Recurrence relations for connection coefficients between q-orthogonal polynomials of discrete variables in the non-uniform lattice X(s) = q2s
Abstract
We obtain the structure relations for q-orthogonal polynomials in the exponential lattice q 2s and from that we construct the recurrence relation for the connection coe cients between two families of polynomials belonging to the classical class of discrete q-orthogonal polynomials. An explicit example is also given.
Full text
RECURRENCE RELATIONS FOR CONNECTION COEFFICIENTS BETWEEN Q-ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES IN THE NON-UNIFORM LATTICE x ( s ) = q 2 s . 1 R. Alvarez-No darse 2 Departamento de Matematicas. Escuela Politecnica Superior. Universidad Carlos III de Madrid. Butarque 15, 28911,Leganes,Madrid. A. Ronveaux 3 Mathematical Physics, Facultes Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium. Key words and phrases: q-Hahn, q-Meixner, q-Charlier and q-Kravchuk p olynomials, discrete p olynomials, q-p olynomials p olynomials. AMS (MOS, 1991) sub ject classication: 33D45 Abstract We obtain the structure relations for q-orthogonal p olynomials in the exp onential lattice q 2 s and from that we construct the recurrence relation for the connection co ecients b etween two families of p olynomials b elonging to the classical class of discrete q-orthogonal p olynomials. An explicit example is also given. 1 Intro duction. Given two families of Polynomials, denoted by P n ( x ) and Q m ( x ), of degree exactly equal to resp ectively n and m , the Connection Problem asks to compute the so-called Connection Coecients C m ( n ) dened by the relation: P n ( x ) = n X m =0 C m ( n ) Q m ( x ) : When b oth families are orthogonal with resp ect to two dierent measures the Connections Co ecients satisfy a relative simple recurrence relation, but mixing in the ( m; n ) table three adjacent m and three adjacent n crossing at ( m; n ). The rst survey on this topic was given by Askey 20 years ago [1]-[2], giving in some cases explicit expression for the Co ecients and discussing also the p ositivity prop erties of these Co ecients. 1 Octob er 29, 1996 2 e-mail renato@dulcinea .uc3m.es Fax:(+431) 6249430 3 e-mail Andre.Ronv[email protected] e Fax(+3281) 724707 1
It was noticed only recently that an additional assumption on the orthogonality measure gives for C m ( n ) a recurrence only in m; n b eing xed. This Orthogonality class is called semi-classical and is very large [11], [7] . The classical (continuous) family: Jacobi, Bessel, Laguerre, Hermite (see for instance [12]) and the classical(discrete) family: Hahn , Kravchuk, Meixner, Charlier (see for instance [13]) are of course included in the semi-classical class. When the orthogonality measure is dened by a weight ( x ), the semi-classical class covers all weights solution of a linear rst order dierential (or dierence) equation with p olynomial co ecients. The key prop erty inside the semi-classical class, in order to obtain a one index (m) recurrence relation for C m ( n ), comes from the existence of a so called Structure Relation, linking linearly the derivative (or dierence) of P n ( x ) times a p olynomial, to a xed combination of P k ( x ). An algorithm has b een given recently building for b oth discrete and continuous classical families (see [3], [15] and [16] ) the explicit recurrences for C m ( n ), solving in many cases theses recurrences with the help of Mathematica [20]. Lo oking for the situation for which a structure relation is known explicitly, we realize that, from the data of Orthogonal Polynomial on the exp onential lattice x ( s ) = q 2 s (a small subset of the q-world). Here we need to p oint out that exits two dierent p oint of view in the study of the q-p olynomials. The rst one, in the framework of the q-basic hyp ergeometric series [6], [8], [9] and the second, in the framework of the theory of dierence equations develop ed by Nikiforov et al. [12], [13], [14]. In this work we will use the second one b ecause it gives us the p ossibility to provide an uniform treatment of several classes of orthogonal p olynomials and, probably, it is the b est way to nd further applications. This pap er shows how to apply the technique to a particular (simple) case: the exp onential lattice, building rst the corresp onding Structure Relations. 2 Structure relations for q-orthogonal p olynomials on the exp onential lattice x ( s ) = q 2 s . Let us to start with the study of some general prop erties of orthogonal p olynomials of a discrete variable in non-uniform lattices. Let b e ~ ( x ( s )) 4 4 x ( s 1 2 ) 5 Y ( s ) 5 x ( s ) + ~ ( x ( s )) 2 4 Y ( s ) 4 x ( s ) + 5 Y ( s ) 5 x ( s ) + Y ( s ) = 0 ; 5 f ( s ) = f ( s ) f ( s 1) ; 4 f ( s ) = f ( s + 1) f ( s ) ; (1) the second order dierence equation of hypergeometric type for some lattice function x ( s ), where 5 f ( s ) = f ( s ) f ( s 1) and 4 f ( s ) = f ( s + 1) f ( s ) denote the backward and forward nite dierence quotients, resp ectively. Here ~ ( x ) and ~ ( x ) are p olynomials in x ( s ) of degree at most 2 and 1, resp ectively, and is a constant. 2
The previous equation (1) can b e obtained from the classical hypergeometric equation ~ ( x ) y 00 ( x ) + ~ ( x ) y 0 ( x ) + y ( s ) = 0 ; via the discretization of the rst and second derivatives y 0 and y 00 in an apropiate lattice [12], [13]. It is b etter to rewrite (1) in the equivalent form (see [13] and [14]) ( s ) 4 4 x ( s 1 2 ) 5 Y ( s ) 5 x ( s ) + ( s ) 4 Y ( s ) 4 x ( s ) + Y ( s ) = 0 ; ( s ) = ~ ( x ( s )) 1 2 ~ ( x ( s )) 4 x ( s 1 2 ) ; ( s ) = ~ ( x ( s )) : (2) The q-orthogonal polynomials P n ( x ( s )) q P n ( s ) q on the exp onential lattice x ( s ) = q 2 s are, for given functions ( s ) and ( s ), the p olynomial (in p owers of x ( s ) = q 2 s ) solution of the second order dierence equation (2). The k-order dierence derivative of the p olynomials P n ( x ( s )) q , dened by v k n ( s ) = 4 4 x k 1 ( s ) 4 4 x k 2 ( s ) ::: 4 4 x ( s ) [ P n ( x ( s )) q ] 4 ( k ) [ P n ( x ( s )) q ] ; and x m ( s ) = x ( s + m 2 ) ; also satisfy the dierence equation of hyp ergeometric typ e of the form ( s ) 4 4 x k ( s 1 2 ) 5 v k n ( s ) 5 x k ( s ) + k ( s ) 4 v k n ( s ) 4 x k ( s ) + k v k n ( s ) = 0 ; (3) where (see [13], page 62, Equation (3.1.29)) k ( s ) = ( s + k ) ( s ) + ( s + k ) 4 x ( s + k 1 2 ) 4 x k 1 ( s ) ; and k = n + k 1 X m =0 4 m ( s ) 4 x m ( s ) : These p olynomial solutions denoted by P n ( x ( s )) q P n ( s ) q satisfy the orthogonality prop erty b 1 X s i = a P n ( x ( s i )) q P m ( x ( s i )) q ( s i ) 4 x ( s i 1 2 ) = nm d 2 n ; (4) where ( x ) is some non-negative function (weight-function), i.e., ( s i ) 4 x ( s i 1 2 ) > 0 ( a s i b 1) ; supp orted in a countable subset of the real line [ a; b ] ( a; b can b e 1 ). The functions ( s ) and k ( s ) are the solutions of the Pearson-typ e dierence equations ([13], Eq.(3.2.9) and (3.2.10) page 64) 4 4 x ( s 1 2 ) [ ( s ) ( s )] = ( s ) ( s ) ; (5) 3
and 4 4 x k ( s 1 2 ) [ ( s ) k ( s )] = k ( s ) k ( s ) (6) and ( s ) satisfy the condition [14]: ( s ) ( s ) x k ( s 1 2 ) j s = a;b = 0 ; 8 k ; l 2 IN ( IN = f 0 ; 1 ; 2 ; ::: g ) : In (4) d 2 n denotes the square of the norm of the corresp onding orthogonal p olynomials. The q-orthogonal p olynomials satisfy a three term recurrence relations (TTRR) of the form x ( s ) P n ( s ) q = n P n +1 ( s ) q + n P n ( s ) q + n P n 1 ( s ) q ; (7) with the initial conditions P 1 ( s ) q = 0 ; P 0 ( s ) q = 1 : It is well known [13]-[14], that the p olynomial solutions of equation (2), denoted by P n ( x ( s )) q , are uniquely determined, up to a normalizing factor B n , by the dierence analog of the Ro drigues formula (see [13] page 66 Eq. (3.2.19) ): P n ( s ) q = B n ( s ) 5 ( n ) n [ n ( s )] 5 ( n ) n = 5 5 x 1 ( s ) 5 5 x 2 ( s ) ::: 5 5 x n ( s ) [ n ( s )] ; (8) where n ( s ) = ( n + s ) Q n k =1 ( s + k ) : These solutions corresp ond to some values of n - the eigenvalues of equation (2), which is computed from ( see [13], page 104 and [14] ) n = 1 2 [ n ] q f ( q n 1 + q n +1 ) ~ 0 + [ n 1] q ~ 00 g ; (9) where ~ ( s ) = ( s ) + 1 2 ~ ( s ) 4 x ( s 1 2 ) and ~ ( s ) = ( s ) (see Eq. (2)). Here [ n ] q denotes the so called q-numbers [ n ] q = q n q n q q 1 = sinh ( hn ) sinh ( h ) ; q = e h : 2.1 The rst structure relation for the q-p olynomials in the lattice x ( s ) = q 2 s . Let us now try to obtain a structure relation for the q-p olynomials in the exp onential lattice x ( s ) = q 2 s . (For the linear lattice see [13] Eq.(2.2.10) page 24.) First of all, we rewrite the Ro drigues equation (8) in another form. We will use the linearity of the op erator 5 ( n ) n , as well as the identity 5 x k ( s ) = q k 5 x ( s ) : Then, a straightforward calculation gives us 4
P n ( s ) q = q n ( n +1) 2 B n ( s ) 5 5 x ( s ) n [ n ( s )] ; 5 5 x ( s ) n = n-times z}| { 5 5 x ( s ) ::: 5 5 x ( s ) : (10) Now, from formulas (5) and (10) we nd 5 n +1 ( s ) 5 x n +1 ( s ) = 5 [ n ( s + 1) ( s + 1)] 5 x n ( s + 1 2 ) = 4 [ ( s ) n ( s )] 4 x n ( s 1 2 ) = n ( s ) n ( s ) : Then by using the Ro drigues formula (8) we obtain P n +1 ( s ) q = B n +1 ( s ) 5 ( n +1) n +1 [ n ( s )] = B n +1 ( s ) 5 ( n ) n 5 n +1 ( s ) 5 x n +1 ( s ) = = B n +1 ( s ) 5 ( n ) n [ n ( s ) n ( s )] = q n ( n +1) 2 B n +1 ( s ) 5 5 x ( s ) n [ n ( s ) n ( s )] : (11) In order to obtain an expression for h 5 5 x ( s ) i n [ n ( s ) n ( s )] we successively apply the formula 5 f ( s ) g ( s ) = f ( s ) 5 g ( s ) + g ( s 1) 5 f ( s ), as well as formulas 4 n ( s ) 4 x ( s ) = q n 0 n ; 5 5 x ( s 1) n = q 2 n 5 5 x ( s ) n : Then, Eq. (11) gives us the following P n +1 ( s ) q = q n ( n +1) 2 B n +1 ( s ) n ( s ) 5 5 x ( s ) n [ n ( s )] + q 2 n 1 [ n ] q 0 n 5 5 x ( s ) n 1 [ n ( s 1)] ! : (12) Using the Ro drigues formula for the dierence derivative of the p olynomial ([13], Eq. (3.2.18) page 66) we nd (notice that 4 x ( s 1) = q 2 4 x ( s )): 5 P n ( s ) q 5 x ( s ) = 4 P n ( s 1) q 4 x ( s 1) = q ( n 1)( n +2) 2 n B n ( s ) ( s ) 5 5 x ( s 1) n 1 [ n ( s 1)] = = q ( n 1)( n 2) 2 n B n ( s ) ( s ) 5 5 x ( s ) n 1 [ n ( s 1)] : Therefore, equation (12) can b e rewritten in the form P n +1 ( s ) q = B n +1 n ( s ) B n P n ( s ) q [ n ] q B n +1 0 n ( s ) n B n 5 P n ( s ) q 5 x ( s ) and then, the following dierentiation formula holds ( s ) 5 P n ( s ) q 5 x ( s ) = n [ n ] q 0 n n ( s ) P n ( s ) q B n B n +1 P n +1 ( s ) q : (13) 5
If we now use the p ower expansion of n ( s ), i.e., n ( s ) = 0 n x n ( s ) + n (0) = 0 n q n x ( s ) + n (0) and the TTRR (7) we obtain the rst structure relation ( s ) 5 P n ( s ) q 5 x ( s ) = ~ S n P n +1 ( s ) q + ~ T n P n ( s ) q + ~ R n P n 1 ( s ) q ; (14) where ~ S n = n [ n ] q q n n B n 0 n B n +1 ; ~ T n = n [ n ] q q n n n (0) 0 n ; ~ R n = n q n n [ n ] q : (15) 2.2 The second structure relation for the q-p olynomials in the lattice x ( s ) = q 2 s . Let us try to obtain now the second structure relation. Firstly, we notice that 4 5 P n ( s ) q 5 x ( s ) = 4 P n ( s ) q 4 x ( s ) 5 P n ( s ) q 5 x ( s ) : Then, by using the dierence equation (2) ( s ) 5 P n ( s ) q 5 x ( s ) = ( s ) 4 P n ( s ) q 4 x ( s ) ( s ) 4 5 P n ( s ) q 5 x ( s ) = = [ ( s ) + ( s ) 4 x ( s 1 2 )] 4 P n ( s ) q 4 x ( s ) + n 4 x ( s 1 2 ) P n ( s ) q : and (14) we nd [ ( s ) + ( s ) 4 x ( s 1 2 )] 4 P n ( s ) q 4 x ( s ) = ~ S n P n +1 ( s ) q + +( ~ T n n 4 x ( s 1 2 )) P n ( s ) q + ~ R n P n 1 ( s ) q ; (16) Now, taking into account that 4 x ( s 1 2 ) = ( q q 1 ) x ( s ), and using the TTRR (7) we nally obtain the second structure relation [ ( s ) + ( s ) 4 x ( s 1 2 )] 4 P n ( s ) q 4 x ( s ) = S n P n +1 ( s ) q + T n P n ( s ) q + R n P n 1 ( s ) q ; (17) where S n = ~ S n ( q q 1 ) n n ; T n = ~ T n ( q q 1 ) n n ; R n = ~ R n ( q q 1 ) n n : (18) 6
3 Recurrence relations for connection co ecients. Let us consider two families of q-p olynomials P n ( x ) and Q n ( x ) b elonging to the class of discrete orthogonal p olynomials in the exp onential lattice x ( s ) = q 2 s . Each p olynomial P n ( x ) can b e represented as a linear combination of the p olynomials Q n ( x ). In particular P n ( x ) = n X m =0 C m ( n ) Q m ( x ) : (19) For the family P n ( x ) we will use the notation 1. ( s ), ( s ) and n for the dierence equation (2) 2. n , n and n for the TTRR (7) co ecients 3. S n , R n and T n for the second structure relation (17) and for the Q n ( x ) 1. ( s ), ( s ) and n for the dierence equation (2) 2. n , n and n for the TTRR (7) co ecients 3. S n , R n and T n for the second structure relation (17) Since the p olynomials of the family P n ( x ) are solutions of the second order dierence equation (2) the action of the dierence op erator of second order ^ L , dened by ^ L = ( s ) 4 4 x ( s 1 2 ) 5 5 x ( s ) + ( s ) 4 4 x ( s ) + n ; on Eq. (19) gives us n X m =0 C m ( n ) " ( s ) 4 4 x ( s 1 2 ) 5 Q m ( x ) 5 x ( s ) + ( s ) 4 Q m ( x ) 4 x ( s ) + n Q m ( x ) # = 0 : (20) Multiplying by ( s ) and using ( s ) 4 4 x ( s 1 2 ) 5 Q m ( x ) 5 x ( s ) = ( s ) 4 Q m ( x ) 4 x ( s ) n Q m ( x ) ; we obtain the relation n X m =0 C m ( n ) ( ( s ) ( s ) ( s ) ( s )) 4 Q m ( x ) 4 x ( s ) + ( n ( s ) ( s ) m ) Q m ( x ) = 0 : (21) In order to eliminate 4 Q m ( x ) 4 x ( s ) , we multiply (21) by ( s ) + ( s ) 4 x ( s 1 2 ) and use the second structure relation (17) for the Q m ( x ) family, obtaining n X m =0 C m ( n ) ( ( s ) ( s ) ( s ) ( s ))( S m Q m +1 ( x ) + R m Q m 1 ( x ) + T m Q m ( x ))+ + ( ( s ) + ( s ) 4 x ( s 1 2 ))( n ( s ) ( s ) m ) Q m ( x ) = 0 : (22) 7
The last step consists to expand the remaining terms of typ e 2 ( s ) Q m ( x ), ( s ) ( s ) Q m ( x ), ( s ) ( s ) Q m ( x ) and ( s ) ( s ) Q m ( x ) in linear combination of Q m ( x ) by using the TTRR (7) rep eatedly for the Q m ( x ) family. After this pro cess, (22) reduces to N X m =0 M m [ C 0 ( n ) ; C 1 ( n ) ; :::; C n ( n )] Q m ( x ) (23) where N = max f n + deg + deg ( ) ; n + 2 deg ( ) ; n + 1 + deg ( ) + deg ( ) ; n + 1 + deg ( ) + deg ( ) ; 1 + deg ( ) + deg ( ) g : Taking into account the linear indep endence of the family Q m ( x ) we obtain the linear system M m [ C 0 ( n ) ; C 1 ( n ) ; :::; C n ( n )] = 0 : (24) These relations contain (linearly) several connection co ecients C i ( n ) dep ending essentially on the degrees of ( s ) and ( s ). In the most general situation they are p olynomials of second degree in x ( s ) = q 2 s . In this case we obtain a relation of the following typ e the linear system we are lo oking for M m [ C m +4 ( n ) ; :::; C m 4 ( n )] = 0 ; (25) which is valid for n greater or equal than the numb er of initial conditions needed to start the recursion ( n 8). Notice that for ( n < 8) the system also gives the solution, but not in a recurrent way. Notice that for the q-Hahn, q-Meixner, q-Charlier and q-Kravchuk p olynomials, as it is show in [13], table 3.3 , page 95, the ( s ) is a p olynomial of second degree in x ( s ) = q 2 s . This implies that for such p olynomials the recurrence relations for the connection co ecient all are of the form (25). Again we want to remark that we are follow the notation intro duced by Nikiforov et al. [13]. 4 Recurrence relations for connection co ecients: A simple example. As we have noticed in the previous section the recurrence relation for connection co ecients for dierent classes of q-p olynomials are to o large (8-terms). Here we will analyze a more simple case. Firstly, notice that in the previous algorithm we have not used the orthogonality prop erty of the p olynomials P n , and only that they satisfy a dierence equation. On the other hand, for the p olynomials Q m we need to have structure relations as well as three term recurrence relations. Let us to show and example in which we will decomp ose a set of p olynomials P n ( s ), satisfying a certain dierence equation of rst order in the lattice x ( s ) = q 2 s , as a linear combination of the orthogonal q-p olynomials dened in the same lattice, i.e., the q-Hahn, q-Meixner, q-Kravchuk and q-Charlier orthogonal p olynomials (see [13], [4] and [17]) 8
Let us dene the quantities ( s ) q and ( s n ) q , dened by ( s ) q = q 2 s 1 q 2 1 = q s 1 [ s ] q (26) and ( s n ) q = ( s ) q ( s 1) q ( s n + 1) q = n 1 Y k =0 q 2 s +2 k 1 q 2 1 (27) The quantities ( s n ) q are closely related to the q-Stirling numbers ~ S q 2 ( n; k ) ; s q 2 ( n; k ) [21] by formulas ( s ) n q = n X k =0 ~ S q 2 ( n; k )( s k ) q ; ( s n ) q = n X k =0 s q 2 ( n; k )( s ) k q (28) and satisfy the following two dierence equations (here, as b efore, x ( s ) = q 2 s ) ( q 2 s 1) 5 ( s n ) q 5 x ( s ) q n +1 [ n ] q ( s n ) q = 0 (29) and ( q 2 s 2 n +2 1) 4 ( s n ) q 4 x ( s ) q n +1 [ n ] q ( s n ) q = 0 : (30) Since ( s n ) q is a p olynomial in x ( s ) = q 2 s , it can b e represented as a linear combination of the p olynomials Q m ( x ), the q-p olynomials in the exp onential lattice. In particular ( s n ) q = n X m =0 C m ( n ) Q m ( x ) : (31) Let us to obtain the recurrence relation for the connection co ecients C m ( n ) b etween the ( s n ) q and the q-Charlier, q-Meixner or q-Kravchuk. (For q-Hahn p olynomials we will consider it separately). In order to do that we apply the op erator ~ L = ( q 2 s 1) 5 5 x ( s ) q n +1 [ n ] q (32) to b oth sides of (31). Using formula (29) ( ~ L ( s n ) q = 0) and multiplying by q 2 s we obtain the following expression 0 = n X m =0 C m ( n ) q 2 s ( q 2 s 1) 5 Q m ( x ) 5 x ( s ) q m +1 [ m ] q q 2 s Q m ( x ) : (33) Taking into account that for q-Charlier, q-Meixner and q-Kravchuk the ( s ) function in (2) coincide with q 2 s ( q 2 s 1) and applying the structure relation (14) and the TTRR (7) to the previous expression we nd 0 = n X m =0 C m ( n ) f A m Q m +1 ( x ) + B m Q m ( x ) + m Q m 1 ( x ) g ; from where we obtain the following TTRR for the connection co ecients C m ( n ) A m 1 C m 1 ( n ) + B m C m ( n ) + m +1 C m +1 ( n ) = 0 ; (34) 9