On the modification of classical orthogonal polynomials: the symmetric case
Abstract
We consider the modifcations of the monic Hermite and Gegenbauer polynomials via the addition of one point mass at the origin. Some properties of the resulting polynomials are studied: three-term recurrence relation, differential equation, ratio asymptotics, hypergeometric representation as well as, for large n, the behaviour of their zeros.
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ON THE MODIFICATIONS OF CLASSICAL ORTHOGONAL POLYNOMIALS: THE SYMMETRIC CASE. 1 R. Alvarez-No darse y and F. Marcellan z Departamento de Matematicas. Escuela Politecnica Superior. Universidad Carlos III de Madrid. Butarque 15, 28911,Leganes,Madrid. July 4, 1997 Key words and phrases: Hermite p olynomials, Gegenbauer p olynomials, discrete measures, zeros, symmetric functionals. AMS (MOS) sub ject classication: 33A65 Abstract We consider the mo dications of the monic Hermite and Gegenbauer polynomials via the addition of one point mass at the origin. Some prop erties of the resulting p olynomials are studied: three-term recurrence relation, dierential equation, ratio asymptotics, hyp ergeometric representation as well as, for large n , the b ehaviour of their zeros. 1 Intro duction. In 1940, H. L. Krall [19] obtained three new classes of p olynomials orthogonal with resp ect to measures which are not absolutely continuous with resp ect to the Leb esgue measure. In fact, his study is related to an extension of the very well known characterization of classical orthogonal p olynomials by S. Bo chner. This kind of measures was not considered in [28 ]. Moreover, in his pap er H. L. Krall obtain that these three new families of orthogonal p olynomials satisfy a fourth order dierential equation. The corresp onding measures are given in the following table. f P n ( x ) g weight function d supp ( ) Laguerre-typ e e x dx + M ( x ) ; M > 0 [0 ; 1 ) Legendre-typ e 2 dx + ( x 1) 2 + ( x + 1) 2 ; > 0[ 1 ; 1] Jacobi-typ e (1 x ) dx + M ( x ) ; M > 0 ; > 1 [0 ; 1] A dierent approach to this sub ject was presented in [18]. 1 y [email protected] z [email protected] MA/UC3M/15/1996 1
The analysis of prop erties of p olynomials orthogonal with resp ect to a p erturbation of a measure via the addition of mass p oints was intro duced by P.Nevai [23 ]. There the asymptotic prop erties of the new p olynomials have b een considered. In particular, he proved the dep endence of such prop erties in terms of the lo cation of the mass p oints with resp ect to the supp ort of the measure. Particular emphasis was given to measures supp orted in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three-term recurrence relation that the corresp onding sequence of orthogonal p olynomials satisfy. The analysis of algebraic prop erties for such p olynomials attracted the interest of several researchers (see [7] for p ositive Borel measures and [21 ] for a more general situation). From the p oint of view of dierential equations see [22]. When two mass p oints are considered, the diculties increase as shows [10]. An interesting application for the addition of two mass p oints at 1 to the Jacobi weight function was analyzed in [17 ]. In this work we will study a generalization of the Hermite and Gegenbauer p olynomials. In fact, we will study the p olynomials orthogonal with resp ect to a mo dication of a symmetric weight function via the addition of one Delta measure at x = 0. It is easy to see that the resulting linear functional is symmetric. In Section 2 we include all the prop erties of the Hermite and Gegenbauer p olynomials which will need. In Section 3 we study the generalized Hermite p olynomials and Section 4 is devoted to the Gegenbauer case. In particular, we obtain their expression in terms of the classical p olynomials, the hyp ergeometric representations, the ratio asymptotics, the second order dierential equation and the three-term recurrerence relation that such generalized p olynomials satisfy. Finally, using the techniques develop ed in [6] and [5],[31 ], we deduce in Section 5 some moments of the distribution of zeros, as well as his semiclassical or WKB density. 2 Some Preliminary Results. In this section we will enclose the basic characteristics of the Hermite and Gegenbauer monic orthogonal p olynomials. For more details see, for instance, [8], [11 ], [24 ], [28 ]. 2.1 Classical Hermite Polynomials. The Hermite p olynomials H n ( x ) are the p olynomial solutions of the second order differential equation y 00 ( x ) 2 xy 0 ( x ) + 2 ny ( x ) = 0 : (1) They satisfy an orthogonal relation of the form Z 1 1 H n ( x ) H k ( x ) e x 2 dx = nk 2 n n ! p ; as well as a three-term recurrence relation (TTRR) xH n ( x ) = H n +1 ( x ) + n 2 H n 1 ( x ) ; (2) 2
and the dierentiation formula ( H n ( x )) ( ) = n ! ( n )! H n ( x ) ; = 1 ; 2 ; 3 ; ::: : (3) Since they are orthogonal with resp ect to a symmetric weight function, then H n ( x ) = ( 1) n H n ( x ) : They are connected with the classical Laguerre p olynomials by relations (see [24] and [28]) H 2 m ( x ) = L 1 2 m ( x 2 ) ; H 2 m +1 ( x ) = xL 1 2 m ( x 2 ) ; and H 2 m (0) = L 1 2 m (0) = ( 1) m (2 m )! 2 2 m m ! ; H 2 m +1 (0) = 0 ; m = 0 ; 1 ; 2 ; ::: : (4) 2.2 Classical Gegenbauer Polynomials. The Gegenbauer p olynomials G n ( x ) are dened G n ( x ) P 1 2 ; 1 2 n ( x ) ; (5) where P ; n ( x ) denotes the classical Jacobi p olynomials [24 ], [28 ]. They satisfy the second order dierential equation (1 x 2 ) y 00 ( x ) (2 + 1) xy 0 ( x ) + n (2 + n ) y ( x ) = 0 ; (6) as well as an orthogonal relation of the form ( > 1 2 ) Z 1 1 G n ( x ) G k ( x )(1 x 2 ) 1 2 dx = nk p n !( n + + 1 2 )( n + 2 ) ( n + + 1)(2 n + 2 ) : They satisfy a three-term recurrence relation (TTRR) xG n ( x ) = G n +1 ( x ) + n (2 + n 1) 4( + n )( + n 1) G n 1 ( x ) ; (7) and the dierentiation formula ( G n ( x )) ( ) = n ! ( n )! G + n ( x ) ; = 1 ; 2 ; 3 ; :::; (8) which is a consequence of the dierentiation formula for the Jacobi p olynomials [24], [28] ( P ; n ( x )) ( ) = n ! ( n )! P + ; + n ( x ) ; = 1 ; 2 ; 3 ; ::: : (9) Since they are orthogonal with resp ect to a symmetric weight function, then G n ( x ) = ( 1) n G n ( x ) : They are connected with the classical Jacobi p olynomials by (see [28 ]) G 2 m ( x ) = 1 2 m P ( 1 2 ; 1 2 ) m (2 x 2 1) ; G 2 m +1 ( x ) = 1 2 m x P ( 1 2 ; 1 2 ) m (2 x 2 1) ; (10) and G 2 m (0) = 2 m P ( 1 2 ; 1 2 ) m ( 1) = ( 1) m ( 1 2 ) m ( m + ) m ; G 2 m +1 (0) = 0 ; m = 0 ; 1 ; 2 ; ::: : (11) 3
3 Generalized Hermite Polynomials. Denition 3.1 The generalized monic Hermite p olynomials H A n ( x ) are the polynomials orthogonal with respect to the linear functional U < U ; P > = < H ; P > + AP (0) ; A 0 ; (12) dened on the set of polynomials IP with real coecients supported on the real line, where H denotes the Hermite functional < H ; P > = Z 1 1 e x 2 P ( x ) dx : (13) In order to obtain the p olynomials H A n ( x ) we consider their Fourier expansion in terms of the classical ones, i.e., H A n ( x ) = n X k =0 a n;k H k ( x ) ; and use the orthogonality prop erty in the same sense that in [2]-[4 ], [21 ]. Nevertheless, we will obtain them by another way. Let us write the symmetric functional U in the form < U ; P > = Z 1 1 e x 2 P ( x ) dx + AP (0) ; A 0 : (14) We will decomp ose the p olynomial H A k ( x ) in two p olynomials H A k ( x ) = p A n ( x 2 ) + xq A m ( x 2 ) ; k = max f 2 n; 2 m + 1 g ; (15) and substitute it in (14). Some straightforward computation gives us 2 Z 1 0 p A n ( x 2 ) p A k ( x 2 ) e x 2 dx + Ap A n (0) p A k (0) + 2 Z 1 0 q A m ( x 2 ) q A l ( x 2 ) x 2 e x 2 dx: If we intro duce in the last expression the change of variables = x 2 we obtain Z 1 0 p A n ( ) p A k ( ) 1 2 e d + Ap A n (0) p A k (0) |{z } p m ( ) = C m L 1 2 ;A m ( x 2 ). + Z 1 0 q A m ( ) q A l ( ) 1 2 e d |{z } q A m = c m L 1 2 m ( x ) : Then, H A 2 m ( x ) = L 1 2 ;A m ( x 2 ) ; H A 2 m +1 ( x ) = xL 1 2 m ( x 2 ) ; m = 0 ; 1 ; 2 ; ::: ; (16) where L ;A m ( x ) denotes the generalized Laguerre-Ko ornwinder p olynomials [4 ], [12]-[15 ], i.e., the p olynomials orthogonal with resp ect to the mo dication of the weight function ( x ) = x e x via the addition of one delta Dirac measure at x = 0. By using the representation formulas for the monic p olynomials L ;A m ( x ) (see [4]) L ;A n ( x ) = L n ( x ) + n d dx L n ( x ) = ( I + n d dx ) L n ( x ) ; (17) n = A ( + 1) n n !( + 1) 1 + A ( +1) n ( n 1)!( +2) ; we obtain the following representation formula for these generalized Hermite p olynomials 4
Prop osition 3.1 The generalized Hermite polynomials H A n ( x ) admit the fol lowing representations in terms of the classical polynomials 1. If n = 2 m , m = 0 ; 1 ; 2 ; ::: , then H A 2 m ( x ) = L 1 2 m ( x 2 ) + B m d dx 2 L 1 2 m ( x 2 ) ; 2 xH A 2 m ( x ) = 2 xH 2 m ( x ) + B m d dx H 2 m ( x ) ; B m = A 1 + A 2( m + 1 2 ) ( m ) ( m + 1 2 ) m ! : (18) 2. If n = 2 m 1 , m = 1 ; 2 ; ::: , then H A 2 m 1 ( x ) = xL 1 2 m 1 ( x 2 ) = H 2 m 1 ( x ) : (19) As we can see from the ab ove prop osition, the p olynomials of o dd degree coincide with the classical ones; then we will only study the p olynomials of even degree. 3.1 The hyp ergeometric representation. Prop osition 3.2 The generalized Hermite polynomials H A 2 m ( x ) are, up to a multiplicative factor, an hypergeometric function 2 F 2 . More precisely, H 2 m ( x ) = ( 1) m (1 2 mB m )( 1 2 ) m 2 F 2 m; 0 + 1 3 2 ; 0 ; x 2 ! ; (20) where 0 = 1 2 mB m 2(1+ B m ) is, in general, a real number. In the case when 0 is a nonpositive integer we wil l take the analytic continuation of the hypergeometric series. Pro of: From the hyp ergeometric representation of Laguerre p olynomials [24], [28 ] L n ( x ) = ( 1) n ( + 1) n 1 F 1 n + 1 ; x ! ; (21) where the hyp ergeometric function p F q is dened by 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 ! ; and ( a ) k denotes the Po chammer symb ol ( a ) 0 := 1 ; ( a ) k := a ( a + 1)( a + 2) ( a + k 1) ; k = 1 ; 2 ; 3 ; ::: : From formula (18) we deduce H A 2 m ( x ) = ( 1) m ( 1 2 ) m " 1 X k =0 ( m ) k ( 1 2 ) k k k ! + B m ( m ) k +1 ( 1 2 ) k +1 k k ! # ; = x 2 : 5
Using ( a ) k +1 = ( a + k + 1)( a ) k we nd H A 2 m ( x ) = ( 1) m ( 1 2 ) m (1 + B m ) 1 X k =0 ( m ) k ( 3 2 ) k k k ! k + 1 2 mB m 1 + B m ; = x 2 : Notice that the expression inside the quadratic brackets is a p olynomial in m of degree 1 of the form [ k + 0 ], where 0 = 1 2 mB m 2(1+ B m ) . Then, from the identities ( a + 1) k = a + k a ( a ) k or ( k + a ) = a ( a + 1) k ( a ) k ; (22) the last expression yields (20). 3.2 Asymptotic of the p olynomials H A 2 m ( x ) . In order to obtain the asymptotic prop erties of the p olynomials H A 2 m ( x ) for large enough m , we rewrite (18) in the form H A 2 m ( x ) H 2 m ( x ) = 1 + B m ( L 1 2 m ( x 2 )) 0 L 1 2 m ( x 2 ) ; (23) where ( L 1 2 n ( x 2 )) 0 denotes the derivative with resp ect to x 2 . If we use the asymptotic formula for the gamma function [1 ] ( ax + b ) p 2 e ax ( ax ) ax + b 1 2 ; x >> 1 ; the following asymptotic expression for the constant B m holds B m 1 2 m : To obtain the asymptotic formula for the ratio H A 2 m ( z ) H 2 m ( z ) we can use the Perron formula (see [29], Eq. (4.2.6) page 133 and [28], Theorem 8.22.3) for the ratio 1 p n ( L n ) 0 ( z ) L n ( z ) of the Laguerre p olynomials ( z 2 IC nf [0 ; 1 ) g ) 1 p n ( L n ) 0 ( z ) L n ( z ) = 1 p z 1 + 1 p n [ C 1 ( + 1 ; z ) C 1 ( ; z ) p z ] + o 1 p n ; where C 1 ( ; z ) = 1 4 p z 3 z + 1 3 z 2 + 1 4 2 . Taking into account that in (23) we have the ratio ( L 1 2 n ( x 2 )) 0 L 1 2 n ( x 2 ) , we need to substitute in the previous expression z ! z 2 . But C 1 ( 1 2 ; z 2 ) = C 1 ( 1 2 ; z 2 ) ; and then, for m large enough H A 2 m ( z ) H 2 m ( z ) = 1 1 2 p m iz 1 iz p m + o 1 m ; z 2 IC n IR : (24) 6
3.3 Second order dierential equation. Here we will obtain an algorithm that allows us to deduce the second order dierential equation (SODE) that the generalized p olynomials satisfy. First of all, notice that b oth classical p olynomials under consideration (Hermite and Gegenbauer) satisfy a SODE ( x ) P 00 n ( x ) + ( x ) P 0 n ( x ) + n P n ( x ) = 0 : In order to obtain the second order dierential equation (SODE) that the generalized Hermite p olynomials satisfy we will rewrite formula (18) in a more convenient form (notice that for Hermite p olynomials ( x ) = 1) 2 x ~ P A 2 m ( x ) = 2 x ~ C P 2 m ( x ) + ( x ) ~ B P 0 2 m ( x ) ; (25) where ~ P A 2 m ( x ) denotes the generalized p olynomial and P 2 m ( x ) denotes the classical one. For the Hermite p olynomials it is easy to check that ~ C = 1 and ~ B = B m . We will show later (see formula (43) from b elow), that there exists for the generalized Gegenbauer p olynomials a similar representation (25), but with ( x ) = 1 x 2 ; ~ C = 1 + mW A m and ~ B = W A m . Next, we will deduce the SODE for these generalized p olynomials. First of all, notice that the SODE which satisfy the classical p olynomials can b e rewritten in the form ( x ) P 00 2 m ( x ) = ( x ) P 0 2 m ( x ) 2 m P 2 m ( x ) : Taking derivatives in (25), multiplying by x and using the ab ove SODE we obtain ( x ) d dx ~ P A 2 m ( x ) = c ( x ) P 2 m ( x ) + d ( x ) d dx P 2 m ( x ) ; c ( x ) = x ~ B n ; d ( x ) = x [2 x ~ C + 0 ( x ) ~ B ] [ ( x ) + x ( x )] : (26) Taking derivatives in (26), multiplying by x ( x ) and using (26), as well as the SODE for the P 2 m ( x ) we get ( x ) 2 d 2 dx 2 ~ P A 2 m ( x ) = e ( x ) P 2 m ( x ) + f ( x ) d dx P 2 m ( x ) ; e ( x ) = ( x )[ xc 0 ( x ) 2 c ( x )] x n d ( x ) ; f ( x ) = x ( x )[ c ( x ) + d 0 ( x )] d ( x )(2 ( x ) + x ( x )] : (27) The expressions (25),(26) and (27) lead to the condition 2 x ~ P A 2 m ( x ) a ( x ) b ( x ) 2 x 2 ( x ) d dx ~ P A 2 m ( x ) c ( x ) d ( x ) 2 x 3 ( x ) d 2 dx 2 ~ P A 2 m ( x ) e ( x ) f ( x ) = 0 ; (28) where a ( x ) = 2 x ~ C and b ( x ) = ( x ) ~ B . Expanding the determinant in (28) by the rst column ~ m ( x ) d 2 dx 2 ~ P A 2 m ( x ) + ~ m ( x ) d dx ~ P A 2 m ( x ) + ~ m ( x ) ~ P A 2 m ( x ) = 0 ; (29) 7
where ~ m ( x ) = ( x ) x 2 [ a ( x ) d ( x ) c ( x ) b ( x )] ; ~ m ( x ) = x [ e ( x ) b ( x ) a ( x ) f ( x )] ; ~ m ( x ) = c ( x ) f ( x ) e ( x ) d ( x ) : (30) If we apply this algorithm for the generalized Hermite p olynomials, for which (see Eq. (18)) ~ C = 1 ; ~ B = B m ; ( x ) = 1 ; we obtain Prop osition 3.3 The generalized Hermite polynomials of even degree satisfy a second order dierential equation ~ m ( x ) d 2 dx 2 H A 2 m ( x ) + ~ m ( x ) d dx H A 2 m ( x ) + ~ m ( x ) H A 2 m ( x ) = 0 ; (31) where ~ m ( x ) = x B m + 2 B 2 m m + 2 x 2 + 2 B m x 2 ; ~ m ( x ) = 2 B m + 2 B 2 m m + B m x 2 2 B 2 m m x 2 2 x 4 2 B m x 4 ; ~ m ( x ) = 4 m x 3 B m 2 B 2 m + 2 B 2 m m + 2 x 2 + 2 B m x 2 : (32) 3.4 The three-term recurrence relation. Prop osition 3.4 The generalized Hermite polynomials satisfy a three-term recurrence relation (TTRR) xH A n ( x ) = H A n +1 ( x ) + A n H A n ( x ) + A n P A n 1 ( x ) ; n 0 H A 1 ( x ) = 0 and H A 0 ( x ) = 1 : (33) This is a consequence of the orthogonality prop erty with resp ect to a p ositive denite functional (see [8 ] or [24]). To obtain the TTRR's co ecients notice that the functional is symmetric and then < U ; xH A n ( x ) H A n ( x ) > = 0, i.e., A n = 0. To obtain the co ecient A n we can analyze the two cases n = 2 m and n = 2 m 1, separately. For the co ecients A n , n = 2 m 1, if we evaluate (33) in x = 0 ( H A 2 m 2 (0) 6 = 0) we obtain A 2 m 1 = H A 2 m (0) H A 2 m 2 (0) = (2 m 1) 2 1 + 2 A ( m 1 2 ) ( m 1) 1 + 2 A ( m + 1 2 ) ( m ) : (34) For the co ecients A n , n = 2 m , this pro cedure in not valid b ecause H A 2 m 1 (0) = 0. For this reason we need to calculate it directly from the denition A 2 m = < U ; xH A 2 m ( x ) H A 2 m 1 ( x ) > < U ; [ H A 2 m 1 ( x )] 2 > : Since H A 2 m 1 ( x ) = H 2 m 1 ( x ), then the denominator is the square norm of the classical Hermite p olynomials d 2 2 m 1 . Let us to calculate the numerator in the ab ove expression. 8
In order to do this we will use the TTRR for the classical Hermite p olynomials, the dierentiation formula (3) as well as formula (18). Then, < U ; xH A 2 m ( x ) H A 2 m 1 ( x ) > = Z 1 1 e x 2 H 2 m 1 ( x ) xH 2 m ( x ) + B m 2 H 0 2 m ( x ) dx = = Z 1 1 e x 2 xH 2 m 1 ( x ) H 2 m ( x ) dx + mB m d 2 2 m 1 ; from which we obtain A 2 m = 2 m + mB m = m (1 + B m ) : (35) Now, notice that A 2 m +1 = H A 2 m +2 (0) H A 2 m (0) = < U ; xH A 2 m +1 ( x ) H A 2 m ( x ) > ( d A 2 m ) 2 : If we calculate the numerator of the ab ove expression we nd < U ; xH A 2 m +1 ( x ) H A 2 m ( x ) > = Z 1 1 e x 2 H 2 m +1 ( x ) xH 2 m ( x ) + B m 2 H 0 2 m ( x ) dx = = Z 1 1 e x 2 xH 2 m +1 ( x ) H 2 m ( x ) dx = 1 2 (2 m + 1) d 2 2 m = ( d A 2 m ) 2 A 2 m +1 : The ab ove formula allows us to calculate the square norm of the generalized Hermite p olynomials. In fact, from the last expression and (34) we obtain 1. If n = 2 m , m = 0 ; 1 ; 2 ; ::: , then ( d A 2 m ) 2 = 1 + 2 A ( m + 3 2 ) ( m +1) 1 + 2 A ( m + 1 2 ) ( m ) (2 m )! p 2 2 m : (36) 2. If n = 2 m + 1, m = 0 ; 1 ; 2 ; ::: , then ( d A 2 m +1 ) 2 = d 2 2 m +1 = (2 m + 1)! p 2 2 m +1 : (37) Notice that, when m = 0, [ d A 0 ] 2 = p + A . This follows from (36) considering the limit when m ! 0 and using that lim x ! 0 ( x ) = 1 . 4 The generalized Gegenbauer p olynomials. Denition 4.1 The generalized monic Gegenbauer p olynomials G ;A n ( x ) are the polynomials orthogonal with respect to the linear functional U < U ; P > = < C G ; P > + AP (0) ; A 0 ; (38) dened on the set of polynomials IP with real coecients, supported on [ 1 ; 1] , where C G denotes the Gegenbauer functional < C G ; P > = Z 1 1 (1 x 2 ) 1 2 P ( x ) dx; > 1 2 : (39) 9
If n = 2 m 1, m = 1 ; 2 ; ::: , then, G ;A 2 m 1 ( x ) G 2 m 1 ( x ) 1 = 0 ; 2 = 2 m (2 2 m ) 4 ( 2 + 2 m + ) ; and the moments are 1 = 0 ; 2 = 2 m 1 2 (2 m 2 + ) : The asymptotic b ehavior of these two moments in b oth cases is 1 = 0 y 2 1 2 + O ( n 1 ) : All o dd moments vanish b ecause our functionals are symmetric. Notice that equation (58) and relation k = ( 1) k k ! Y k ( y 1 ; y 2 ; 2 y 3 ; :::; ( k 1)! y k ) provide us a general metho d to obtain all the moments r = 1 n y r , but it is highly non-linear and cumb ersome. This is a reason why we use it only for the computation of the moments of low order. We want to remark here that the metho d describ ed ab ove allows a recurrent computation of the moments of any desired order and it can b e implemented in any computer algebra system. See, for instance, [27], [33] where the corresp onding symb olic programs were used to compute the moment of p olynomial solutions of fourth-order dierential equations. 5.2 The semiclassical density distribution of of zeros. Next, we will analyze the so-called semiclassical or WKB approximation (see [5],[31] and references contained therein). Denoting the zeros of ~ P A n ( x ) by f x n;k g n k =1 we can dene its distribution function as n ( x ) = 1 n n X k =1 ( x x n;k ) : (59) We will follow the metho d presented in [31 ] in order to obtain the WKB density of zeros, which is an approximate expression for the density of zeros of solutions of any second order linear dierential equation with p olynomial co ecients a 2 ( x ) y 00 + a 1 ( x ) y 0 + a 0 ( x ) y = 0 (60) The main result is established in the following Theorem 5.1 Let S ( x ) and ( x ) be the functions S ( x ) = 1 4 a 2 2 f 2 a 2 (2 a 0 a 0 1 ) + a 1 (2 a 0 2 a 1 ) g ; (61) ( x ) = 1 4[ S ( x )] 2 ( 5[ S 0 ( x )] 2 4[ S ( x )] S 00 ( x ) ) = P ( x; n ) Q ( x; n ) ; (62) where P ( x; n ) and Q ( x; n ) are polynomials in x as wel l as in n . If the condition ( x ) << 1 holds, then, the semiclassical or WKB density of zeros of the solutions of (60) is given by W K B ( x ) = 1 q S ( x ) ; x 2 I IR ; (63) in every interval I where the function S ( x ) is positive. 16
-200-100 100 200 Classical Hermite 20 40 60 80 -200-100 100 200 Generalized Hermite 20 40 60 80 Figure 1: WKB density of zeros of the H A n ( x ). The pro of of this Theorem can b e found in [5 ], [31 ]. Now we can apply this result to our dierential equation. Using the co ecients of the equation (56) we obtain that for suciently large n , ( x ) n 1 . From the ab ove Theorem the corresp onding WKB density of zeros of the p olynomials ~ P A n ( x ) follows. The computations are very long and cumb ersome. For this reason we provide a little program using Mathematica [30] and some graphics representation for the W K B ( x ) function. We will analyze only the p olynomials of even degree, i.e., ~ P A 2 m ( x ). 5.2.1 Hermite p olynomials H A 2 m ( x ) . In this case from (61) and (63) w k bcl as ( x ) = p R ( x ) ( B m + 2 B 2 m m + 2 x 2 + 2 B m x 2 ) ; R ( x ) = 6 B m 3 B 2 m + 24 B 2 m m + 8 B 3 m m 32 B 3 m m 2 4 B 4 m m 2 + 16 B 4 m m 3 8 B m x 2 9 B 2 m x 2 32 B m m x 2 32 B 2 m m x 2 + 4 B 3 m m x 2 + +32 B 2 m m 2 x 2 + 32 B 3 m m 2 x 2 4 B 4 m m 2 x 2 + 4 x 4 + 12 B m x 4 8 B 2 m x 4 + +16 m x 4 + 32 B m m x 4 + 8 B 2 m m x 4 8 B 3 m m x 4 4 x 6 8 B m x 6 4 B 2 m x 6 : If we take the limit A ! 0, we recover the classical expression [31], [32 ] w k b ( x ) = p 1 + 4 m x 2 : Notice that, since B m 1 2 m , w k b ( x ) has the asymptotic form asy mp w k b ( x ) = p 2 + x 2 + 4 m x 2 x 4 x : In Figure 1 we represent the WKB density of zeros for our generalized Hermite p olynomials. We have plotted the Density function for dierent values of n (from top to b ottom) n = 2 10 4 ; 1 : 5 10 4 ; 10 4 ; 10 3 . Notice that the value of the mass do esn't play a crucial role, since for n >> 1 B m 1 2 m , indep endently of A . It is imp ortant to take into account that our generalized p olynomials have a lot of zeros near the origin. This follows from the fact that w k b ( x ) have, asymptotically, a singular p oint at x = 0. 5.2.2 Gegenbauer p olynomials G ;A 2 m ( x ) . 17
WKB Density -1 -0.5 0.5 1 Classical Gegenbauer 20000 40000 60000 80000 -1 -0.5 0.5 1 Gegenbauer-Koornwinder 10000 20000 30000 40000 50000 60000 70000 -1 -0.5 0.5 1 Classical Legendre 20000 40000 60000 80000 -1 -0.5 0.5 1 Legendre-Koornwinder 10000 20000 30000 40000 50000 60000 70000 Figure 2: WKB density of zeros of the G ;A n ( x ). In this case the expression is very large and we will provide only the limit case when A ! 0. In this case we recover the classical expression [31], [32 ] w k b ( x ) = p 2 + 16 m 2 + 4 + 16 m + x 2 16 m 2 x 2 16 m x 2 4 2 x 2 2 (1 x 2 ) : For Legendre generalized p olynomials we have, asymptotically, asy mp w k b ( x ) = p R asy mp ( x ) (1 x ) (1 + x ) (1 x 2 + m x 2 4 m 2 x 2 + 4 m 3 x 2 ) ; where R asy mp ( x ) = 7 8 m + 4 m 2 12 x 2 + 2 m x 2 8 m 2 x 2 16 m 3 x 2 + 64 m 4 x 2 32 m 5 x 2 + +7 x 4 + 7 m x 4 + 6 m 2 x 4 + 44 m 3 x 4 112 m 4 x 4 + 80 m 5 x 4 96 m 6 x 4 + 64 m 7 x 4 2 x 6 10 m 2 x 6 4 m 3 x 6 + 16 m 4 x 6 32 m 5 x 6 + 96 m 6 x 6 64 m 7 x 6 : In Figure 2 we represent the WKB density of zeros for our generalized Gegenbauer p olynomials. Notice that the value of the mass do esn't play a crucial role, since for n >> 1, W m 1 2 m 2 , indep endently of A . We have plotted the Density function for dierent values of the degree of the p olynomials (from top to b ottom) n = 2 10 4 ; 1 : 5 10 4 ; 10 4 ; 10 3 for two dierent cases: the generalized Legendre p olynomials ( = 1 2 ) and the generalized Gegenbauer with = 5. 5.2.3 Numerical Exp eriments. As we can see in Figures 1 and 2, the zeros of the classical and generalized p olynomials have the same b ehaviour. In order to convince ourself that really the inuence of the masses is very small we compare the numb er N of zeros in a small interval, say [ 1 10 ; 1 10 ], 18
by computing the quantity N = Z 1 10 1 10 ( x ) dx Z 1 10 1 10 w k b ( x ) dx; which gives us (approximately) the numb er of zeros on such an interval. The result of such a calculation for the Legendre L A 2 m ( x ) and Hermite H A 2 m ( x ) families is given in table 1. Table 1. The numb er of zeros in [ 1 10 ; 1 10 ] of the L A 2 m ( x ) and H A 2 m ( x ) p olynomials. 2 m L 2 m ( x ) L A 2 m ( x ) H 2 m ( x ) H A 2 m ( x ) 100000 12753.7 12753.8 40.2634 38.8898 200000 25507.5 25507.5 56.941 55.4299 300000 38261.2 38261.2 69.7382 68.047 400000 51014.9 51014.9 80.5268 78.6623 500000 63768.6 63768.7 90.0317 88.0054 600000 76522.3 76522.4 98.6247 96.4471 700000 89276. 89276.1 106.527 104.207 800000 102030. 102030. 113.882 111.428 900000 114783. 114784. 120.79 118.208 1000000 127537. 127537 127.324 124.62 ACKNOWLEDGEMENTS The research of the authors was supp orted by Direccion General de Investigacion Cientca y Tecnica (DGICYT) of Spain under grant PB 93-0228-C02-01. The authors thank to the referee for its useful suggestions and remarks and also for p ointing out references [27 ] and [33 ]. References [1] M. Abramowitz and I. Stegun Editors: Handb o ok of Mathematical Functions. Dover Publications, Inc., New York, 1972. [2] R. Alvarez-No darse and F. Marcellan: A generalization of the classical Laguerre polynomials. Rend. Circ. Mat. Palermo. Serie I I, 44 , (1995), 315-329. [3] R. Alvarez-No darse and F. Marcellan: A generalization of the classical Laguerre polynomials II. Zeros, Second order dierential equation and three-term recurrence relation. Appl. Analysis. (1996). (in press) 19
[4] R. Alvarez-No darse and F. Marcellan: The limit relations between generalized orthogonal polynomials. Indag. Math. (1997) (in press) [5] E.R. Arriola, A. Zarzo, and J.S. Dehesa: Spectral Properties of the biconuent Heun dierential equation. J. Comput. Appl. Math. 37 (1991), 161-169. [6] E. Buenda, S. J. Dehesa, and F. J. Galvez: The distribution of the zeros of the polynomial eigenfunctions of ordinary dierential operators of arbitrary order. In Orthogonal Polynomials and their Applications. M.Alfaro et al. Eds. Lecture Notes in Mathematics. Vol. 1329 , Springer-Verlag, Berlin, 1988, 222-235. [7] T. S. Chihara: Orthogonal polynomials and measures with end point masses. Ro cky Mount. J. of Math. 15 (3), (1985), 705-719. [8] T. S. Chihara: An intro duction to orthogonal p olynomials. Gordon and Breach. New York. 1978. [9] N. Dradi: Sur l'adjonction de deux masses de Dirac a une forme lineaire reguliere quelconque. These Do ctorat de l'Universite Pierre et Marie Curie. Paris, 1990. [10] N. Dradi and P. Maroni: Sur l'adjonction de deux masses de Dirac a une forme reguliere quelconque. In Polinomios ortogonales y sus aplicaciones. A. Cachafeiro and E. Go doy Eds. Actas del V Simp osium. Universidad de Santiago. Vigo 1988, 83-90. [11] A. Erdelyi, A. Magnus, F. Ob erhettinger and F. Tricomi: Higher Transcendental Functions. McGraw-Hill Bo ok Co. New York, Vol 2 , 1953. [12] J. Ko eko ek and R. Ko eko ek : On a dierential equation for Koornwinder's generalized Laguerre polynomials. Pro c. Amer. Math. So c. 112 , (1991), 1045-1054. [13] R. Ko eko ek and H. G. Meijer: A generalization of Laguerre polynomials. SIAM J. Math. Anal. 24 (3), (1993), 768-782. [14] R. Ko eko ek: Koornwinder's Laguerre Polynomials. Delft Progress Rep ort. 12 , (1988), 393-404. [15] R. Ko eko ek: Generalizations of the classical Laguerre Polynomials and some qanalogues. Thesis, Delft University of Technology. 1990. [16] R.Ko eko ek: Dierential equations for symmetric generalized ultraspherical polynomials. Transact. of the Amer. Math. So c. 345 , (1994), 47-72. [17] T. H. Ko ornwinder: Orthogonal polynomials with weight function (1 x ) (1 + x ) + M ( x + 1) + N ( x 1) . Canad. Math. Bull. 27 (2), (1984), 205-214. [18] A. M. Krall: Orthogonal Polynomials satisfying fourth order dierential equations. Pro c. Roy. So c. Edinburgh, 87 , (1981), 271-288. [19] H. L. Krall: On Orthogonal Polynomials satisfying a certain fourth order dierential equation. The Pennsylvania State College Bulletin, 6 , (1940), 1-24. [20] H. K. Kwon and S. B. Park: Two point masses perturbation of regular moment functionals. Indag. Math. (1997) (In press) 20
[21] F. Marcellan and P. Maroni: Sur l'adjonction d'une masse de Dirac a une forme reguliere et semi-classique. Ann. Mat. Pura ed Appl. IV , CLXI I, (1992), 1-22. [22] F. Marcellan and A. Ronveaux: Dierential equations for classical type orthogonal polynomials. Canad. Math. Bull. 32 , (4), (1989), 404-411. [23] P. Nevai: Orthogonal Polynomials. Memoirs of the Amer. Math. So c. 213 , Providence, Rho de Island, 1979. [24] A.F. Nikiforov and V. B. Uvarov: Sp ecial Functions of Mathematical Physics. Birkhauser Verlag, Basel, 1988. [25] E.D. Rainville: Sp ecial Functions. Chelsea Publishing Company, New York, 1971. [26] J. Riordan: An Intro duction to Combinatorial Analysis. Wiley, New York, 1958. [27] A. Ronveaux, A. Zarzo, and E. Go doy: Fourth-order dierential equations satised by the generalized co-recursive of al l classical orthogonal polynomials. A study of their distribution of zeros. J. Comput. Appl. Math. 59 (1995), 295-328. [28] G. Szego: Orthogonal Polynomials. Amer. Math. So c. Collo q. Publ., 23 , Amer. Math. So c., Providence, Rho de Island, 1975 (4th edition). [29] W. Van Assche: Asymptotics for Orthogonal Polynomials. Lecture Notes in Mathematics Vol. 1265 . Springer-Verlag, Berlin, 1987. [30] S.Wolfram: Mathematica. A system for doing Mathematics by Computer. Addison-Wesley Publishing Co., New York, 1991. [31] A.Zarzo and J.S. Dehesa: Spectral Properties of solutions of hypergeometric-type differential equations. J.Comput.Appl.Math. 50 (1994), 613-623. [32] A.Zarzo, J.S. Dehesa, and R.J. Ya ~nez: Distribution of zeros of Gauss and Kummer hypergeometric functions: A semiclassical approach. Annals Numer. Math. 2 (1995) 457-472. [33] A. Zarzo, A. Ronveaux, and E. Go doy: Fourth-order dierential equation satised by the associated of any order of al l classical orthogonal polynomials. A study of their distribution of zeros. J. Comput. Appl. Math. 49 (1993), 349-359. 21