Distribution of zeros of discrete and continuous polynomials from their recurrence relation

ZERO DISTRIBUTIONS OF DISCRETE AND CONTINUOUS POLYNOMIALS FROM THEIR RECURRENCE RELATION. R.  Alvarez-No darse 1 Instituto Carlos I de Fsica Teorica y Computacional Universidad de Granada. E-18071 Granada, Spain and Departamento de Analisis Matematico. Facultad de Matematicas. Universidad de Sevil la. c/ Tara, s/n, E-41012 Sevil la, Spain and Jes us S. Dehesa 2 Instituto Carlos I de Fsica Teorica y Computacional and Departamento de Fsica Moderna. Universidad de Granada. E-18071 Granada, Spain. Key words and phrases: orthogonal p olynomials, three-term recurrence relation, distribution of zeros, moments of zeros, sp ectral asymptotics. PACS subject classification: 02.30.Gp . AMS (MOS 1991) subject classification: 33C45, 42C05 Abstract The hyp ergeometric polynomials in a continous or a discrete variable, whose canonical forms are the so-called classical orthogonal p olynomial systems, are ob jects which naturally app ear in a broad range of physical and mathematical elds from quantum mechanics, the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of Sturm-Liouville dierential and dierence equations. Often, they are encountered in the form of a three term recurrence relation (TTRR) which connects a polynomial of a given order with the p olynomial of the contiguous orders. This relation can b e directly found, in particular, by use of Lanczos-typ e metho ds, tight-binding mo dels or the application of the conventional discretisation pro cedures to a given dierential op erator. Here the distribution of zeros and its asymptotic limit, characterized by means of its moments around the origin, are found for the continuous classical (Hermite, Laguerre, Jacobi, Bessel) p olynomials and for the discrete classical (Charlier, Meixner, Kravchuk, Hahn) p olynomials by means of a general pro cedure which (i) only requires the three-term recurrence relation and (ii) avoids the often high-brow subleties of the p otential theoretic considerations used in some recent approaches. The moments are given in an explicit manner which, at times, allows us to recognize the analytical form of the corresp onding distribution. 1 Intro duction. The hyp ergeometric p olynomials in a continous [10, 39] or a discrete [40 , 39] variable are objects not only interesting per se and b ecause of its abundant applications in many areas of mathematics ranging from angular momentum algebra and probability theory to numerical analysis and the theory of Sturm-Liouville dierential and dierence equations, the theory of random matrices and the study of sp eech signals, but also b ecause they help us to interpret and characterize numerous natural phenomena encountered, e.g. in the quantum mechanical description of physical and chemical systems, the theory of vibrating strings and the study of random walks with discrete time pro cesses, as p ointed out by several authors [10 , 2, 3, 4, 7, 18, 19 , 35, 40, 39, 42 , 43, 44]. 1 E-mail address [email protected]; Fax No. +34-958-242862 2 E-mail address [email protected]; Fax No. +34-958-242862 2 ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION The global b ehaviour of the zeros of the discrete and continuous classical orthogonal p olynomials in b oth nite and asymptotic cases has received a great deal of attention from the early times [22 , 27 , 45 ] of approximation theory up to now [8, 9, 20 , 23 , 24 , 31 , 32 , 33 , 34 , 36 , 37, 41, 42, 46, 47, 48, 50]. Indeed, numerous interesting results have b een found from the different characterizations (explicit expression, weight function, recurrence relation, second order dierence or dierential equation) of the p olynomial. See [16] for a survey of the published results up to 1977; more recent discoveries are collected in [49] and [31] for continuos and discrete p olynomials, resp ectively. Still now, however, there are op en problems which are very relevant by their own and b ecause of its numerous applications to a great variety of classical systems [29, 35] as well as quantum-mechanical systems whose wavefunctions are governed by orthogonal p olynomials in a \discrete" [2, 3, 40, 43, 44 ] or a \continuous" variable [5 , 18, 19 , 39]. In this pap er the attention will b e addressed to the problem of determination of the moments of the distribution density of zeros for a classical orthogonal p olynomial of a given order n in b oth discrete and continuous cases as well as its asymptotic values (i.e, when n ! 1 ), which fully characterize the limiting distribution of zeros of those p olynomials, in an explicit and exact manner. At times, the analytical form of the distribution asso ciated to the calculated moments is recognized. This problem is solved for a general system of p olynomials, dened by the recurrence relation given by (2.1) and (2.2) b elow, which includes all classical orthogonal families in the discrete (Hahn, Meixner, Kravchuk, Charlier) and continuous (Hermite, Laguerre, Jacobi, Bessel) cases. We have used a metho d [12 , 16, 17] which is based only on the three-term recurrence relation satised by the involved p olynomials. This metho d, which will b e describ ed in Section 2, is of general vality since no p eculiar constraints are imp osed up on the co ecients of the recurrence relation. It was found in a context of tridiagonal matrices [6, 13, 14 , 15] and it has b een already used for the study of the distribution of zeros of qp olynomials [1, 11, 16 ]. Some of the results found here have b een previously obtained by other means and are disp ersely published, what will b e mentioned in the appropiate place; they are included here for completeness, for illustrating the go o dness of our pro cedure or b ecause they are not accessible for the general reader [16]. Then, in Section 3, expresions for the moments of the discrete distribution of zeros of any discrete and continuous classical p olynomials of arbitrary, but xed, degree n are given in a closed and compact form; the explicit values for the rst few moments of lowest order are also shown. Finally, in Section 4, the limiting distribution of zeros of all classical p olynomials is describ ed by means of its moments and, at times, its analytical form is shown. 2 Density of zeros of a general p olynomial system from its recurrence relation. Basic to ols We will consider here a general system of p olynomials f P n g 1 n =0 dened by the following three-term recurrence relation P n ( x ) = ( x  a n ) P n  1 ( x )  b 2 n  1 P n  2 ( x ) P  1 ( x ) = 0 ; P 0 ( x ) = 1 ; n  1 (2.1) where the co ecients a n and b 2 n  1 are rational functions in n dened by ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 3 a n =  X i =0 c i n   i  X i =0 d i n   i  Q  ( n ) Q  ( n ) ; b 2 n =  X i =0 e i n   i  X i =0 f i n   i  Q  ( n ) Q  ( n ) : (2.2) The parameters dening a n and b 2 n are supp osed to b e real. In the case when the e i and f i are such that b 2 n > 0 for n  1, then, Favard's theorem [10 ] assures the orthogonality of the p olynomials f P n g 1 n =0 and we will say that the relation (2.1) denes a sequence of orthogonal p olynomials. Here we will collect the to ols which allow us to nd the moments of the distribution of zeros (Theorem 1), and its asymptotic values (Theorem 2), of the p olynomials which ob ey a three-term recurrence relation of the form (2.1). These results, previously found in a context of tridiagonal matrices [12, 16 , 17 ], constitute an alternative metho d to compute the prop erties of the sp ectral moments of the orthogonal p olynomials directly from the three-term recurrence co ecients ( a n ; b n ). They are used to obtain the distribution of zeros of the discrete and continuous classical orthogonal p olynomials for b oth nite and asymptotic cases in Section 3 and 4, resp ectively. Theorem 1 The sp ectral moments (Dehesa [12 , 16]) Let f P k ; k = 0 ; 1 ; :::; n; ::: g , be a system of polynomials dened by the recurrence relation (2.1), which is characterized by the sequences of numbers f a n g and f b n g . Let the quantities  0 = 1 ;  0 ( n ) m = 1 n Z b a x m  n ( x ) dx; m = 1 ; 2 ; :::; n (2.3) be the normalized-to-unity spectral moments of the polynomial P n , i.e., the moments around the origin of the discrete density of zeros  n dened by  n ( x ) = 1 n n X i =1  ( x  x n;i ) ; (2.4) f x n;i ; i = 1 ; 2 ; :::; n g being the zeros of that polynomial. It is full led that  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 a r 0 1 i b 2 r 1 i a r 0 2 i +1 b 2 r 2 i +1 : : : b 2 r j i + j  1 a r 0 j +1 i + j ; m = 1 ; 2 ; :::; n: (2.5) Or, in a compact form,  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 2 4 j +1 Y k =1 a r 0 k i + k  1 3 5 2 4 j Y k =1 ( b 2 i + k  1 ) r 0 k 3 5 ; m = 1 ; 2 ; :::; n ; (2.6) where s denotes the number of non-vanishing r i which are involved in each partition of m . The rst summation runs over al l partitions ( r 0 1 ; r 1 ; :::; r 0 j +1 ) of the number m such that 1. R 0 + 2 R = m , where R and R 0 denote the sums R = [ m 2 ] X i =1 r i and R 0 = [ m 2 ]  1 X i =1 r 0 i , or [ m 2 ]  1 X i =1 r 0 i + 2 [ m 2 ] X i =1 r i = m : (2.7) 4 ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 2. If r i = 0 ; 1 < i < [ m 2 ] , then r k = r 0 k = 0 for each k > i and 3. [ m 2 ] = m 2 or [ m 2 ] = m  1 2 for m even or odd, respectively. The factorial coecient F is dened by F ( r 0 1 ; r 1 ; r 0 2 ; :::; r 0 p  1 ; r p  1 ; r 0 p ) = = m ( r 0 1 + r 1  1)! r 0 1 ! r 1 ! 2 4 p  1 Y i =2 ( r i  1 + r 0 i + r i  1)! ( r i  1  1)! r i ! r 0 i ! 3 5 ( r p  1 + r 0 p  1)! ( r p  1  1)! r 0 p ! ; (2.8) with the convention r 0 = r p = 1 . For the evaluation of these coecients, we must take into account the fol lowing convention F ( r 0 1 ; r 1 ; r 0 2 ; r 2 :::; r 0 p  1 ; 0 ; 0) = F ( r 0 1 ; r 1 ; r 0 2 ; r 2 :::; r 0 p  1 ) : This theorem was initially found in a context of Jacobi matrices [12, 16 ]. A straightforward calculation gives  0 1 = 1 n " n X i =1 a i # ;  0 2 = 1 n " n X i =1 a 2 i + 2 n  1 X i =1 b 2 i # ;  0 3 = 1 n " n X i =1 a 3 i + 3 n  1 X i =1 b 2 i ( a i + a i +1 ) # ;  0 4 = 1 n " n X i =1 a 4 i + 4 n  1 X i =1 b 2 i ( a 2 i + a i a i +1 + a 2 i +1 + 1 2 b 2 i ) + 4 n  2 X i =1 b 2 i b 2 i +1 # ; (2.9) for the the rst four sp ectral moments. Recently, it has b een shown [25, 26 , 34] that the moments given by Eq. (2.6) may b e represented as the so-called Lucas p olynomials of the rst kind in several variables, each dep ending on the recurrence co ecients ( a n ; b n ) in a certain manner. Theorem 2 The asymptotic values for the moments (Dehesa [16, 17]) Let f P k ; k = 0 ; 1 ; :::; n; ::: g be a system of polynomials dened by the recurrence relation (2.1), which is characterized by the sequences of numbers f a n g and f b n g . Let  ,   and   the asymptotic zero distribution functions of the polynomial P n dened as fol lows  ( x ) = lim n !1  n ( x ) ;   ( x ) = lim n !1  n  x n 1 2 (    )  ;   ( x ) = lim n !1  n  x n (    )  : (2.10) Here,  n is given by Eq. (2.4), and the moments of the functions  ,   , and   are  0 m = lim n !1  0 ( n ) m ;  00 m = lim n !1  0 ( n ) m n m 2 (    ) ;  000 m = lim n !1  0 ( n ) m n m (    ) : (2.11) Then, according to the dierent behaviour of the asymptotic zero distribution, the polynomial system f P k g 1 k =0 may be subdivided in the seven fol lowing classes ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 5 1. Class  <  and  <  . The polynomials belonging to this class has a spectrum of zeros characterized by the quantities  0 0 = 1 ;  0 m = 0 ; m = 1 ; 2 ; ::: 2. Class  <  and  =  . The polynomials in this class are such that 8 > > > < > > > :  0 2 m =  e 0 f 0  m 2 m m ! ;  0 2 m +1 = 0 ; m = 0 ; 1 ; 2 ; ::: 3. Class    and  >  . The polynomials in this class are such that 8 > > > < > > > :  00 2 m = 1 m (    ) + 1  e 0 f 0  m 2 m m ! ;  00 2 m +1 = 0 ; m = 0 ; 1 ; 2 ; ::: 4. Class  =  and  <  . The polynomials in this class are such that  0 m =  c 0 d 0  m ; m = 0 ; 1 ; 2 ; ::: 5. Class  =  and  =  . The polynomials in this class are such that  0 m = [ m 2 ] X i =0  c 0 d 0  m  2 i  e 0 f 0  i 2 i i ! m 2 i ! ; m = 0 ; 1 ; 2 ; ::: 6. Class  >  and    . The polynomials in this class are such that  000 m = 1 m (    ) + 1  c 0 d 0  m ; m = 0 ; 1 ; 2 ; ::: 7. Class  >  and  >  . Here three cases may be distinguised: (a) Case    > 1 2 (    ) . The polynomials in this subclass are such that (see case 6)  000 m = 1 m (    ) + 1  c 0 d 0  m ; m = 0 ; 1 ; 2 ; ::: (b) Case    = 1 2 (    ) . The polynomials in this subclass are such that  000 m = 1 m (    ) + 1 [ m 2 ] X i =0  c 0 d 0  m  2 i  e 0 f 0  i 2 i i ! m 2 i ! ; m = 0 ; 1 ; 2 ; ::: (c) Case    < 1 2 (    ) . The polynomials in this subclass are such that (see case 3) 8 > > > < > > > :  00 2 m = 1 m (    ) + 1  e 0 f 0  m 2 m m ! ;  00 2 m +1 = 0 ; m = 0 ; 1 ; 2 ; ::: 6 ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 3 The sp ectral moments of the classical p olynomials. 3.1 Classical discrete p olynomials. Let us compute the moments around the origin of the distribution of zeros of a p olynomial of a given degree, which b elong to one of the four classical families (Charlier, Meixner, Kravchuk and Hahn) of orthogonal p olynomials in a discrete variable. They are given in terms of the parameters which characterize the three-term recurrence relation of the corresp onding family. Alternative expressions for these quantities may b e obtained from the explicit expressions of the p olynomial [49 ]. 3.1.1 Charlier Polynomials. The Charlier p olynomials c  n ( x ) satisfy a three-term recurrence relation (2.1) with the co e- cients [40] a n = n +   1 ; b 2 n = n: (3.1) Then, Theorem 1 leads to the expression  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 j +1 Y k =1 [ i + k  2 +  ] r 0 k j Y k =1 [( i + k  1)  ] r k ; for the sp ectral moments of the p olynomial c  n ( x ). The rst four moments are  0 ( n ) 1 = n + 2   1 2 ;  0 ( n ) 2 = ( n  1) ( 2 n  1) 6 + 2 ( n  1)  +  2 ;  0 ( n ) 3 = ( n  1) 2 n 4 + 3( n  1) 2  + 9 ( n  1)  2 2 +  3 ;  0 ( n ) 4 = n 2 (10 + 3 n (2 n  5))  1 30 + 4( n  1) 3  + 2 ( n  1) (6 n  7)  2 + 8 ( n  1)  3 +  4 : 3.1.2 Meixner Polynomials. The Meixner p olynomials m  ; n ( x ) are dened by the three-term recurrence relation (2.1) with co ecients [40] a n = ( n  1)(1 +  ) +  1   ; b 2 n = n ( n  1 +  ) (1   ) 2 : (3.2) Application of Theorem 1 gives the value  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 j +1 Y k =1  ( i + k  2)(1 +  ) +  1    r 0 k   j Y k =1  ( i + k  1)  ( i + k  2 +  ) (1   ) 2  r k ; for the m -th order sp ectral moment of the Meixner p olynomial of n -th degree. A simple calculation gives  0 ( n ) 1 = 1 +   2    n (1 +  ) 2 (   1) ; ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 7  0 ( n ) 2 =  1  3 n + 2 n 2 +  2  1 + 6 (   1)   3 n + 6  n + 2 n 2  + 2  ( n  1) (6  + 4 n  5)  6 (   1) 2 ;  0 ( n ) 3 = 1 4 (1   ) 3 h  2  2  2 + 2   (1 +  ) ( n  1) + (1 +  ) 2 ( n  1) n  ( n  1 +  (2  + n  1)) + +2  ( n  1)  6  2  + 3 (1 +  ) ( n  2) ( n  1) +  (4 n  5 +  (8 n  13))  i ; for the three sp ectral moments of lowest order. 3.1.3 Kravchuk Polynomials. The Kravchuk p olynomials k n ( x; p; N ) satisfy a three-term recurrence relation of the typ e (2.1) with co ecients [40 ] a n = N p + (1  2 p )( n  1) ; b 2 n = np (1  p )( N  n + 1) : (3.3) In this case, Theorem 1 leads to  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 j +1 Y k =1 [ N p + (1  2 p )( i + k  2)] r 0 k   j Y k =1 [( i + k  1) p (1  p )( N  i  k + 2)] r k ; (3.4) for the m -th order sp ectral moment of the n -th degree Kravchuk p olynomial. From this general expression, it is straightforward to nd the values  0 ( n ) 1 =  1 2 + n  1 2  p  + p + N p;  0 ( n ) 2 = ( n  1) (2 n  1) 6 + 2 ( n  1) ( N  n + 1) p + ( N  n + 1) ( N  2 n + 2) p 2 ;  0 ( n ) 3 = 1 4  ( n  1) 2 n + 12( n  1) 2 ( N  n + 1) p  6 ( n  1) ( N  n + 1) (5 n  3 ( N + 2)) p 2   4 ( N  n + 1)  6 + n (5 n  11) + 5 N  5 nN + N 2  p 3  ;  0 ( n ) 4 = 1 30  n 2 (10 + 3 n (  5 + 2 n ))  1  + 4( n  1) 3 ( N  n + 1) p + +2 ( n  1) ( N  n + 1) ( n (6 N  9 n + 22) + 7 (2 + N )) p 2   4 ( n  1) ( N  n + 1)  7 n 2 + 2 (2 + N ) (3 + N )  2 n (9 + 4 N )  p 3 + + ( N  n + 1)  14 n 3  (2 + N ) (3 + N ) (4 + N ) + n (3 + N ) (20 + 9 N )  n 2 (50 + 21 N )  p 4 ; for the four moments of lowest order of the distribution of zeros of the p olynomials k n ( x; p; N ). 3.1.4 Hahn Polynomials. The Hahn p olynomials h ; n ( x; N ) satisfy a three-term recurrence relation of the form (2.1) with co ecients [40] a n = (  + 1)( N  1)(  +  ) + ( n  1)(2 N +     2)(  +  + n ) (  +  + 2 n )(  +  + 2 n  2) ; b 2 n = n ( N  n )(  +  + n )(  + n )(  + n )(  +  + N + n ) (  +  + 2 n  1)(  +  + 2 n ) 2 (  +  + 2 n + 1) : (3.5) 8 ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION They constitute a nite family of orthogonal p olynomials, dened for the degrees n < N ( N is the numb er of p oints in the discrete set). Applying Theorem 1, one obtains  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1   j +1 Y k =1  (  + 1)( N  1)(  +  )+( i + k  2)(2 N +     3)(  +  + i + k  1) (  +  + 2 i + 2 k  4)(  +  + 2 i + 2 k  2)  r 0 k   j Y k =1  ( i + k  1)( N  i  k + 1)(  +  + i + k  1)(  + i + k  1)(  + i + k  1)(  +  + N + i + k  1) (  +  + 2 i + 2 k + 1)(  +  + 2 i + 2 k  2) 2 (  +  + 2 i + 2 k  3)  r k ; for the m -th order sp ectral moment of the Hahn p olynomial of degree n . This general expression for m = 1 ; 2 reduces to  0 ( n ) 1 =   + n (  2 + 2 N +    ) + (  1 + 2 N )  2 (2 n +  +  ) ;  0 ( n ) 2 = 1 6 (  1 + 2 n +  +  ) (2 n +  +  ) 2   2 n 5 + n 4 (4  6   6  )  2 n 3 (  4 +   3 N (3 N + 3  4  )  11  + 3( N + 3  )  )+ +(  +  )(  2 + (1 + 6( N  1) N )(   1)  +  (2   12 N  1  ))+ +2 n 2 (  2 +  (9 + (   4)  ) + 3    (  10 + 3  )   (3   8)  2 + +  3 + 6 N 2 (  + 3   1) + 6 N (1  3  +  2 + 2(   2)    2 ))+ + n (  3  3  4  + 3  2 (3 + 4 N (   1) +  ) + 3  (6(1  N ) N +  + +2 N (4 N  5)  + (1  2 N )  2 )   (4 + 6( N  1) N + 6(2 + 3( N  3) N )  + +3(3 + 2 N )  2 ))  A very imp ortant sp ecial sub class of the Hahn p olynomials is when  =  = 0; it is the Chebyshev system of discrete p olynomials t n ( x; N ). In this case the recurrence co ecients (2.2) b ecome a n = N  1 2 and b 2 n = n 2 ( N 2  n 2 ) 4(4 n 2  1) , and the sp ectral moments have the simpler expression  0 ( n ) m = 1 n X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) n  s X i =1 j +1 Y k =1  ( N  1) 2  r 0 k j Y k =1 " ( i + k  1) 2 [ N 2  ( i + k  1) 2 ] 4[4(2 i + 2 k  1) 2  1] # r k ; and  0 ( n ) 1 = N  1 2 ;  0 ( n ) 2 = 2 n 2  n 3 + n (3 N  2) 2  6 ( N  1) N  2 24 n  12 ;  0 ( n ) 3 = ( N  1)  (2  n ) n 2 + (2  4 n ) N + (5 n  4) N 2  16 n  8 ;  0 ( n ) 4 = 1 240( 2 n  1) 2 (2 n  3)  24  112 n + 280 n 2  414 n 3 + 206 n 4 + 52 n 5  60 n 6 + +9 n 7  360 n 2 N + 1140 n 3 N  960 n 4 N + 240 n 5 N  360 N 2 + 1470 nN 2   1590 n 2 N 2  90 n 3 N 2 + 630 n 4 N 2  150 n 5 N 2 + 720 N 3  2820 nN 3 + 3360 n 2 N 3   1200 n 3 N 3  360 N 4 + 1350 nN 4  1530 n 2 N 4 + 525 n 3 N 4  ; ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION 9 for m = 1 ; 2 ; 3 ; 4. For convenience let us also consider the rescaled p olynomials T n ( x; N )   N  1 2   n t n  N  1 2 ( x + 1) ; N  ; (3.6) which form an orthogonal system with resp ect to the discrete set of the p oints  x k =  1 + 2 k  1 N  1 ; k = 0 ; 1 ; 2 ; :::   [  1 ; 1] : They satisfy a recurrence relation of the form (2.1) with co ecients a n = 0 ; b 2 n = n 2 ( N 2  n 2 ) ( N  1) 2 (4 n 2  1) : (3.7) Then, the moments of its distribution of zeros are given by  0 ( n ) m = 8 > > > > < > > > > : 2 n k X p =1 0 @ X ( m ) F (0 ; r 1 ; 0 ; r 2 ; :::; 0 ; r p ) 1 A n  p X i =1 p Y k =1 " ( i + k  1) 2 [ N 2  ( i + k  1) 2 ] ( N  1) 2 [4( i + k  1) 2  1] # r k ; m = 2 k 0 ; m = 2 k  1 ; so that the rts few non-vanishing moments are given by  0 ( n ) 2 = ( n  1)  3 N 2  n 2 + n  1  3 (2 n  1) ( N  1) 2 ;  0 ( n ) 4 = 1 15 (2 n  3) (2 n  1) 2 ( N  1) 4 h 45 n 3 N 4 + 98 n  200 n 2 + 276 n 3  274 n 4 + 172 n 5  60 n 6 + 9 n 7 + +90 N 2  360 nN 2 + 510 n 2 N 2  360 n 3 N 2 + 150 n 4 N 2  30 n 5 N 2  45 N 4 + 150 nN 4  150 n 2 N 4  21 i : 3.2 Classical continuous p olynomials. Let us now consider the classical orthogonal p olynomials in the a continuous variable: Hermite, Laguerre, Jacobi and Bessel. Here we compute the sp ectral moments of a p olynomial of a given degree b elonging to one of these classical continuous families. Let us mention that the rst few moments of lowest order were previously obtained by use of a general highly-non-linear recurrence relationship generated from the second-order dierential equation satised by the p olynomials under consideration [8, 9, 23 ]. As well, they can b e also found by means of the explicit expression of the p olynomials [34, 49]. 3.2.1 Hermite Polynomials. The Hermite p olynomials H n ( x ) are dened by the three-term recurrence relation (2.1) with co ecients [39] a n = 0 ; b 2 n = n 2 : (3.8) 16 ZERO DISTRIBUTIONS OF POLYNOMIALS FROM THEIR RECURRENCE RELATION as already found in the literature by other means [7 , 16, 36 , 24, 47 ]. 4.2.2 Laguerre Polynomials. For this case, the recurrence co ecients given by (3.9) are of the form (2.2) with parameters  = 1,  = 0,  = 2 and  = 0, as well as ( c 0 ; d 0 ) = (2 ; 1) and ( e 0 ; f 0 ) = (1 ; 1). Then, the Laguerre p olynomials L  n ( x ) b elong to the class 7b describ ed in Theorem 2; so that, its asymptotical distribution of zeros has the moments  000 m = 1 m + 1 [ m 2 ] X i =0 2 m  2 i 2 i i ! m 2 i ! = 1 m + 1 2 m m ! ; m = 0 ; 1 ; 2 ; ::: ; (4.10) which characterize another sp ecial case of Beta distribution [30 , Vol. 2, p. 210]. This result has b een previously obtained in the literature [16, 36]. This indicates that the contracted density of zeros of the Laguerre p olynomials with large degree n is given by   x n  = 1 2   x n   1 2  4  x n  1 2 ; 0  x n  4 : Similar analytical expressions, which coincide with this one for very large values of n , have b een derived in the WKB framework [50 ], by use of random matrix metho ds [7] and also in [24]. 4.2.3 Jacobi Polynomials. From (3.10) one notices that the recurrence co ecients of these p olinomials b ehave as a n =  2   2 4 n 4 + O ( n ) ; b 2 n = 4 n 4 + O ( n 3 ) 16 n 4 + O ( n 3 ) : These expressions are of the form (2.2) with parameters  = 0,  = 2 and  =  = 4, as well as ( e 0 ; f 0 ) = (4 ; 16). Then, Jacobi p olynomials b elong to class 2 as decrib ed in Theorem 2; so that, the moments of the asymptotic density of zeros are 8 > > > < > > > :  0 2 m =  1 2  2 m 2 m m ! ;  0 2 m +1 = 0 ; m = 0 ; 1 ; 2 ; ::: (4.11) This corresp onds to the so-called arc-sin density [22 ]  ( x ) = 1  p 1  x 2 ;  1  x  1 : (4.12) 4.2.4 Bessel p olynomials. From (3.11) one notices that the recurrence co ecients of these p olinomials b ehave as a n =  2  4 n 4 + O ( n ) ; b 2 n =  4 n 2 + O ( n ) 16 n 4 + O ( n 3 ) : These expressions are of the form (2.2) with parameters  = 0,  = 2,  = 2 and  = 4. Then, Bessel p olynomials b elong to class 1 as decrib ed in Theorem 2; so that, the moments of the asymptotic density of zeros are (  0 0 = 1 ;  0 m = 0 ; m = 1 ; 2 ; ::: (4.13) which corresp ond to a delta-Dirac density [23]. 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