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Journal of Computational and Applied Mathematics 99 (1998) 143–154

Laguerre–Freud equations for the recurrence coecients of D!
semi-classical orthogonal polynomials of class one
M. Foupouagnignia;∗ , M.N. Hounkonnoua , A. Ronveauxb
b

a
Institut de Mathematiques et de Sciences Physiques, B.P. 613 Porto-Novo, Benin
Mathematical Physics, Facultes Universitaires Notre-Dame de la Paix, Rue de Bruxelles 61,
B-5000 Namur, Belgium

Received 30 October 1997; received in revised form 15 May 1998

Abstract
The Laguerre–Freud equations giving the recurrence coecients n ,
n of orthogonal polynomials with respect to a D!
semi-classical linear form are derived. D! is the dierence operator. The limit when ! → 0 are also investigated recovering
c 1998 Elsevier Science
known results. Applications to generalized Meixner polynomials of class one are also treated.
B.V. All rights reserved.

AMS classication: 33C25
Keywords: Laguerre–Freud equations; Semi-classical orthogonal polynomials

1. Introduction
Let L be a regular linear form in the vector space of all complex polynomials of one variable.
By ‘regular linear form L’ [13] we mean that there exists (Pn )n¿0 a sequence of monic orthogonal
polynomials with respect to L, i.e.,
degree of Pn = n;
hL; Pn Pm i = 0;

n ¿ 0;
n 6= m;

hL; Pn Pn i =
6 0;

n ¿ 0;

∗

(1)

Corresponding author. E-mail: foupouag@syfed.bj.refer.org. Research supported by: Deutscher Akademischer
Austauschdienst (DAAD), Kennedy Allee 50, 53175 Bonn, R.F.A.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 5 2 - 6

144

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

where hL; Pi denotes the value of the linear form L applied to P. The three-term recurrence relation
satised by the monic family (Pn )n¿0 will be written as
Pn+1 (x) = (x − n )Pn (x) −
n Pn−1 (x);
P0 (x) = 1;

P1 (x) = x − 0 ;

n ¿ 1;

n 6= 0; n ¿ 1:

(2)

The dierence operator D! is dened by:
D! P(x) =

P(x + !) − P(x)
;
!

! 6= 0; D1 =
; D−1 = 3:

Denition 1 ([6, 12, 13]). The linear form L is said to be D! semi-classical if L is regular and
there exist two polynomials of degree ¿ 1 and  such that
D! (L) = L;

(3)

where
h L; Pi = hL; Pi;

hD! L; Pi = −hL; D−! Pi:

(4)

Moreover, if L is D! semi-classical, the class of L, denoted cl(L), is dened as [13]
cl(L) = min{max(−2 + degree of ; −1 + degree of

)};

where the minimum is taken over all pairs of polynomials , and

(5)
of degree at least one, satisfying

D! (L) = L:
The following characterization of the class of the D! semi-classical functionals [9, 13] follows from

denition 1.
If L is a D! semi-classical functional satisfying (3), then L is said to be of class s if and only
if for any root c of the polynomial , one of these two conditions is satised:
(i) rc−! 6= 0,
(ii) hL; c−! i =
6 0,
where
s = max{−2 + degree of ; −1 + degree of
(x) = (x − c)c (x);

};
(6)

(x) − c (x) = (x − c + !)c−! (x) + rc−! :
The two coupled algebraic equations giving n and
n are called the Laguerre–Freud equations,
introduced for the rst time in [2]. In [1], Belmehdi has given them explicitly for D(= d=dx) semiclassical polynomials of class s = 1. Let us recall that when  is of degree maximum 2 and
of
degree 1, the corresponding polynomials are called (discrete) classical and of course the n ;
n are

well known [4, 10, 14].
The aim of this work is to give the Laguerre–Freud equations when the operator D = d=dx is
replaced by the dierence operator D! [4, 13]. D! semi-classical orthogonal polynomials appear
already in [13], but n and
n have been computed only in the classical case.

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

145

We suppose that the linear form L is semi-classical and we present here the derivation of the
Laguerre–Freud equations for the class s = 1 which means that polynomials  and are restricted to
(x) = a0 + a1 x + a2 x2 ;

(x) = b0 + b1 x + b2 x2 + b3 x3 :

(7)

The Laguerre–Freud equations are deduced like in [2] from the two obvious relations deduced
now from

D! (L) = L;

(8)

hD! (L); Pn Pn i = hL; Pn Pn i;

(9)

hD! (L); Pn Pn+1 i = hL; Pn Pn+1 i:

(10)

If the rules (4) are the same for D and D! , then the main dierences (and diculties) come from
the more complicated product rule:
D! [P(x)Q(x)] = P(x + !)D! Q(x) + Q(x)D! P(x);

(11)

introducing shifted polynomial P(x + !).
Rules (4) and (11) transform the Eqs. (9) and (10) into

hL; Pn (x − !)D−! Pn (x)i + hL; Pn D−! Pn i = −hL; Pn Pn i;

(12)

hL; Pn (x − !)D−! Pn+1 (x)i + hL; Pn+1 D−! Pn i = −hL; Pn Pn+1 i:

(13)

2. Basic iteration scheme
In order to express the coecients of the orthogonal polynomial Pn in terms of the recurrence
coecients n and
n , let us compute them recursively from the iteration process given by the
following lemma.
Lemma 2. All basic coecients Tn; i in the expansion of
Pn (x) =

n
X

Tn; i x n−i

(14)

i=0

can be computed recursively from the relations:
T1; 1 = − 0 ;
Tn; 0 = 1;

n ¿ 0;

Tn+1; 1 = Tn; 1 − n ;

(15)
n ¿ 1;

(16)

Tn+1; j = Tn; j − n Tn; j−1 −
n Tn−1; j−2 ;

Tn+1; n+1 = − n Tn; n −
n Tn−1; n−1 ;

2 6 j 6 n;

n ¿ 1:

(17)
(18)

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M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

These basic relations are easily proved by identication from:
xPn (x) =

n
X

Tn; i x n+1−i =

i=0

n+1
X

Tn+1; i x n+1−i + n

n
X

Tn; i x n−i +
n

Tn−1; i x n−1−i :

i=0

i=0

i=0

n−1
X

Use of Eqs. (15)–(17) gives
Tn+1; 1 = −

n
X

i ;

n ¿ 0;

(19)

i=0

X

Tn+1; 2 =

i j −

06i¡j6n

Tn+1; 3 = −

n
X

i ;

X

i j k +

n
X

i−1
i ;

X

(
i j + i
j ) + 0

16i¡j6n

06i¡j¡k6n

−

n ¿ 1;

(20)

i=1

n ¿ 2:

n
X

i

i=1

(21)

i=1

All other terms can be computed in the same way but for class s = 1, only these three terms will
be used.
Let us emphasize that the two terms: Tn; 1 and Tn; 2 are already given in [4]; the computation of the
higher order coecients allows to generate Laguerre–Freud equations for any arbitrary class s¿1.
These coecients play the role (but in a simpler way) of the Turan determinants introduced in [2]
showing the interest in Laguerre–Freud equations.

3. Intermediate coecients
The structure constants [1, 13] will rst be obtained in terms of the Tn; i and the coecients of
polynomials  and dening the linear form L.
In order to do that, we rst need to expand the polynomials Pn (x + !) and D! Pn in terms of the
Tn; i and also to control the action of the linear form L on polynomial x n+k Pn via the coecients
Tn; i .
3.1. Coecients An; i and A∗n; i
These coecients appear in the following expansions:
Pn (x + !) =

n
X

An; i (!)x n−i ;

(22)

i=0

D! Pn (x) =

n
X
i=1

A∗n; i (!)x n−i :

(23)

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

147

Both coecients are related to Tn; i via the relation
A∗n; i (!) =

An; i (!) − Tn; i
;
!

(24)

and are explicitly given by
An; i (!) =


i 
X
n−k
k=0

A∗n; i (!) =

i−k


i−1 
X
n−k
k=0

i−k

!i−k Tn; k ;
!i−k−1 Tn; k ;

0 6 i 6 n;
1 6 i 6 n:

(25)
(26)

3.2. Coecients Bnk
The coecients Bnk appear from the action of the linear form L on x n+k Pn (x):
Bnk = hL; x n+k Pn i:

(27)

Expansion of x n+k Pn (x) in terms of the Tn; i and use of the basic norm:
I0; n = hL; Pn Pn i;

(28)

taking care that
Bn0 = hL; x n Pn i = hL; Pn Pn i = I0; n ;

(29)

and using the orthogonality property give the relations easily derived:
Bn1 = −Tn+1; 1 I0; n ;

(30)

Bn2 = (Tn+1; 1 Tn+2; 1 − Tn+2; 2 )I0; n ;

(31)

Bn3

(32)

= [Tn+1; 1 (Tn+3; 2 − Tn+2; 1 Tn+3; 1 ) + Tn+3; 1 Tn+2; 2 − Tn+3; 3 ]I0; n ;

Bnk = −

k
X

Tn+k; i Bnk−i :

(33)

i=1

All other coecients Bnk can be computed in the same way.
The connection between the Bnk and the coecients Cj;kk introduced in [2]:
x n+k Pn (x) =

2n+k
X

Cj;n+k
n Pj (x);

(34)

j=0

is obviously
Bnk = C0;n+k
n I0; 0 :

(35)

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M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

3.3. Structure relation
As any semi-classical orthogonal family, Pn satises a structure relation [13] which for the class
s = 1, and taking into account Denition 1, reduces to
n+2
X

(x)D−! Pn (x) =

n; i (!)Pi (x):

(36)

i=n−2

The aim is again to represent the structure constants n; i in terms of the Tn; i . Using the orthogonality
rules, Eqs. (28) and (36), we obtain:
n; i (!)I0; i = hL; Pi D−! Pn i:

(37)

Use of the orthogonality rules in Eq. (37) gives
n; i (!)I0; i =

*

L; Pi (x)(x)

"n−i+3
X

#+

A∗n; k (−!)x n−k

k=1

(38)

:

Using Eqs. (7), (19)–(21), (26), (27), (30)–(32) and (38), we compute the structure relation constants n; n+2 , n; n+1 , n; n , n; n−1 as
n; n+2 (!) = nb3 ;

(39)
"

n; n+1 (!) = nb2 + −
"

n; n (!) = nb1 + −
+
+
n; n−1 (!) =

( 

n
3

(

nn2

 

n
2

 

n
2

2

! −

! + nn +

n−1
X

#

 

n
2

!n − (n − 1)!

(i ) + 2

"

+ 3

n−1
X

n−1
X

i2

n
2



−! 
"

n−1
X

)

i b3

i + 2

i=0

n−1
X
i=1

i (i−1 + i ) +
n

n−1
X

"

!b1 + −(n − 1)!

n−1
2



i + n(n−1 + n )
n b3

n−1
X

i +

i=0

i j + (n − 1)

!2

(41)

i b3 ;

#

i=0

X

)

i + n
n b2

i=1

06i¡j6n−1

+

+ n

#

i=1

 

n−1
X
i=0

n−1
X

(40)

i b3 ;

i b2

i=0

"

#

i=0

+ n(
n +
n+1 ) +

i=0

−

! + n(n + n+1 ) +

i=0

n−1
X

n−1
X

n−1
X

 

i=0

i −

 

n
4

!

2

#

b2

i2 + (2n − 3)

i=0

n−1
X

n
3

n−1
X
i=1

#

! 3 b3 :

i +

 

n
2



n  b3
(42)

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

149

The aim of the following is to avoid the computation of Tn; 4 in order to have simple expression for
n; n−2 and n; n−1 .
One easily shows, using the derivative rule:
Pi (x)D−! Pn (x) = D−! [Pi (x + !)Pn (x)] − Pn (x)D! Pi (x):

(43)

Using Eq. (37) and taking into account Eqs. (4), (8) and (43), we obtain:
n; i (!)I0; i = −h (x)L; Pi (x + !)Pn (x)i − h(x)L; Pn (x)D! Pi (x)i:

(44)

Using Eqs. (7), (19)–(21), (26), (27), (30)–(32) and (44), we compute the structure relation constants n; n−1 ; n; n−2 as
n; n−1 (!) = −a1
n −
n [(n − 1)! + n−1 + n ]a2 − (n − 1)
n b2
"

− (n − 2)(n−1 + n ) +

n
X

i +

i=0

n; n−2 (!) = −[a2 + (n − 2)b3 ]
n−1
n :



n−1
2

 #

!
n b3 ;

(45)
(46)

This representation of the structure relation transforms Eqs. (12) and (13), respectively, into
n; n−2 (!)hL; Pn (x − !)Pn−2 (x)i + n; n−1 (!)hL; Pn (x − !)Pn−1 (x)i
+2n; n (!)I0; n = −h L; Pn Pn i;

(47)

n+1; n−1 (!)hL; Pn (x − !)Pn−1 (x)i + n+1; n (!)I0; n + n; n+1 (!)
n+1 I0; n
= − h L; Pn Pn+1 i:

(48)

The right-hand side of each equation containing
already given in [2], i.e.:

is the same as in the continuous case s = 1,

h L; Pn Pn i = (n )I0; n + a2 (
n +
n+1 )I0; n ;

(49)

h L; Pn Pn+1 i = [a1 + a2 (n + n+1 )]
n+1 I0; n :

(50)

4. Final form of the Laguerre–Freud equations
The D! term in both Laguerre–Freud equations in the form (12) and (13) are now eliminated and
replaced by n; i Pi terms. The remaining quantities to be computed are therefore
hL; Pn (x − !)Pi (x)i =

*

L; Pi

n−i
X
k=0

An; k (−!)x

n−k

+

;

all quantities and all constants being written in terms of the Tn; i .

(51)

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M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

Replacing the structure constants n; i (n − 2 6 i 6 n) given by Eqs. (41), (45) and (46), Eq. (47),
and using also Eqs. (25)–(27) and (49)–(51), the rst Laguerre–Freud equation reads
(n ) + 4b3

n−1
X

i + 2

i=1

+

 

n
2

n−1
X

n (i ) + !

i=0

n−1
X

n (i ) + 2

i=0

 

n
! 2 b3
3

!2 a2 = −(a2 + 2nb3 )(
n +
n+1 );

(52)

with
a (x) =

(x) − (a)
:
x−a

Replacing the structure constants n+1; n−1 , n+1; n and n; n+1 given by Eqs. (40), (42) and (46), Eq.
(48), and using also Eqs. (25)–(27) and (49)–(51), the second Laguerre–Freud equation reduces to
n
X

"

(i ) + (2n + 1)
n+1 + 2

i=0

n
X

+ 3b3

n
X

#

i b2

i=1

i (i−1 + i ) + 2
n+1 nn +

i=1

−!

X



n+1
2

i j + n

06i¡j6n

+

" 

n
2

!

2

n
X



"

!b1 + −n!

n
X

i i + (2n − 1)

i −

i=0

n+1
4

i +

i=0

i=0



n
X

n
X
i=1



!

!

i b 3

i=0

+ n!a2
n+1 −


n
X

3

#



n+1
3



!

2

#

b2



i + n
n+1  b3

b3 + [a1 + a2 n ]
n+1

= −[a2 + (2n + 1)b3 ]n+1
n+1 :

(53)

Let us emphasize that we can also obtain the second Laguerre–Freud equation by identication of
the two expressions of n; n−1 given by Eqs. (42) and (45).
The rst equation gives
n+1 linearly in terms of
j and j (1 6 j 6 n).The second equation is
used in order to compute n+1 from the previous
n+1 via the two non linear terms
(2n + 1)
n+1 n+1 b3

and

−
n+1 n+1 a2 :

The nonlinearities in the second equation only exemplify the fundamental barrier between semiclassical and classical situation in which both a2 and b3 are zero. From the nonlinearities, both
relations must therefore be used simultaneously starting with
0 =

M1
;
M0

a2
1 = − (0 );

(54)

M0 and M1 are the moments of order 0 and of 1 of the linear form L. In the classical case, the
equations can be decoupled [13].

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

151

5. Limiting situation and particular cases
1. The starting D! Laguerre–Freud equations reduce to the D Laguerre–Freud equations already
considered in [2] as ! tends to 0. The D! Laguerre–Freud equations (52) and (53) reduce exactly
to the equations given in [2] relabelling the coecients of  and in a proper way:
(n ) + 4b3

n−1
X

i + 2

i=1

n
X

(i ) + 3b3

n−1
X

n (i ) = −(a2 + 2nb3 )(
n +
n+1 );

i=0

n
X

"

i (i−1 + i ) + (2n + 1)
n+1 + 2

i=1

i=0

+2
n+1 nn +

n
X

n
X
i=1

!

#

i b2

i b3 + [a1 + a2 n ]
n+1

i=0

= −[a2 + (2n + 1)b3 ]n+1
n+1 :

(55)

The ! dependence is of order 2 in the rst equation (52) and of order 3 in the second one (53),
and does not aect the nonlinear term
n+1 n+1 already mentioned.
2. The Laguerre–Freud equations obtained in (52) and (53) contain obviously the classical cases
when a2 = b3 = 0. In the other cases, let us use the notation of [13] so that we can compare more
easily with the results in [13]:
(x) = ax2 + bx + c

and

(x) = px + q:

(56)

Eqs. (52) and (53) reduce to
(n ) + 2a

n−1
X

i + 2nb + 2nan = −n!p;

(57)

i=0

n
X

"

(i ) + (2n + 1)
n+1 + 2

i=0

"

+ −n!

n
X
i=0

n
X
i=1

i +



n+1
3



!

2

#

#

i a −





n+1
!b
2

a = −p
n+1 :

(58)

Rewriting the second equation with n → n − 1 and subtracting we get:
(n ) + [p + (2n + 1)a]
n+1 − [p + (2n − 3)a]
n
−n!b − an!n − a!

n−1
X
i=0

i + a!

2

 

n
= 0:
2

(59)

Using symbolic computation with Maple V.4 [3], we have checked positively that for classical discrete orthogonal polynomials, coecients n and
n , given explicitely in terms of polynomials  and
appearing in the discrete Pearson equation [7, 8], are solutions of Eqs. (57) and (58) (with ! = 1).
Eqs. (57) and (59) are exactly the equations derived in the thesis [13] taking into account the D!
derivative of the linear form given by Eq. (4) and the one used in [13]. Let us remark, however,

152

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

that in [13] the
n equation is obtained using the so-called D! representation, expanding a classical
orthogonal polynomials Pn as a sum of (maximum three) D! Pi (i = n + 1; n; n − 1). This technique
cannot be extended to the class 1, because of the nonexistence of such a representation for semiclassical orthogonal polynomials of class s¿0.
6. Applications
6.1. Generalized Meixner
These polynomials with q parameters were introduced in [11] in order to show the quasiorthogonality character of the D! derivative (with ! = 1). The weight  is given by
q
i Y
(i) =
(i + j );
(i!) q j=1

(0¡¡1; j ¿0); i = 0; 1; 2; : : : :

(60)

These polynomials are denoted by mn(; ) where  = (1 ; : : : ; q ), which reduce of course to the
well-known classical Meixner polynomials when  is the scalar  (q = 1).
When q = 2; 1 6= 1 and 2 6= 1; this family is semi-classical of class s = 1 with
(x) = x2

and

(x) = ( − 1)x2 + (1 + 2 )x + 1 2 :

(61)

Of course when 1 = 2 = 1, the class reduces to 0 and the polynomials, a particular case of the
Meixner polynomials, are called discrete Laguerre polynomials [4]:
lan (x) = mn(1; ) (x):

(62)

We have checked, positively, the Laguerre–Freud equations when ! → 1 with the known n ;
n of
the discrete Laguerre polynomials. Let us also emphasize that for q = 2 and for arbitrary positive
1 and 2 , the weight given by Eq. (60), is not a polynomial modication of the Meixner weight,
except when 1 or 2 is an integer.
Replacing in Eqs. (52) and (53) ! by one and polynomials  and given by Eq. (61), we obtain
Laguerre–Freud equations for generalized Meixner polynomial of class s = 1:
(1 − )(
n +
n+1 ) = ( − 1)

 

n
2

+

n2



+ ((1 + )n

+ (1 + 2 ))n + (1 + )

n−1
X

i

i=0

+ (1 + 2 ) n + 1 2 ;
(1 − )(n + n+1 )
n+1 = −n

n
X

(63)

i + ((1 + )n + (1 + 2 ) + 1)
n+1

i=0

+



n+1
3



+

n
X
i=0

i2 + 2

n
X
i=1

i ;

(64)

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

with initial values
M1 1 2 2 F1 (1 + 1 ; 1 + 2 ; 2; )
0 =
=
;
M0
2 F1 (1 ; 2 ; 1; )

1 =

(0 )
:
1−

153

(65)

6.2. Concluding remarks
(i) The polynomials have been computed for generalized Meixner polynomial of class one up
to n = 10 from the n ;
n generated by the Laguerre–Freud equations given above and also from
the Hankel representation of polynomials which requires the computation of the moments Mj up to
j = 19. These moments were computed from the moment recurrence relation for generalized Meixner
polynomial:
(1 − )Mk+2 = 1 2 Mk + (1 + 2 )Mk+1 −

k
X
j=1

(−1) j

 

k
j

Mk+2−j :

(66)

The polynomial coecients in both approaches are written in terms of M0 and M1 using the initial
values of the Laguerre–Freud recurrence given by Eqs. (65). Polynomials obtained in these two
ways coincide of course and the Laguerre–Freud approach is obviously more ecient.
(ii) Using Eqs. (63) and (64), we have also computed numerically with Maple V.4, coecients
n and
n up to n = 100 000, for several values of coecients 1 ; 2 and .
The result of the plot for all cases indicates that the sequences
n =n2 , n =n are convergent. Assuming
that they converge, their limits, a() and b(), are obtained using Maple V.4 and Eqs. (63), (64)
with the approximations:
n ∼
= a() n2 and n ∼
= b()n, for n large. We obtain
a() =


;
(1 − )2

b() =

1+
:
1−

(67)

By the same way using numerical and symbolic computation with Maple V.4 and analysis of Eqs.
(63) and (64) [5], the following asymptotic behaviour is observed for coecients n and
n .
6.3. Conjecture [5]
The coecients of the three-terms recurrence relation of the generalized Meixner polynomial of
class one are given by
1+
(1 + 2 − 1)
n+
+ (1 − 1) (2 − 1)U (n);
1−
1−
 (n + 1 − 1)(n + 2 − 1)
n =
− (1 − 1) (2 − 1)V (n);
(1 − )2

n =

(68)

where U (n) and V (n) are two positive sequences converging to zero.
Acknowledgements
Part of this work was done when one of the authors (M. F.) was visiting the Konrad–Zuse–
Zentrum fur Informationstechnik Berlin (Z.I.B.). M. F. wishes to acknowledge Professor Wolfram

154

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 99 (1998) 143–154

Koepf and Boing Harald for their warm hospitality and their help in computing aspect of this work,
Professors Francisco Marcellan and Walter Van Assche for their careful reading of the manuscript
and their useful comments. M. F. also thanks the Institute Z.I.B. and Professor Antonio J. Duran for
partially nancing his participation to the meeting ‘VIII Simposium Sobre Polinomios Ortogonales
y Aplicaciones’ (Sevilla, September 1997) where this paper was presented.
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