1 Connection problem

c 1997 M. Alfaro et al. (Eds.) Proceedings of the

International Workshop on Orthogonal Polynomials in Mathematical Physics. Leganes, 24-26 June, 1996.

ORTHOGONAL POLYNOMIALS: CONNECTION AND LINEARIZATION COEFFICIENTS A. Ronveaux Abstract 1 Let fPn (x)g1 n=0 and fQm (x)gm=0 be two families of orthogonal polynomials. The linearization problem involves only one family via the relation: i+j X

Pi (x) Pj (x) =

k=ji?j j

Lijk Pk (x)

and the connection problem mixes both families:

Pn (x) =

n X

m=0

Cm (n) Qm(x):

In many cases, it is possible to build a recurrence relation involving only m , satis ed by the linearization coecients Lijk and connection coecients Cm (n) , in particular if the families Pn (x) and Qm (x) are classical: continuous or discrete. We intend to examine these coecients from the point of view of recurrence properties, emphasizing mainly the structure of these recurrence relations. This article contains some results already published, in print or in preparation, by A. Ronveaux, A. Zarzo, E. Godoy, I. Area, R. A lvarez-Nodarse, S. Belmehdi, N. M. Hounkonnou and S. Lewanowicz.

1 Connection problem. 1 Let fPn (x)g1 n=0 and fQm (x)gm=0 be two sequences of polynomials of degree exactly equal to n. The so-called Connection Problem asks to nd the coecients Cm (n) in the expansion:

Pn (x) =

n X

m=0

Cm (n) Qm(x) ;

(1)

and goes back to Stirling [35] as pointed out by Askey in his 1975 survey [4]. Let us represent the polynomials Pn (x) and Qm (x) in the monomial basis fxk g1 k=0 , and analyze the structure of the connection coecients Cm (n) as depending on several assumptions on families Pn (x) and Qm (x)

Pn(x) =

n X

k=0

an;k xk ;

Qm(x) =

m X

k=0

am;k xk ;

an;k ; an;k 2 IR :

(2)

Other bases like x[n] , (x)n or Bernstein basis can also be used, and polynomials Pn (x) and Qm (x) can be de ned by some properties or equations. So let us use in this article the following de nitions:

1. Falling factorial or Stirling polynomial [1] n P m n m x = x(x ? 1)


(x ? n + 1) = Sn x , where Snm are Stirling numbers of rst kind [ ]

[35].

(

)

m=0

131

(

)

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2. Rising factorial or Pochhammer symbols [1] (x)n = x(x + 1)


(x + n ? 1), (x) = 1 . 3. Appell family fPn (x)g [7] 0

Pn0 (x) = (n)Pn?1 (x)

4. Orthogonal family fPn (x)g [7, 8]

Monic polynomial Pn (x) of degree n generated by the three-term Recurrence Relation (R.R.)

n0;

xPn (x) = Pn+1 (x) + nPn(x) + n Pn?1 (x) ; P?1 (x) = 0 ; P0 (x) = 1 ;

n 6= 0 ; n  0 :

5. Classical orthogonal polynomials (continuous) [25, 26]

Polynomial solution Pn (x) of degree n of the second order Dierential Equation (D.E.):

L ;n[Pn (x)]  Pn00 + Pn0 ? n 0n? = Pn = 0 ; 2

(

1) 2

r =  + r0 ,  = (x) polynomial of degree 2: Jacobi (J), Bessel (B); of degree 1: Laguerre (L); of degree 0: Hermite (He), and    (x) polynomial of rst degree.

6. Classical orthogonal polynomial (discrete) [25, 26]

Polynomial solution Pn (x) of degree n of the second order Dierence Equation

D ;n[Pn(x)]  
rPn + 
Pn ? n 0n? = Pn = 0 2

(

1) 2

  (x) polynomial of degree 2: Hahn (H) and Hahn-Eberlein (H-E), and polynomial of degree 1: Krawtchouk (K), Meixner (M), Charlier (C).  =  (x) polynomial of rst degree. (
f (x) = f (x + 1) ? f (x) ; rf (x) =
f (x ? 1))

7. Semi-classical orthogonal family (continuous and discrete) [11, 24]

Family of orthogonal polynomial Pn (x) which satis es a structure relation (S.R.)

(x)LPn (x) =

n+X t?1 k=n?s?1

k;nPk (x)

(k;n constants)

where L = d=dx (continuous) or L =
(discrete). For t = 0; 1; 2 and s = 0 semi-classical orthogonal polynomials (O.P.) are classical O.P.

8. Hildebrandt classical polynomial (continuous) [12]

Polynomial solution Pn (x; k) of degree n of the Dierential Equation:

(x)Pn00 (x; k) + k?n(x)Pn0 (x; k) ? nk0 ?n=2?1=2 Pn(x; k) = 0 : These polynomials: H-J, H-B, H-L, H-He, reduce to the family #5. (orthogonal) when k = n , and to the family Pn(r) (x) , when k ? n = r ( Pn(r) (x) , r-th derivative of Pn(x) ; [Pn+r (x; n + r)](r) = Pn (x; n + r) ). For xed k , Pn (x; k) is an Appell family.

9. Bernstein Basis (continuous) [34] Bin (x) =

!

n n xi(1 ? x)n?i = X (?1)j ?i nj i j =i

!

!

j xj ; i

0in:

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Proceedings of the IWOP'96

10. Bernstein Basis (discrete) [34] bni(N; x) =

!

n x[i] (N ? x)[n?i] ; 0  i  n  N i N [i]

The bases Bin(x) and bni (N; x) are of degree n ( 0  i  n  N ) for all i , contrary to the previous polynomials Pn (x) and Qn (x) of degree equal to n . The vectors P~nt = (P0 ;


; Pn ) and Q~ tn = (Q0 ;


; Qn ) de ne now a lower triangular connection matrix [Cn;m ] P~n = [Cn;m ]Q~ n with Cn;m = Cm(n) (3) which has a speci c structure (see Table 1) following the assumptions #1 to #10. In general, we work with monic polynomials ( an;n = an;n = 1 ). In that case, the set of all connection matrix of size n +1 by n +1 forms a multiplicative group called T (n +1; IR) , subgroup of SL(n + 1; IR) . Let us emphasize the structure of the connection coecients in two following cases: i) assumption #4 (families Pn and Qn both orthogonal) and ii) assumption #7 (families Pn and Qn both semi-classical). It is important to realize rst (and to prove easily by derivation of the S.R.) that the existence of a S.R. for the family Pn (x) implies that Pn (x) is solution of a second order dierential (or dierence) equation:

L ;n[Pn(x)] = 0 ;

( D2;n [P (x)] = 0 ): (4) These operators L2;n and D2;n are of course more general than the operators considered in the assumptions #5 and #6. 2

1 i) fPn (x)g1 n=0 and fQm (x)gm=0 are both orthogonal families. (assumption #4). Both families of polynomials involved in the connection problem (1) satisfy a three-term recurrence relation which, considering monic polynomials without loss of generality, are respectively:

x Pn (x) P?1 (x) x Qm (x) Q?1 (x)

= = = =

Pn+1 (x) + n Pn (x) + n Pn?1 (x) ; n  0 ; 0; P0 (x) = 1 ; Qm+1 (x) + m Qm(x) + m Qm?1 (x) ; m  0 ; 0; Q0 (x) = 1 ;

(5) (6)

where n 6= 0 and m 6= 0 . Then, multiplication of (1) by x , the use of (5) and (6) and expansion of Pn+1 , Pn and Pn?1 in terms of Qm by means of (1), generates the \cross-rule": ( m ? n ) Cm (n) + Cm?1 (n) + m+1 Cm+1 (n) ? Cm (n + 1) ? n Cm (n ? 1) = 0

(7)

which is valid for m = 0; 1;


; n + 1 , with the convention Ck (j ) = 0 when k < 0 or k > j . Notice that this \cross-rule" mixes the three adjacent coecients ( m?1; m; m+1 ) of the row n , crossing at the (m; n) position the column m of the aforementioned [Cn;m ]{matrix. Any coecient Cm (n) can be computed from this \cross-rule" starting from the initial condition: Cn (n) = 1 (monic polynomials), Cn?1 (n) = an ? an and Cn?2 (n) = bn ? bn ? (an ? an )an?1 , where an and bn (respectively an and bn ) are the rst coecients in the expansions of Pn (x) and Qn (x) , an = aa ?1 , bn = aa ?2 . n;n

n;n

n;n

n;n

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Pn(x) =

n X m=0

Cm(n)Qm (x)

Pn(x)

Qm (x)

arbitrary

arbitrary

Connection matrix [Cn;m ] = [Cm (n)] no structure (except Cn(n) 6= 0 ) hook structure

xn

x m] [

sn(m) = sn(m??1 1) + msn(m?)1

sn(m) = Stirling numbers of second kind hook structure

xn

xm

[ ]

sn(m) = sn(m??1 1) ? (n ? 1)sn(m?)1 sn(m) = Stirling numbers of rst kind

8 > >
> :

Appell

Cm+1 (n + 1) = k(m; n)Cm (n)

orthogonal

orthogonal

cross structure

semi-classical O.P.

semi-classical O.P.

row structure

xn

semi-classical O.P.

row structure

Hildebrandt classical

semi-classical O.P.

row structure

Lr;n[Pn (x)] = 0 > > : Dr;n [Pn (x)] = 0

semi-classical O.P.

row structure

primitive of semi-classical

semi-classical O.P.

row structure

Bernstein basis

no structure

(continuous or discrete)

(full square matrix)

Appell

8 > >