The distribution of zeros of general q-polynomials

THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS. 1 R.  Alvarez-No darse 2 Departamento de Matematicas. Escuela Politecnica Superior. Universidad Carlos III de Madrid. c/ Butarque 15, 28911, Leganes, Madrid, Spain and Instituto Carlos I de Fsica Teorica y Computacional Universidad de Granada. E-18071 Granada, Spain. E. Buenda Departamento de Fsica Moderna. Facultad de Ciencias. Universidad de Granada. E-18075 Granada, Spain. and J.S. Dehesa 3 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: q-orthogonal p olynomials, three-term recurrence relation, distribution of zeros, moments. PACS subject classification: 02.20.+b ; 02.30.+g ; 03.65.Fd . AMS (MOS 1991) subject classification 33D45, 33E30 Abstract A general system of q-orthogonal p olynomials is dened by means of its three-term recurrence relation. This system encompasses many of the known families of q-p olynomials, among them the q-analog of the classical orthogonal p olynomials. The asymptotic density of zeros of the system is shown to b e a simple and compact expression of the parameters which characterize the asymptotic b ehavior of the co ecients of the recurrence relation. This result is applied to sp ecic classes of p olynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey & Wilson, Al Salam-Carlitz and the celebrated q-little and q-big Jacobi. 1 Intro duction. In the last decade an increasing interest on the so called q-orthogonal p olynomials (or basic orthogonal p olynomials) is observed ( for a review see [1 ], [2] and [3]). The reason is not only of purely intrinsic nature but also b ecause of the so many applications in several areas of Mathematics ( e.g., continued fractions, eulerian series, theta functions, elliptic functions,...; see for instance [4] and [5 ]) and Physics ( e.g., angular momentum [6 ] and [7 ] and its q-analog [8]-[11 ], q-Shrodinger equation [12] and q-harmonic oscillators [13]-[19 ]). Moreover, it is well known the connection b etween the representation theory of quantum algebras (Clebsch-Gordan co ecients, 3j and 6j symb ols) and the q-orthogonal p olynomials, (see [20 ], [21] (Vol. I I I), [22 ], [23 ], [24] ), and the imp ortant role that these q-algebras play in physical applications (see for instance [26 ]-[31 ] and references therein). However, the distribution of zeros of these p olynomials remains practically unknown to the b est of our information. The present pap er continues, corrects and considerably extends the investigation of the asymptotic b ehavior of zeros of the q-p olynomials initiated by one of us [32 ]. This is done by the consideration of a general system of q-p olynomials which include most of the 1 This pap er app ear in the J. Phys. A: Math. Gen. 30 (1997) 6743-6768 2 E-mail address [email protected]; Fax No. +34-1-6249430 3 E-mail address [email protected]; Fax No. +34-58-242862 THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 2 q-p olynomials encountered in the literature and then, study of its distribution density of zeros as well as the corresp onding asymptotic limit. The metho d of pro of used is very straightforward. It is based on an explicit formula for the moments-around-the-origin of the discrete density of zeros of a p olynomial with a given degree in terms of the co ecients of the three term recurrence relation [37], as describ ed in Lemma 1 given b elow. This metho d was previously employed to normal (non-q) p olynomials where recurrence co ecients are given by means of a rational function of the degree [38], as well as to corresp onding Jacobi matrices [39] encountered in quantum mechanical description of some physical systems. The pap er is structured as follows. Firstly, in section 2, one intro duces a general set of qp olynomials f P n ( x ) q g N n =0 by means of its three-term recurrence relation. Section 3 contains the main results which refer to the discrete density of zeros (i.e. the numb er of zeros p er unit of zero interval) of the p olynomial P n ( x ) q , n b eing a suciently large value, and to its asymptotical limit (i.e., when n ! 1 ). Both discrete and asymptotic densities of zeros are supp osed to b e characterized by the knowledge of all their moments. These results are given in the form of four theorems. Theorem 1 gives the b ehavior of the moments of the discrete density of zeros in terms of the parameters dening the recurrence relation. The asymptotic density of zeros is given by Theorems 2, 3 and 4 in a similar way. Pro ofs and detailed discussion of these theorems are contained in Sections 4 and 5 resp ectively. The utmost eort has b een concentrated on searching for an appropiate asymptotic density of zeros to obtain as much information as p ossible ab out the asymptotic distribution of zeros of the new p olynomials. Finally, Section 6 contains application of theorems 1, 2, 3 and 4 formulated in section 3 to several known families of q-p olynomials. 2 The general system of q-orthogonal p olynomials. The general system of q-orthogonal p olynomials f P n ( x ) q g N n =0 is dened by the 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 (1) with the co ecients a n and b 2 n  1 given by a n = A X m =0 g m X i =0  ( m ) i n g m  i ! q d m n A 0 X m =0 0 @ h m X i =0  ( m ) i n h m  i 1 A q e m n  a num n a den n b 2 n = B X m =0 0 @ k m X i =0  ( m ) i n k m  i 1 A q f m n B 0 X m =0 0 @ l m X i =0  ( m ) i n l m  i 1 A q s m n  ( b num n ) 2 ( b den n ) 2 (2) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 3 where q is an arbitrary p ositive real numb er bigger than 1. Further, the following general requeriments on the real parameters dening a n and b 2 n will b e assumed: 1. All memb ers of the sequence f  ( m ) i ; 0  i  h m g A 0 m =0 , f  ( m ) i ; 0  i  l m g B 0 m =0 do not vanish simultaneously. So we assure a n and b 2 n not to b e innite for all n . 2. The parameters f  ( m ) i ; 0  i  k m g B m =0 and f  ( m ) i ; 0  i  l m g B 0 m =0 are such that b 2 n > 0 for n  1. Then Favard's theorem assures the orthogonality of the p olynomials f P n ( x ) q g N n =0 . 3. The following inequalities are veried: q d 0 > q d 1 > : : : > q d A ; q e 0 > q e 1 > : : : > q e A 0 q f 0 > q f 1 > : : : > q f B ; q s 0 > q s 1 > : : : > q s B 0 (3) and g 0 > g 1 > : : : > g m ; h 0 > h 1 > : : : > h m k 0 > k 1 > : : : > k m ; l 0 > l 1 > : : : > l m (4) Conditions (3) and (4) do not obviously imply any loss of generality. Here one should also p oint out that the p olynomials discussed in reference [32] are instances of the p olynomials (1)-(2) corresp onding to the values g m = k m = h m = e m = l m = s m = 0 for all m . 3 Main Results. Before collecting the main results of this work, let us describ e Lemma 1 which is the basic to ol to nd them. Lemma 1 Let f P N ( x ) g be a system of polynomials P N ( x ) dened by the recurrence relation (1), which is characterized by the secuences of numbers f a n g and f b n g . Let the quantities  0 = N ;  0 ( N ) m = Z b a x m  N ( x ) dx; m = 1 ; 2 ; :::; N (5) be the non-normalized-to-unity spectral moments of the polynomials P N ( x ) , i.e., the moments around the origin of the discrete density of zeros  N ( x ) , dened by  N ( x ) = N X i =1  ( x  x N ;i ) ; (6) f x N ;i ; i = 1 ; 2 ; :::; N g being the zeros of that polynomial. It is full led that  0 ( N ) m = X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) N  t 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 ; (7) for m = 1 ; 2 ; :::; N . The sumation X ( m ) 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 = j X i =1 r i and by R 0 = j  1 X i =1 r 0 i , or j  1 X i =1 r 0 i + 2 j X i =1 r i = m (8) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 4 2. If r s = 0 ; 1 < s < j , then r k = r 0 k = 0 for each k > s and 3. j = m 2 or j = m  1 2 for m even or odd respectively. The factorial coecient F are 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 ! ; (9) with the convention r 0 = r p = 1 . For the evaluation of these coecients, we must take into account the fol lowing convenction 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 ) In (7), t denotes the number of non-vanishing r i which are involved in each partition of m . This Lemma was initially found in a context of Jacobi matrices [37 ]-[38 ]. Just to understand the practical use of the Lemma, let us give the rst three sp ectral moments  0 1 = N X i =1 a i ;  0 2 = N X i =1 a 2 i + 2 N  1 X i =1 b 2 i ;  0 3 = N X i =1 a 3 i + 3 N  1 X i =1 b 2 i ( a i + a i +1 ) : (10) In the following, the main results of this work are collected in the form of four theorems. The rst of them refers to the discrete density of zeros (6) of the p olynomials dened by (1)-(2) and the other three concern with the asymptotic density of zeros, i.e., when the degree of the p olynomial tends towards innity. Throughout the pap er the symb ol  means behaves as . Theorem 1 Let P N ( x ) q , very large N , be a polynomial dened by the expressions (1)-(4). The moments f  0 ( N ) m ; m = 1 ; 2 ; :::; N g of the non-normalized density of zeros  N ( x ) = P N i =1  ( x  x N ;i ) of the polynomial P N ( x ) q have the fol lowing behavior 1. If d 0  e 0 = 1 2 ( f 0  s 0 ) = 0 , three cases occur: (a) If g 0  h 0 > 1 2 ( k 0  l 0 ) . Then  0 ( N ) m  "  (0) 0  (0) 0 # m N ( g 0  h 0 ) m +1 : (11) (b) If g 0  h 0 = 1 2 ( k 0  l 0 ) . Then  0 ( N ) m  X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R N 1 2 ( k 0  l 0 ) m +1 : (12) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 5 (c) If g 0  h 0 < 1 2 ( k 0  l 0 ) . Then  0 ( N ) m  "  (0) 0  (0) 0 # m 2 N 1 2 ( k 0  l 0 ) m +1 : (13) 2. If d 0  e 0 6 = 0 and/or f 0  s 0 6 = 0 , two cases occur: (a) i. If d 0  e 0 < 0 and f 0  s 0 < 0 in such a way that  1 6 = 0 . Then  0 ( N ) m  X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) q   2 ( l n q ) M "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R d M d  M 1 q  1 1  q  1 ! ; (14) where d M d  M 1 denotes the M derivative with resp ect to  1 . ii. If d 0  e 0 = 0 and f 0  s 0 < 0 and g 0  h 0 = k 0  l 0 = 0 . Then  0 ( N ) m  X ( m ) F ( r 0 1 ; 0 ; :::; 0 ; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 N : (15) iii. If d 0  e 0 < 0 and f 0  s 0 = 0 and g 0  h 0 = k 0  l 0 = 0 . Then  0 ( N ) m  X ( m ) F (0 ; r 1 ; :::; r j ; 0) "  (0) 0  (0) 0 # R N : (16) (b) If d 0  e 0 > 0 and/or f 0  s 0 > 0 , three dierent subcases may occur, namely: i. d 0  e 0 > 1 2 ( f 0  s 0 ) . Then  0 ( N ) m  "  (0) 0  (0) 0 # m q m ( N +1)( d 0  e 0 ) q m ( d 0  e 0 )  1 N ( g 0  h 0 ) m : (17) ii. If d 0  e 0 = 1 2 ( f 0  s 0 ) . Then three diferent types stil l come up: A. If g 0  h 0 > 1 2 ( k 0  l 0 ) , then  0 ( N ) m  "  (0) 0  (0) 0 # m q m ( N +1)( d 0  e 0 ) q m ( d 0  e 0 )  1 N ( g 0  h 0 ) m : (18) B. If g 0  h 0 = 1 2 ( k 0  l 0 ) , then  0 ( N ) m  X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R   q  2 + m ( N +1  t )( d 0  e 0 ) q m ( d 0  e 0 )  1 N m ( g 0  h 0 ) : (19) C. If g 0  h 0 < 1 2 ( k 0  l 0 ) , then  0 ( N ) m  "  (0) 0  (0) 0 # m 2 q ( d 0  e 0 ) mN q ( d 0  e 0 ) m  1 N 1 2 ( k 0  l 0 ) m : (20) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 6 iii. d 0  e 0 < 1 2 ( f 0  s 0 ) . Then  0 ( N ) m  "  (0) 0  (0) 0 # m 2 q 1 2 ( f 0  s 0 ) mN q 1 2 ( f 0  s 0 ) m  1 N 1 2 ( f 0  s 0 ) m : (21) The sumation X ( m ) and the parameter t are as dened in Lemma 1. Besides, the parameters  1 ,  2 and M are as fol lows:  1 = [( d 0  e 0 )  1 2 ( f 0  s 0 )] R 0 + m 2 ( f 0  s 0 ) (22)  2 = ( d 0  e 0 ) j X k =1 k r 0 k +1 + 2( f 0  s 0 ) j  1 X k =1 k r k +1 (23) M = [( g 0  h 0 )  1 2 ( k 0  l 0 )] R 0 + m 2 ( k 0  l 0 ) (24) The pro of of this theorem is shown in Section 4. Theorem 2 Let P N ( x ) q be a polynomial dened as in Theorem 1 with the aditional condition ( d 0  e 0 ) = 1 2 ( f 0  s 0 ) = 0 . (i.e. case 1) Let  ( x ) ,   1 ( x ) and   2 ( x ) be the asymptotic (i.e. when N ! 1 ) densities of zeros of the polynomial P N ( x ) q dened by  ( x ) = lim N !1  N ( x ) ;   1 ( x ) = lim N !1 1 N  N  x N ( g 0  h 0 )  ;   2 ( x ) = lim N !1 1 N  N  x N 1 2 ( k 0  l 0 )  (25) and their corresponding moments are as fol lows:  0 m = lim N !1  0 ( N ) m ;   m (1) = lim N !1  0 ( N ) m N ( g 0  h 0 ) m ;   m (2) = lim N !1  0 ( N ) m N ( k 0  l 0 ) m 2 (26) for m = 0 ; 1 ; 2 ; ::: respectively. Here  N ( x ) denotes the (discrete) density of zeros of the polynomial P N ( x ) q . It turns out that  0 m = 1 ; m  0 (27) and 1. If g 0  h 0 > 1 2 ( k 0  l 0 ) . Then   m (1) = "  (0) 0  (0) 0 # m ; m  0 (28) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 7 2. If g 0  h 0 = 1 2 ( k 0  l 0 ) . Then   m (2) = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R ; m  0 (29) 3. If g 0  h 0 < 1 2 ( k 0  l 0 ) . Then   m (2) = "  (0) 0  (0) 0 # m 2 ; m  0 : (30) Here the coecients F and the symbol of summation X ( m ) are as in Theorem 1. Theorem 3 Let P N ( x ) q be a polynomial dened as in Theorem 1 with the aditional condition ( d 0  e 0 )  0 and 1 2 ( f 0  s 0 )  0 . (i.e. subcase 2a) Let  ( x ) and  1 ( x ) be the asymptotic densities of zeros of the polynomial P N ( x ) q dened by  ( x ) = lim N !1  N ( x );  1 ( x ) = lim N !1 1 N  N ( x ) (31) and their corresponding moments are as fol lows:  0 m = lim N !1  0 ( N ) m ;  0 m (1) = lim N !1  0 ( N ) m N (32) for m  0 , respectively. It turns out that: 1. If d 0  e 0 < 0 and f 0  s 0 < 0 in such a way that  1 6 = 0 . Then  0 m = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) q   2 ( l n q ) M "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R d M d  M 1 q  1 1  q  1 ! ; (33) and  0 0 (1) = 1 ;  0 m (1) = 0 ; m  1 : (34) 2. If d 0  e 0 = 0 and f 0  s 0 < 0 and g 0  h 0 = k 0  l 0 = 0 . Then  0 m = 1 ; m  0 (35)  0 m (1) = 8 > > > > > < > > > > > : 1 m = 0 X ( m ) F ( r 0 1 ; 0 ; :::; 0 ; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 m  1 : (36) 3. If d 0  e 0 < 0 and f 0  s 0 = 0 and g 0  h 0 = k 0  l 0 = 0 . Then  0 m = 1 ; m  0 (37)  0 m (1) = 8 > > > > < > > > > : 1 m = 0 X ( m ) F (0 ; r 1 ; 0 ; :::; r j ; 0) "  (0) 0  (0) 0 # R m  1 : (38) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 8 Here the coecients F and the symbol of summation X ( m ) and the parameters  1 ,  2 and M are as in Theorem 1. Theorem 4 Let P N ( x ) q be a polynomial dened as in Theorem 1 with the aditional condition ( d 0  e 0 ) > 0 and/or 1 2 ( f 0  s 0 ) > 0 . (i.e. subcase 2b) Let  ( x ) ,   1 ( x ) ,   2 ( x ) ,   3 ( x ) ,  ++ 1 ( x ) ,  ++ 2 ( x ) and  ++ 3 ( x ) be the asymptotic densities of zeros of the polynomial P N ( x ) q given by  ( x ) = lim N !1  N ( x ) ; (39)   1 ( x ) = lim N !1  N xq  ( d 0  e 0 ) N N ( g 0  h 0 ) ! ;   2 ( x ) = lim N !1  N xq  ( d 0  e 0 ) N N 1 2 ( k 0  l 0 ) ! ;   3 ( x ) = lim N !1  N xq  1 2 ( f 0  s 0 ) N N 1 2 ( k 0  l 0 ) ! ; (40)  ++ 1 ( x ) = lim N !1 ( m ) q ( mN ) q  N xq  ( d 0  e 0  1) N N ( g 0  h 0 ) ! ;  ++ 2 ( x ) = lim N !1 ( m ) q ( mN ) q  N xq  ( d 0  e 0  1) N N 1 2 ( k 0  l 0 ) ! ;  ++ 3 ( x ) = lim N !1 ( m ) q ( mN ) q  N xq  1 2 ( f 0  s 0  2) N N 1 2 ( k 0  l 0 ) ! ; (41) and their corresponding moments are as fol lows:  0 m = lim N !1  0 ( N ) m (42)   m (1) = lim N !1  0 ( N ) m N ( g 0  h 0 ) q ( d 0  e 0 ) mN   m (2) = lim N !1  0 ( N ) m N 1 2 ( k 0  l 0 ) q ( d 0  e 0 ) mN   m (3) = lim N !1  0 ( N ) m N 1 2 ( k 0  l 0 ) q 1 2 ( f 0  s 0 ) mN (43)  ++ m (1) = lim N !1 ( m ) q ( mN ) q  0 ( N ) m N ( g 0  h 0 ) q ( d 0  e 0  1) mN  ++ m (2) = lim N !1 ( m ) q ( mN ) q  0 ( N ) m N 1 2 ( k 0  l 0 ) q ( d 0  e 0  1) mN  ++ m (3) = lim N !1 ( m ) q ( mN ) q  0 ( N ) m N 1 2 ( k 0  l 0 ) q 1 2 ( f 0  s 0  2) mN (44) for m  0 , respectively, and where symbol ( n ) q denotes the q-basic numb er ( n ) q = q n  1 q  1 ; (45) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 9 related with the q-numbers [ n ] q = q n  q  n q  q  1 by formula ( n ) q = q n  1 2 [ n ] q 1 2 . It turns out that  0 m = 1 ; m  0 (46) and 1. d 0  e 0 > 1 2 ( f 0  s 0 ) . Then   m (1) = 8 > > > > < > > > > : 1 m = 0 "  (0) 0  (0) 0 # m q m ( d 0  e 0 ) q m ( d 0  e 0 )  1 m  1 : (47) Also,  ++ m (1) = 8 > < > : 1 m = 0 ( q m  1)   m (1) m  1 (48) 2. If d 0  e 0 = 1 2 ( f 0  s 0 ) . Then three diferent situation come up: (a) If g 0  h 0 > 1 2 ( k 0  l 0 ) . Then the moments   m (1) and  ++ m (1) have the same values as in the previous case., i.e., as formulas (47) and (48). (b) If g 0  h 0 = 1 2 ( k 0  l 0 ) , then   m (1) = 8 > > > > > < > > > > > : 1 m = 0 X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R q  2 + m (1  t )( d 0  e 0 ) q m ( d 0  e 0 )  1 m  1 (49) Also,  ++ m (1) = 8 > < > : 1 m = 0 ( q m  1)   m (1) m  1 : (50) (c) If g 0  h 0 < 1 2 ( k 0  l 0 ) , then   m (2) = 8 > > > > < > > > > : 1 m = 0 "  (0) 0  (0) 0 # m 2 1 q ( d 0  e 0 ) m  1 m  1 (51) Also,  ++ m (2) = 8 > < > : 1 m = 0 ( q m  1)   m (2) m  1 (52) 3. d 0  e 0 < 1 2 ( f 0  s 0 ) . Then   m (3) = 8 > > > > < > > > > : 1 m = 0 "  (0) 0  (0) 0 # m 2 1 q 1 2 ( f 0  s 0 ) m  1 m  1 (53) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 16 then the dominant term in the (m)-summation of the expression (71) is the one corresp onding to the partition (0 ; m; 0 ; :::; 0). Therefore R 0 = 0, R = m 2 , t = 1,  2 = 0, M = 1 2 ( k 0  l 0 ) and  0 ( N ) m  F (0 ; m; 0 ; :::; 0) "  (0) 0  (0) 0 # m 2 q 1 2 ( f 0  s 0 ) mN q 1 2 ( f 0  s 0 ) m  1 N 1 2 ( f 0  s 0 ) m ; which coincides with (21) since F (0 ; m; 0 ; :::; 0) = 1. This completely proves the Theorem 1. As a conclusion to this section we provide the scheme with all dierent p osibilities obtained in this section. Scheme: The caracterization of general q-p olynomials by its sp ectral prop erties. 1 : d 0  e 0 = 1 2 ( f 0  s 0 ) 8 > < > : ( a ) g 0  h 0 > 1 2 ( k 0  l 0 ) ( b ) g 0  h 0 = 1 2 ( k 0  l 0 ) ( c ) g 0  h 0 < 1 2 ( k 0  l 0 ) 2 : d 0  e 0 6 = 0 f 0  s 0 6 = 0 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : ( a ) d 0  e 0  0 f 0  s 0  0 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ( i ) ( d 0  e 0 < 0 f 0  s 0 < 0  1 6 = 0 ( ii ) ( d 0  e 0 = 0 f 0  s 0 < 0 g 0  h 0 = k 0  l 0 = 0 ( iii ) ( d 0  e 0 < 0 f 0  s 0 = 0 g 0  h 0 = k 0  l 0 = 0 ( b ) d 0  e 0 > 0 and=or f 0  s 0 > 0 8 > > > > > > > > > > < > > > > > > > > > > : ( i ) d 0  e 0 > 1 2 ( f 0  s 0 ) ( ii ) d 0  e 0 = 1 2 ( f 0  s 0 ) 8 > < > : A ) g 0  h 0 > 1 2 ( k 0  l 0 ) B ) g 0  h 0 = 1 2 ( k 0  l 0 ) C ) g 0  h 0 < 1 2 ( k 0  l 0 ) ( iii ) d 0  e 0 < 1 2 ( f 0  s 0 ) 5 Searching for a normalized density of zeros. In this Section the asymptotic distribution of zeros of the p olynomial P N ( x ) q dened by Eqs. (1)-(4) will b e discussed. In particular Theorems 2-4 will b e proved. The starting p oint will b e Theorem 1. From Theorem 1, one observes that the moments  0 ( N ) m of the (non-normalized) density of zeros  N ( x ) dep ends on N as follows: N am +1 in case 1 ; C onstant in sub case 2(a)i ; N in sub cases 2(a)ii-2(a)iii ; N am q bmN in case 2b ; (72) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 17 where the constants a and b are known and distinct for each case. Obviously we would like to have a normalized density of zeros  nor m N ( x ). The usual way to have it is to imp ose that the moment of order zero b e equal unity, what p ermits to write  nor m N ( x ) = 1 N  N ( x ) ; (73) whose moments ~  0 ( N ) m will b e related to those of  N ( x ) by ~  0 ( N ) m = 1 N  0 ( N ) m ; m  0 : (74) Then, from (72) and (74) it is clear that the N -dep endence of the moments of the normalized to unity density of zeros is given by N am in case 1 ; N  1 in sub case 2(a)i ; C onstant in sub cases 2(a)ii-2(a)iii ; N am  1 q bmN in case 2b ; (75) As said b efore, we are interested in the asymptotic density of zeros. If this is dened by  ( x ) = lim N !1  N ( x ) ; (76) then taking into account that  0 ( N ) m have a N -dep endence of the form (72), its moments  0 m given by  0 m = lim N !1  0 ( N ) m will b e innity in case 1, sub cases 2(a)ii and 2(a)iii and in case 2b; and constant given by (14) in sub case 2(a)i. Therefore, the expressions (27), (33), (35) and (37) of theorems 2, 3 and 4, resp ectively, have b een proved. If one wants to have some information ab out the asymptotic distribution of zeros in case 1, sub cases 2(a)ii and 2(a)iii and in case 2b, one needs to intro duce a normalization factor and/or a scaling factor into the density  N ( x ) in the sense disscused in Eq. (55) and (56). Let us rst think of a scaled density. For the case 1 there is no scaling factor D which leads to an asymptotic density of zeros whose moments have non-zero, nite values unless the scaling factor b e of the form D = N  a  1 m but this is not useful since it would oblige to dene a dierent scaled asymptotic density function for each moment. Contrary to this, for the case 2b one can consider scaling factor D = N  a q  bN and dene the discrete density of zeros given by   N ( x ) =  N  x q bN N a  and the asymptotic density of zeros given by   ( x ) = lim N !1  N  x q bN N a  (77) whowe moments   m are acording to (56), as follows   m = lim N !1  0 ( N ) m q mbN N am : (78) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 18 From (72) and (78), it is clear that all the quantities   m have nite values. It is only missing to take the parameters a and b for the dierent sub cases of 2b. For the sub cases 2(b)i, 2(b)iiA and 2(b)iiB it turns out that a = g 0  h 0 and b = d 0  e 0 . Then, as in expression (77), one can dene the asymptotic density function   1 ( x ) in the form   1 ( x ) = lim N !1  N xq  ( d 0  e 0 ) N N ( g 0  h 0 ) ! ; (79) whose moments   m (1) given by   m (1) = lim N !1  0 ( N ) m N ( g 0  h 0 ) m q ( d 0  e 0 ) mN ; (80) have, according to (17) and (18), the values (for m  1)   m (1) = "  (0) 0  (0) 0 # m q m ( d 0  e 0 ) q m ( d 0  e 0 )  1 (81) in the sub cases 2(b)i and 2(b)iiA, and, according to (19), the values   m (1) = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R q  2 + m (1  t )( d 0  e 0 ) q m ( d 0  e 0 )  1 (82) in the sub case 2(b)iiB. Remark that the expressions (81) and (82) are identical to (47) and (49) of Theorem 4, resp ectively. Similary, for the sub cases 2(b)iiC it turns out that a = 1 2 ( k 0  l 0 ) and b = d 0  e 0 . Then, as in expression (77), one denes the asymptotic density function   2 ( x ) by (40), whose moments   m (2) given by (43) have, according to (20), the values given by (51). Finally, for the sub case 2(b)iii one has the density   3 ( x ) dened by (40), whose moments   m (3), given by (43), have according to (21), the values given by (53). For the entire case 2b it happ ens that, according to (78) and since  0 ( N ) 0 = N ,   0 =   0 (1) =   0 (2) =   0 (3) = 1 ; as in Theorem 4 is also p ointed out. Let us now searshed for a normalized to unity asymptotic density of zeros. The simplest way is to dene it as  1 ( x ) = lim N !1  nor m N ( x ) = lim N !1 1 N  N ( x ) ; (83) where the Eq. (73) has b een used. Its moments given by  0 0 (1) = 1 ;  0 m (1) = lim N !1 1 N  0 ( N ) m ; m  1 ; (84) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 19 have, taking into account (75), the following values  0 0 (1) = 1  0 m (1) = 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 1 m  1 in cases 1 and 2b 0 m  1 sub case 2(a)i X ( m ) F ( r 0 1 ; 0 ; :::; 0 ; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 m  1 sub case 2(a)ii X ( m ) F (0 ; r 1 ; 0 ; :::; r j ; 0) "  (0) 0  (0) 0 # R m  1 sub case 2(a)iii (85) Then expressions (34)-(38) of Theorem 3 has b een demonstrated. So, Theorem 3 has entirely proved. For the case 1 and sub case 2b one would like to have information more useful than that expressed by (85), keeping the normalization to unity of the density  1 ( x ) given by (83). Therefore one has to compress the sp ectrum of zeros by intro ducing a scaling factor. In the case 1 it is very easy to nd that factor by lo oking at the expression (75): it is D = N  a . Then one denes from (75) and (83) the density function   ( x ) = lim N !1  nor m N  x N a  = lim N !1 1 N  N  x N a  (86) whose moments are according to (56) and (84) as   0 = 1 ;   m = lim N !1  0 ( N ) m N am +1 ; m  1 : (87) From (72) and (87) it is obvious that the quantities   m have nite values. One has only to take the values of a in the dierent sub cases of the case 1. For the sub case 11, a = g 0  h 0 ; then here it is convenient to dene, according to (86), the following asymptotic density of zeros   1 ( x ) = lim N !1 1 N  N  x N g 0  h 0  ; whose moments are, according to (87) and (11), as follows   0 (1) = 1;   m (1) = "  (0) 0  (0) 0 # m ; m  1 ; which is the expression (28) of Theorem 2. For the sub cases 12 and 13, it turns out that a = 1 2 ( k 0  l 0 ), which denes the following asymptotic density of zeros   2 ( x ) = lim N !1 1 N  N  x N 1 2 ( k 0  l 0 )  ; whose moments have, according to (87) and (12), the values (   0 (2) = 1)   m (2) = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) "  (0) 0  (0) 0 # R 0 "  (0) 0  (0) 0 # R ; m  1 THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 20 for the sub case 12, and, according to (87) and (13), the values   0 (2) = 1;   m (2) = "  (0) 0  (0) 0 # m 2 ; m  1 ; for the sub case 13. Remarks that the last two expressions coincide with the expressions (29) and (30) of Theorem 2, resp ectively. Then this Theorem has b een entirely proved. For the sub case 2b the scaled normalization to unity asymptotic density function of the form (86) would also have all its moments of order other than zero equal to innite. No other scaling factor would b e able to make nite these moments unless D = N  a + 1 m q mN , but this factor is of usefulness for reasons already discussed. Therefore one is obliged to change the normalization factor in this sub case. Here the discrete density of zeros  + N ( x ) is normalized so that its moments are dened by  +( N ) m = q m  1 q mN  1  0 ( N ) m ; m  0 ; i.e., that  + N ( x ) = ( m ) q ( mN ) q  N ( x ) (88) when ( m ) q and ( mN ) q are q-numb ers dened by Eq.(45). This normalization factor has the following relevant prop erty: It tends to N  1 if m ! 0 and q ! 1. In particular, this implies that  +( N ) 0 = 1 : Furthermore, for the case 2b under consideration it turns out that the N  dep endence of  +( N ) m is as N am q ( b  1) mN . this dep endence suggest to analize the asymptotic sp ectrum of zeros by means of the asymptotic density function dened by  ++ ( x ) = lim N !1  N  x N a q ( b  1) N  ; (89) whose moments  ++ m are given by  ++ m = lim N !1  +( N ) m N am q ( b  1) mN = lim N !1 ( q m  1)  0 ( N ) m ( q mN  1) N am q ( b  1) mN : (90) Taking into account this expression together with the values (17)-(21) of  0 ( N ) m given in the Theorem 1, one observes that for the sub cases 2(b)i, 2(b)iiA and 2(b)iiB the parameter a and b take the values a = g 0  h 0 ; b = d 0  e 0 and the appropiate asymptotic density of zeros is, according to (88)-(89), the function  ++ 1 ( x ) given by (41) in Theorem 4. For the sub case 2(b)iiC it turns out that a = 1 2 ( k 0  l 0 ) ; b = d 0  e 0 : Then, the appropiate asymptotic density of zeros for this sub case is, according to (88)-(89), the function  ++ 2 ( x ) given by (41) in Theorem 4. THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 21 Finally for the case 2(b)iii a = 1 2 ( k 0  l 0 ), b = 1 2 ( f 0  s 0 ) and the appropiate asymptotic density of zeros is, according to (88)-(89), the function  ++ 3 ( x ) given by (41) in Theorem 4. Now the Eq. (90) and the values (17)-(21) for  0 ( N ) m gives in a straightforward manner the moments  ++ m (1),  ++ m (2) and  ++ m (3) ot the asymptotic density functions  ++ 1 ( x ),  ++ 2 ( x ) and  ++ 3 ( x ). Indeed, the values of these quantities are given by the Eqs. (48) for the sub cases 2(b)i, 2(b)iiA, (50) for the sub case 2(b)iiB, (52) for the sub case 2(b)iiC and (54) for the sub case 2(b)iii, resp ectively. This entirely proves Theorems 2-4. 6 Applications. In this Section we will use the theorems obtained in the two previous sections to investigate the sp ectral prop erties of several known families of orthogonal q-p olynomials. Let us make the observation that for a nite p olynomial sequence (e.g. Hahn, Racah and Kravchuk p olynomials), i.e., when the degree n of the p olynomial is b ounded by a xed parameter N (not to b e confused with the same letter previously used as generic degree of p olynomials), it is assumed that N is suciently large and 1 << n  N so that Eq. (61) b e fullled. The q-Hahn p olynomials h ; n ( q  x ; N ) . The q-Hahn p olynomials h ; n ( q  x ; N ) play a fundamental role in the Representation Theory of the q-Algebras S U q (2) and S U q (1 ; 1) (see [20], [22], [21 ]). They also app ear in numerous physical applications since e.g. the Clebsh-Gordan Co ecients of the q-Algebras S U q (2) and S U q (1 ; 1) are prop ortional to them. The theory and applications of the q-Hahn and classical Hahn p olynomials have some close parallels. So, e.g. q-Hahn and classical Hahn p olynomials app ears in the analysis of functions on the lattice of subspaces of a nite vector space and the lattice of subsets of a nite set, resp ectively. These p olynomials verify the recurrence relation [3] (page 59) h ; n ( q  x ; N ) = [ q  x  (1  A n  1  C n  1 )] h ; n  1 ( q  x ; N ) + B n  1 h ; n  2 ( q  x ; N ) ; (91) where B n = A n  1 C n , and the A and C parameters are A n =  1   q 1+ n   1    q 1+ n   1  q  N + n  (1    q 1+2 n ) (1    q 2+2 n ) ; C n =   q n (1  q n ) ( 1   q n )  q  N    q 1+ n  (1    q 2 n ) (1    q 1+2 n ) : Let us also p oint out that  n +1 A n =  n (92) where  n is the leading co ecient of the p olynomial. The comparison of Eqs. (91) and (1) gives that a num n =  (0) 0 q 3 n =  2  (1 +  ) q N +1 q 3 n ; a den n =  (0) 0 q 4 n =  2  2 q N q 4 n : and ( b num n ) 2 =  (0) 0 q 7 n =  4  3 q  N q 7 n ; ( b den n ) 2 =  (0) 0 q 8 n =  4  4 q 8 n : Then, g m = h m = k m = l m = 0 for all m = 0 ; 1 ; ::N and d 0 = 3 ; e 0 = 4 ; f 0 = 7 ; s 0 = 8 : This is the case d 0  e 0 < 0 and f 0  s 0 < 0, i.e., case 2(a)i. Therefore, Eqs. (33) and (34) of Theorem 3 give us the moments THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 22  0 m = X ( m ) F ( r 0 1 ; r 1 ; :::; r j ; r 0 j +1 ) q  P j k =1 k r 0 k +1  2 P j  1 k =1 k  q (1 +  )   R 0   q N ( q + q  1 )  R 1 q m 2  1 (93) for the asymptotic density of zeros  ( x ) dened by Eq. (31), and  0 m (1) = 8 > < > : 1 m = 0 0 m  1 (94) for the corresp onding asymptotic quantity  1 ( x ) given by Eq. (31). q-Kravchuk p olynomials k p n ( q  x ; N ) . The matrix elements of the representations T l of the U q ( sl 2 ) quantum algebra are prop ortional to the q-Kravchuk p olynomials (see [21], Vol. I I I, page 64). According to [3] (page 76) and taking into account Eq. (92) the three term recurrence relation of these p olynomials can b e expressed as k p n ( q  x ; N ) = [ q  x  (1  A n  1  C n  1 )] k p n  1 ( q  x ; N ) + B n  1 k p n  2 ( q  x ; N ) ; (95) where B n  1 = A n  1 C n  1 , and A n = (1 + p q n )  1  q  K + n  (1 + p q 2 n ) (1 + p q 1+2 n ) ; C n =  p q  1  K +2 n (1  q n )  1 + p q K + n  (1 + p q 2 n ) ( 1 + p q  1+2 n ) : The comparison with (1) gives that a num n =  (0) 0 q 3 n = pq ( pq N  1) q 3 n ; a den n =  (0) 0 q 4 n = p 2 q N q 4 n : and ( b num n ) 2 =  (0) 0 q 6 n = p 3 q  2 q 6 n ; ( b den n ) 2 =  (0) 0 q 8 n = q  3 p 4 q N q 8 n : Then, g m = h m = k m = l m = 0 for all m = 0 ; 1 ; ::N and d 0 = 3 ; e 0 = 4 ; f 0 = 6 ; s 0 = 8 : This is the case d 0  e 0 =  1 < 0 and f 0  s 0 =  2 < 0, i.e., case 2(a)i. Therefore, Eq. (34) of Theorem 3 gives us the values  0 m (1) = 8 > < > : 1 m = 0 0 m  1 (96) for the moments of the asymptotic density of zeros  1 ( x ). Furthermore, since  1 = 1 2 ( f 0  s 0 ) =  m ,  2 =  ( P j k =1 k r 0 k +1  4 P j  1 k =1 k r k +1 ) and M = 0, Eq. (33) of Theorem 3 gives us  0 m = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) q   2 " q ( pq N  1) pq N # R 0 " q 1  N p # R 1 q m  1 ; m  0 ; (97) for the moments of the sp ectral quantity  ( x ) dened by Eq. (31). THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 23 q-Racah p olynomials R n (  ( x ) ; ;  ;  ;  ) .  ( x ) = q  x +   q x +1 . It is well known the imp ortant role that the 6j symb ols play in the quantum angular momentum theory (see [6]). It is known that the q-analog of the Racah co ecients (6j symb ols) for the qalgebra U q ( sl 2 ) are prop ortional to the q-Racah p olynomials (see [21 ], Vol. I I I, page 70). From the three term recurrence relation of these p olynomials [3] (page 53), as well as Eq. (92), we can rewrite [3] Eq.(3.15.3) in the form  ( x ) R n  1 (  ( x ) ; ;  ;  ;  ) = R n (  ( x ) ; ;  ;  ;  )+ +[1 +   q  (1  A n  1  C n  1 )] R n  1 (  ( x ) ; ;  ;  ;  ) + B n  1 R n  2 (  ( x ) ; ;  ;  ;  ) ; (98) where B n  1 = A n  1 C n  1 , and A n =  1   q 1+ n   1    q 1+ n   1    q 1+ n   1   q 1+ n  (1    q 1+2 n ) (1    q 2+2 n ) ; C n = q (1  q n ) (    q n ) (1   q n ) (     q n ) (1    q 2 n ) ( 1    q 1+2 n ) : The comparison with (1) gives that a num n =   (0) 0 q 3 n = q  (  +  +  +   +   +  +   +    ) q 3 n ; a den n =   (0) 0 q 4 n =  2  2 q 4 n : and ( b num n ) 2 =  (0) 0 q 8 n = q  4  4   q 8 n ; ( b den n ) 2 =  (0) 0 q 8 n =  4  4 q 8 n : Then, g m = h m = k m = l m = 0 for all m = 0 ; 1 ; ::N and d 0 = 3 ; e 0 = 4 ; f 0 = 8 ; s 0 = 8 : This is the case d 0  e 0 =  1 < 0 and f 0  s 0 = 0, i.e., case 2(a)iii. Therefore, Eq (16) of Theorem 3 yield the moments  0 m (1) = 8 > > > < > > > : 1 m = 0 X ( m ) F (0 ; r 1 ; 0 ; :::; r j ; 0) [ q   ] R m  1 (99) for the asymptotic densities of zeros  1 ( x ) dened by Eq. (31). q-Askey & Wilson p olynomials p n ( x; a; b; c; d ) . According to [3 ] (page 51) and Eq. (92), the three term recurrence relation for the q-Askey & Wilson p olynomials can b e rewritten as xp n  1 ( x; a; b; c; d ) = p n ( x; a; b; c; d ) + 1 2 [ a + a  1  ( A n  1 + C n  1 )] p n  1 ( x; a; b; c; d )+ + B n  1 p n  2 ( x; a; b; c; d ) ; (100) where B n  1 = A n  1 C n  1 , and A n =  1  a b c d q  1+ n  (1  a b q n ) ( 1  a c q n ) ( 1  a d q n ) a (1  a b c d q 2 n ) (1  a b c d q  1+2 n ) ; THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 24 C n = a  1  bcq  1+ n   1  b d q  1+ n   1  c d q  1+ n  (1  q n ) (1  a b c d q  2+2 n ) ( 1  a b c d q  1+2 n ) : The comparison with (1) gives a num n =   (0) 0 q 3 n = q abcd ( abc + abd + acd + bcd + q ( a + b + c + d )) q 3 n ; a den n =   (0) 0 q 4 n = 2 a 2 b 2 c 2 d 2 q 4 n : and ( b num n ) 2 =  (0) 0 q 8 n = a 4 b 4 c 4 d 4 q 8 n ; ( b den n ) 2 =  (0) 0 q nn = a 4 b 4 c 4 d 4 q 8 n : Then, g m = h m = k m = l m = 0 for all m = 0 ; 1 ; ::N and d 0 = 3 ; e 0 = 4 ; f 0 = 8 ; s 0 = 8 : This is the case d 0  e 0 =  1 < 0 and f 0  s 0 = 0, i.e., case 2(a)iii. Therefore, Eq. (38) of Theorem 3 gives us the moments  0 m (1) = 8 > > > < > > > : 1 m = 0 X ( m ) F (0 ; r 1 ; 0 ; :::; r j ; 0) m  1 : (101) for the asymptotic density of zeros  1 ( x ) dened by Eq. (31). Al Salam and Carlitz p olynomials u  n ( x ) and v  n ( x ) . In dealing with the q-harmonic oscillator, Askey and Suslov [16 ] have intro duced the q-p olynomials u  n ( x ) =   n q  n ( n  1) 2 U (   ) n ( x ) : where U   n ( x ) are the so called Al Salam and Carlitz p olynomials. These p olynomials satisfy the recurrence relation [16] xu  n  1 ( x ) = u  n ( x ) + (1   ) q n  1 u  n  1 ( x ) + q n  2 (1  q n  1 ) u  n  2 ( x ) ; (102) which is of the typ e (1) with the co ecients a num n =   (0) 0 q n = (1   ) q  1 q n ; a den n = 1 ; and ( b num n ) 2 =  (0) 0 q 2 n = q  1 q 2 n ; ( b den n ) 2 =  (0) 0 q s 0 n = 1 : Then, g m = h m = k m = l m = 0 for all m = 0 ; 1 ; ::N and d 0 = 1 ; e 0 = 0 ; f 0 = 2 ; s 0 = 0 : This is the case d 0  e 0 = 1 and f 0  s 0 = 2, i.e., case 2(b)iiB. Therefore, Eqs. (49) and (50) of Theorem 4 give us the moments   m (1) = 8 > > > > < > > > > : 1 m = 0 X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) [1   ] R 0  R q  2 q m  1 m  1 (103) THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS 25 and  ++ m (1) = 8 > > > < > > > : 1 m = 0 X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) [1   ] R 0  R q  2 m  1 (104) (where  2 = P j k =1 k r 0 k +1 + 4 P j  1 k =1 k r k +1  mt ) corresp onding to the asymptotic quantities   1 ( x ) and  ++ 1 ( x ), resp ectively. It has b een encountered [15 ] that another class of Al Salam and Carlitz p olinomials, to b e denoted by v  n ( x ), is related also to the q-oscillator. So, it seems natural to search for its distribution of zeros. These p olynomials satisfy the relation [15 ] xv  n  1 ( x ) = v  n ( x ) + ( q +  ) q  n  2 v  n  1 ( x ) + q  n  3 ( q  n  1  1) v  n  2 ( x ) : (105) Therefore, d 0 =  1 ; e 0 = 0 ; f 0 =  1 ; s 0 = 0. This corresp onds to the case 2(a)i. Then, Eq. (34) of Theorem 3 gives us the moments  0 m (1) = 8 > < > : 1 m = 0 0 m  1 : (106) for the asymptotic density  ( x ). Furthermore, since  1 =  1 2 ( R 0 + m ),  2 =  ( P j k =1 k r 0 k +1  2 P j  1 k =1 k r k +1 ) and M = 0, Eq. (33) of Theorem 3 gives  0 m = X ( m ) F ( r 0 1 ; r 1 ; :::; r 0 j +1 ) q   2 h q  1 ( q +  ) i R 0  R q  m q 1 2 ( R 0 + m )  1 (107) for the moments of the normalized -to- 1 N sp ectral quantity  1 ( x ) dened by Eq. (31). The little q-Jacobi p olynomials p n ( x; a; b ) . The little q-Jacobi p olynomials p n ( x; a; b ) play a fundamental role (see e.g. [20]) in the Representation Theory of the q-Algebra U q ( sl 2 ) b ecause they are the matrix elements of the representations T l (see [21 ], Vol. I I I, page 51). They satisfy the three term recurrence relation [3] (page 59) p n ( x; a; b ) = [ x + A n  1 + C n  1 ] p n  1 ( x; a; b ) + B n  1 p n  2 ( x; a; b ) ; (108) where A and C parameters are given by A n = q n  1  a q 1+ n   1  a b q 1+ n  (1  a b q 1+2 n ) (1  a b q 2+2 n ) ; C n = a q n (1  q n ) ( 1  b q n ) (1  a b q 2 n ) ( 1  a b q 1+2 n ) ; and B n = A n  1 C n . This relation is of the typ e (1) with the co ecients a num n =  (0) 0 q 3 n =  ab (1 + a ) q 3 n ; a den n =  (0) 0 q 4 n = a 4 b 4 q 4 n : and ( b num n ) 2 =  (0) 0 q 6 n = a 3 b 2 q 6 n ; ( b den n ) 2 =  (0) 0 q 8 n = q a 4 b 4 q 8 n :