h 0 (1) + B 2 h 0 ( ? 1) = 0 ; A 2 h 0 (1) ? B 2 h 0 ( ? 1) = 0 : (39) As A 2 6 = 0 and B 2 6 = 0 = ) h 0 (1) = h 0 ( ? 1) = 0, hence h 0 ( x ) = ( x 2 ? 1) r ( x ). The minimal choice of r ( x ) is, in this situation, r ( x ) 1. Therefore h ( x ) = x 3 3 ? x + a (40) or, equivalently, h ( x ) = x 3 ? 3 x + b: (41) In order to op erate with h ( x ) we put b = 0. In such a way we can guarantee that h ( x ) = x 3 ? 3 x leads to the searched symmetric op erator on IP , when A 2 6 = 0 and B 2 6 = 0. This fact allows to write a seven-term recurrence relation for Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ). In fact, from ( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = n +3 X j =0 nj Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) : (42) and taking into account that nj = < ( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) > < Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) > = = < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; ( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) > < Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) > = 0 ; if j < n ? 3 ; (43) 8
( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = n +3 X j = n ? 3 nj Q ; ;A 1 ;B 1 ;A 2 ;B 2 j ( x ) ; (44) where n;n ? 3 = < ( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) > < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) > = = < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; ( x 3 ? 3 x ) Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) > < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) > = = < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) > < Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) ; Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ? 3 ( x ) > > 0 : (45) 5 Representation as hyp ergeometric series. Here we will prove the following prop osition Prop osition 3 The orthogonal polynomial Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) is, up to a constant factor, a generalized hypergeometric series. More precisely, Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = 2 n ? 3 ( + 3) n ? 3 4 (0) ( n + + + 1) n 6 F 5 ? n;n + + +1 ; 0 +1 ; 1 +1 ; 2 +1 ; 3 +1 +3 ; 0 ; 1 ; 2 ; 3 1 ? x 2 ! ; (46) where 4 (0) is given in (52) and the coecients ? 0 , ? 1 , ? 2 and ? 3 are the zeros of a polynomial of fourth degree at k (see formula (49) from below). In general, they are complex numbers. If for some i = 0 ; 1 ; 2 ; 3 , ? i is a negative integer number we need to take the analytic continuation of the hypergeometric series (46) . The representation (46) can b e considered as a generalization of the representation as hyp ergeometric series of the Jacobi p olynomials. Pro of: Using (4)-(5) we can rewrite (29) as follows Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = A n P ; n ( x ) + nB n P +1 ; +1 n ? 1 ( x ) + nC n P +1 ; +1 n ( x ) + nD n P +1 ; +1 n ? 2 ( x )+ + n ( n ? 1) E n P +2 ; +2 n ? 2 ( x ) + n ( n ? 1) F n P +2 ; +2 n ? 1 ( x ) + n ( n ? 1) G n P +2 ; +2 n ? 3 ( x ) ; (47) where A n = 1 ? nC n ; B n = n ? n + C n +1 ; +1 n ? 1 ; C n = ? ( n + n ) ; D n = C n +1 ; +1 n ? 1 ; E n = n ? ! n + F n +2 ; +2 n ? 2 ; F n = n + ! n ; G n = F n +2 ; +2 n ? 2 : (48) Substituting the hyp ergeometric representation of the Jacobi p olynomials (7) in (47) we nd Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) = 2 n ? 3 ( + 3) n ? 3 ( n + + + 1) n 1 X k =0 " 8 A n ( n + )( k + + 1)( k + + 2) ? ? 4 B n ( n + )( k ? n )( k + n + + + 1)( k + + 2)+ + 8 nC n ( n + )( n + + 1)( k + n + + + 1)( k + n + + + 2)( k + + 2) (2 n + + + 1)(2 n + + + 2) + 9
pp In this section we will apply the so-called semiclassical or WKB approximation (see [7], [29] and references therein) to nd the WKB density of zeros of the p olynomials Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ). Let us denote these zeros by f x n;i g n i =1 . Then the corresp onding distribution function of zeros is given by n ( x ) = 1 n n X i =1 ( x ? x n;i ) : (84) Here we will use the metho d presented in [29] in order to obtain the WKB density of zeros, which gives an approximate analytic expression for the density of zeros of the solutions of any linear second order dierential equation with p olynomial co ecients. In particular we will consider (60) ~ ( x ) y 00 + ~ ( x ) y 0 + ~ ( x ) y = 0 : (85) The key step is the following Theorem 3 ([29]) Let S ( x ) and ( x ) be the functions S ( x ) = 1 4 ~ ( x ) 2 h 2 ~ ( x ) 2 ~ ( x ) ? ~ 0 ( x ) + ~ ( x ) ? 2 ~ 0 ( x ) ? ~ ( x ) i ; (86) ( x ) = 1 4[ S ( x )] 2 ( 5[ S 0 ( x )] 2 4[ S ( x )] ? S 00 ( x ) ) = P ( x; n ) Q ( x; n ) ; (87) where P ( x; n ) and Q ( x; n ) are polynomials in x as wel l as in n . If the condition sup x 2 X j ( x ) j << 1 holds, then the semiclassical or WKB density of zeros of the solutions of (85) is given by W K B ( x ) = 1 q S ( x ) ; x 2 X IR ; (88) in every interval X where the function S ( x ) is positive. Using the ab ove algorithm, the computations have b een p erformed by using the symb olic computer algebra package Mathematica [28]. First of all we check the conditions of the Theorem nding that in the considered case n ? 1 , so the Theorem can b e applied for n large enough. The explicit expression for W K B ( x ) given by (88) is extremely large and we will omit it here. It is straightforward to see that if we take the limit A 1 ; A 2 ; B 1 ; B 2 ! 0 in the resulting expression for W K B ( x ) we recover the classical expression for the Jacobi p olynomials [29]. We will provide here some graphics for the normalized W K B ( x ) function. In Figure 1 the WKB density of zeros for the Jacobi-Sob olev-typ e orthogonal p olynomials app ears. We have used the formulas (60), (69), (86) and (88) and plotted the normalized Density function for n = 10 4 in four dierent cases with several values of the parameters and ( = = 0, = = ? 1 2 , = = 5 and nonsymmetric case = 0 and = 1). In Figure 2 app ears the WKB density of zeros for the same values of and and n = 10 5 . In Figure 3 we represent the WKB density of zeros for n = 10 6 and the same values of the parameters and . Finally, in Figure 4 is shown W K B ( x ), for n = 10 7 with the ab ove values of and . Clear, in each Figure from the b ottom to the top, is distinguishible the case = = 0, while the remaining cases b ehave almost equal. Some numerical tests based on the computation of the numb er N of zeros in the interval ( ? 1 10 ; 1 10 ) by using the expression N R 1 = 10 ? 1 = 10 W K B ( x ) dx for b oth families of orthogonal p olynomials P ; n ( x ) and Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) show that their global sp ectral prop erties are the same. This result is in accordance with the next one. 16
n g n zero of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) . Then n ?! 1 p 1 ? x 2 ; in the weak star topology. Pro of: From (6) and (69), we get A n = 1 ? + + 4 n + o 1 n ; E n = 1 + 5 + 2 ? 5 ? 2 n 4 + o 1 n 4 ; B n = 2( ? ) n 2 + 2 ? (19 + (7 + )) + 2 (5 + ) ? 5 2 ? 3 + 3 (4 + ) n 4 + o 1 n 4 C n = 2 n 2 ( + + 4) + o 1 n 2 ; D n = 1 2 n 2 ( + + 6) + o 1 n 2 ; F n = 12 + (5 + ) + (5 + ) 2 n 4 + o 1 n 4 ; G n = 2(12 + (5 + ) + (5 + )) n 4 + o 1 n 4 : Using (47), jj Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) jj [ ? 1 ; 1] A n jj P ; n ( x ) jj [ ? 1 ; 1] + nB n jj P +1 ; +1 n ? 1 ( x ) jj [ ? 1 ; 1] + nC n jj P +1 ; +1 n ( x ) jj [ ? 1 ; 1] + nD n jj P +1 ; +1 n ? 2 ( x ) jj [ ? 1 ; 1] + n ( n ? 1) E n jj P +2 ; +2 n ? 2 ( x ) jj [ ? 1 ; 1] + n ( n ? 1) F n jj P +2 ; +2 n ? 1 ( x ) jj [ ? 1 ; 1] + n ( n ? 1) G n jj P +2 ; +2 n ? 3 ( x ) jj [ ? 1 ; 1] ; (89) where jj jj [ ? 1 ; 1] denotes the sup-norm in the interval [-1, 1]. Because of jj P ; n ( x ) jj 1 n [ ? 1 ; 1] 1 2 (see [27]), we deduce lim n !1 jj Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) jj 1 n [ ? 1 ; 1] 1 2 : (90) Thus, from Theorem 2.1 in [10] n ?! 1 p 1 ? x 2 : (91) ACKNOWLEDGEMENTS Part of this work was provided during the stay of the second author in the University of Amsterdam. He is very grateful to the Department of Mathematics of the University of Amsterdam for his kind hospitality. The research of the rst author (JA) was supp orted by a grant of Ministerio de Educacion y Cultura (MEC) of Spain. The research of the three rst authors (JA, RAN and FM) was supp orted by Direccion General de Ense ~nanza Sup erior (DGES) of Spain under grant PB 96-0120-C03-01. The authors are very grateful to the unknown referees for their helpful remarks and for help us to correct some missprints and errors and signicantly improve the pap er. 17
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-1 -0.5 0.5 1 0.5 1 1.5 2 2.5 Figure 1: WKB Density of zeros for n = 10 4 of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) with x 2 [ ? 0 : 99 ; 0 : 99] . -1 -0.5 0.5 1 0.5 1 1.5 2 2.5 Figure 2: WKB Density of zeros for n = 10 5 of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) with x 2 [ ? 0 : 99 ; 0 : 99] . -1 -0.5 0.5 1 0.5 1 1.5 2 Figure 3: WKB Density of zeros for n = 10 6 of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) with x 2 [ ? 0 : 986 ; 0 : 986]. -1 -0.5 0.5 1 0.5 1 1.5 2 2.5 Figure 4: WKB Density of zeros for n = 10 7 of Q ; ;A 1 ;B 1 ;A 2 ;B 2 n ( x ) with x 2 [ ? 0 : 986 ; 0 : 986] . Figure 5: Comparison of the numerical computation results. 20