Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol101.Issue1-2.1999:

Journal of Computational and Applied Mathematics 101 (1999) 231–236

Fourth order q-dierence equation for the rst associated of the
q-classical orthogonal polynomials
M. Foupouagnigni a; 1 , A. Ronveaux b , W. Koepf c; ∗
a

b

Institut de Mathematiques et de Sciences Physiques. B.P. 613 Porto-Novo, Benin
Mathematical Physics, Facultes Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium
c
HTWK Leipzig, Fachbereich IMN, Postfach 66, D-04251 Leipzig, Germany
Received 8 June 1998

Abstract
We derive the fourth-order q-dierence equation satised by the rst associated of the q-classical orthogonal polynomials.
The coecients of this equation are given in terms of the polynomials  and  which appear in the q-Pearson dierence
c 1999
equation Dq ( ) =  dening the weight  of the q-classical orthogonal polynomials inside the q-Hahn tableau.
Elsevier Science B.V. All rights reserved.

1991 MSC: 33C25
Keywords: q-Orthogonal polynomials; Fourth-order q-dierence equation

1. Introduction

The fourth-order dierence equation for the associated polynomials of all classical discrete polynomials were given for all integers r (order of association) in [5], using the properties of the Stieltjes
functions of the associated linear forms.
On the other hand, the equation for the rst associated (r = 1) of all classical discrete polynomials
was obtained in [13] using a useful relation proved in [2]. In this work, mimicking the approach
used in [13] we give a single fourth-order q-dierence equation which is valid for the rst associated
of all q-classical orthogonal polynomials. This equation is important for some connection coecient
problems [10], and also in order to represent nite modications inside the Jacobi matrices of the
q-classical starting family [14]. q-classical orthogonal polynomials involved in this work belong to
∗
1

Corresponding author. E-mail: koepf@imn.htwk-leipzig.de.
Research supported by: Deutscher Akademischer Austauschdienst (DAAD).

c 1999 Elsevier Science B.V. All rights reserved.

0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 2 5 - 8

232

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 101 (1999) 231–236

the q-Hahn class as introduced by Hahn [8]. They are represented by the basic hypergeometric series
appearing at the level 3 2 and not at the level 4 3 of the Askey–Wilson orthogonal polynomials.
The orthogonality weight  (dened in the interval I ) for q-classical orthogonal polynomials is
dened by a Pearson-type q-dierence equation
Dq () = ;

(1)

where the q-dierence operator Dq is dened [8] by
Dq f(x) =

f(qx) − f(x)
;

(q − 1)x

x 6= 0; 0 ¡q ¡ 1;

(2)

and Dq f(0) := f′ (0) by continuity, provided that f′ (0) exists.  is a polynomial of degree at most
two and  is polynomial of degree one.
The monic polynomials Pn (x; q), orthogonal with respect to , satisfy the second-order q-dierence
equation
Q2; n [y(x)] ≡ [(x)Dq D1=q + (x)Dq + q; n Id ]y(x) = 0;

(3)

an equation which can be written in the q-shifted form
[(1 + 1 t1 )Tq 2 − ((1 + q)1 + 1 t1 − q; n t12 )Tq + q1 Id ]y(x) = 0;

(4)

′′

1 − qn
; [n]q =
= −[n]q  + [n − 1] 1
;
1−q
q 2q
≡ (qi x); i ≡ (qi x); ti ≡ t (qi x); t (x) = (q − 1)x

(5)

with
q; n
i





′

and the geometric shift Tq dened by
Tq i f(x) = f(qi x);

Tq 0 ≡ Id ( ≡ identity operator):

(6)

(1)
2. Fourth-order q-dierence equation for the rst associated Pn−1
(x; q) of the q-classical
orthogonal polynomial
(1)
The rst associated of Pn−1 (x; q) is a monic polynomial of degree n − 1, denoted by Pn−1
(x; q),
and dened by
(1)
Pn−1
(x ; q ) =

1

0

Z

I

Pn (s; q) − Pn (x; q)
(s) dq s;
s−x

(7)

R

where
0 is given by
0 = I (s) dq s and the q-integral is dened in [7].
The polynomials Pn (x; q) ≡ Pn(0) (x; q) and Pn(1) (x; q) satisfy also the following three-term recurrence
relation [4] for r = 0 and r = 1, respectively,
(r)

(r)
Pn+1
(x; q) = (x − n+r )Pn(r) (x; q) −
n+r Pn−1
(x; q);

P0(r) (x; q) = 1;

P1(r) (x; q) = x − r :

n ¿ 1;

(8)

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 101 (1999) 231–236

233

Relation (7) can be written as
(1)

Pn−1
(x; q) = (x)Qn (x; q) − Pn (x; q)(x)Q0 (x; q);

(9)

where
1
Qn (x; q) =
0 (x)

Z

I

Pn (s; q)
(s) dq s:
s−x

It is well-known [15] that Qn (x; q) also satises Eq. (3); hence by (9)
" (1)

#
Pn−1 (x; q)
Q2; n
+ Pn (x; q)Q0 (x; q) = 0:
(x)

(10)

In a rst step, we eliminate (x) and Q0 (x; q) in Eq. (10) using Eqs. (1) and (3) for Pn (x; q). This
can be easily carried out using a computer algebra system — we used Maple V Release 4 [3] —
and gives the relation
h
i 

( +  t )Q∗
P (1) (x; q) = eT + fI P (x; q);
(11)
1

1 1

q

n−1

2; n−1

d

n

with

Q2;∗ n−1 = 2 Tq 2 − ((1 + q)1 + 1 t1 − q; n t12 )Tq + q( + t )Id ;
′′
e=
− ′ ((1 + q)1 + 1 t1 − q; n t12 )t1 ;
2
 ′′



′
f= −
−  ((q + 1)1 + 1 t1 )t1 :
2




(12)

In a second step, we use Eqs. (11), (12) and the fact that the polynomials Pn (x; q) satisfy Eq. (3),
again. This gives — after some computations with Maple V.4 — the operator Q2;∗∗n−1 annihilating
the right-hand side of Eq. (11),
Q∗∗ = ( +  t )[q2 A + (1 + q) +  t ]T2 − [q3 A ( +  t ) + A ( + qA )]T
2; n−1

3

3 3

1

2

2 2

2

+ q1 [q A2 + (1 + q)3 + 3 t3 )] Id ;

q

1

2

2 2

3

2

1

q

(13)

where A(x) = (1 + q)(x) + (x)t (x) − q; n t (x)2 and Aj ≡ Aj (x) ≡ A(q j x); j = 1; 2; 3:
We therefore obtain the factorized form of the fourth-order q-dierence equation satised by each
(1)
Pn−1 (x; q),
Q∗
(1)
(x; q)] = 0:
(14)
Q2;∗∗n−1 2 2; n−1 2 2 [Pn−1
q (q − 1) x
3. Limiting situations, comments and example

(1) Since limq→1 Dq = d = d x; from Eqs. (12) and (13), we recover by a limit process the factorized
(1)
(x) of the
form of the fourth-order dierential equation satised by the rst associated Pn−1
(continuous) classical orthogonal polynomials Pn−1 [12],
(15)
Q∗∗c Q∗c [P (1) (x)] = 0;
2; n−1

2; n−1

n−1

234

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 101 (1999) 231–236

with
Q2;∗ n−1
d2
d
=

+ (2′ − ) + (′′ − ′ + n )Id ;
2
2
2
2
q→1 q (q − 1) x
dx
dx
∗∗
Q
d2
1
d
c
=
Q2;∗∗n−1
lim 2 2; n−1 2 2 =  2 + (′ + ) + (′ + n )Id ;
4(x) q→1 q (q − 1) x
dx
dx

Q2;∗cn−1 = lim

′′

where n ≡ limq→1 q; n = −n [(n − 1) 2 + ′ ].
(2) If the polynomials  and  are such that ′′ = 2′ [12–14], then the right-hand side of Eq. (11)
(1)
is equal to zero, and the rst associated Pn−1
satises the second (instead of fourth)-order
dierence equation
(1)
Q2;∗ n−1 [Pn−1
(x; q)] = 0:

For the little q-Jacobi polynomials pn (x; a; b|q) [1, 9]
 (x ) =

x(x − 1)
;
q

 (x ) =

1 − aq + (abq2 − 1)x
;
q(q − 1)

and for the big q-Jacobi polynomials Pn (x; a; b; c; q) [1, 9]
(x) = acq − (a + c)x +

x2
;
q

(x) =

cq + aq(1 − (b + c)q) + (abq2 − 1)x
;
q(q − 1)

the constant ′′ − 2′ is equal to 2(1 − abq)= (q − 1). Therefore, the rst associated of the little
q-Jacobi polynomials (resp. big q-Jacobi polynomials) is still in the little q-Jacobi (resp. big
q-Jacobi) family when abq = 1.
Computations involving the coecients n and
n (see Eq. (8)) given in [1, 6, 11] and use
of Maple V.4 generate the following relations between the monic little q-Jacobi (resp. monic
big q-Jacobi) polynomials and their respective rst associated


pn(1) x; a;
Pn(1)



1
x 1
; ; aq |q ;
|q =(aq)n pn
qa
aq a






(16)

x 1
1
; c; q =(a)n Pn
; ; aq; c q; q :
x; a;
qa;
a a






(17)

(3) The results given in this paper (see Eqs. (11) and (13)), which agree with the ones obtained
using the Stieltjes properties of the associated linear form [6], can be used for connection
(1)
problems, expanding the rst associated Pn−1
in terms of Pn , in the same spirit as in [10]. We
have also computed the coecients of the fourth-order q-dierence equation satised by the
rst associated of the q-classical orthogonal polynomials appearing in the q-Hahn tableau. In
particular, from the big q-Jacobi polynomials, we derive by limit processes [9] the fourth-order
dierential (resp. q-dierence) equation satised by the rst associated of the classical (resp.
q-classical) orthogonal polynomials.

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 101 (1999) 231–236

235

(4) For the little q-Jacobi polynomials for example, the operators Q2;∗ n−1 and Q2;∗∗n−1 are given below,
with the notation:  = qn .
Q∗ = qx[(q2 x − 1)T2 − −1 (− − a + q2 xab2 + qx)T + a(−1 + bqx)I ];
2; n−1

q

q

d

Q2;∗∗n−1 = −1 q4 x2 [qa(−1 + bq4 x)(q3 xab + q3 xab2 + q2 x + q2 x − q − qa −  − a)Tq 2
− −1 (q5 x2 + a2 + q2 − q2 x2 − q3 xab3 + q7 x2 a2 b2 3
− q3 xa2 b3 − q5 xab3 + q2 a2 2 − q5 xab2 − q5 xa2 b2 + q2 a2
− q5 xa2 b3 − q2 xa − q4 xa − q2 x − q4 x − q3 xa + q5 x2 
− q3 x + q7 x2 a2 b2 4 + q6 x2 ab − q4 xa2 b3 + qa2 2 − q2 xa2

+ 2q6 x2 ab2 + q6 x2 ab3 + 2qa2 + 2 − q4 xab3 )Tq
+ (−1 + qx)(q4 xab + q4 xab2 + q3 x + q3 x − q − qa −  − a)Id ]:
Acknowledgements

Part of this work was done when two of the authors M.F. and A.R. were visiting the KonradZuse-Zentrum fur Informationstechnik Berlin. They thank W. Koepf and W. Neun for their warm
hospitality.
References
[1] I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Inversion problems in the q-Hahn tableau, J. Symb. Computation (1999),
to appear.
[2] N.M. Atakishiyev, A. Ronveaux, K.B. Wolf, Dierence equation for the associated polynomials on the linear lattice,
Zt. Teoret. Mat. Fiz. 106 (1996) 76–83.
[3] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, S.M. Watt, Maple V Language Reference
Manual, Springer, New York, 1991.
[4] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
[5] M. Foupouagnigni, A.W. Koepf, A. Ronveaux, The fourth-order dierence equation for the associated classical
discrete orthogonal polynomials, Letter, J. Comput. Appl. Math. 92 (1998) 103–108.
[6] M. Foupouagnigni, A. Ronveaux, M.N. Hounkonnou, The fourth-order dierence equation satised by the associated
orthogonal polynomials of the Dq -Laguerre-Hahn Class (submitted).
[7] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 35,
Cambridge University Press, Cambridge, 1990.

[8] W. Hahn, Uber
Orthogonalpolynome, die q-Dierenzengleichungen genugen, Math. Nachr. 2 (1949) 4 –34.
[9] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report
Fac. of Technical Math. and Informatics 94-05 T.U. Delft, revised version from February 1996, available at: http:
www.can.nl= ∼ renes.
[10] S. Lewanowicz, Results on the associated classical orthogonal polynomials, J. Comput. Appl. Math. 65 (1995)
215–231.
[11] J.C. Medem, Polinomios ortogonales q-semiclasicos, Ph.D. Dissertation, Universidad Politecnica de Madrid, 1996.
[12] A. Ronveaux, Fourth-order dierential equations for numerator polynomials, J. Phys. A: Math. Gen. 21 (1988)
749–753.

236

M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 101 (1999) 231–236

[13] A. Ronveaux, E. Godoy, A. Zarzo, I. Area, Fourth-order dierence equation for the rst associated of classical discrete
orthogonal polynomials, Letter, J. Comput. Appl. Math. 90 (1998) 47–52.
[14] A. Ronveaux, W. Van Assche, Upward extension of the Jacobi matrix for orthogonal polynomials, J. Approx. Theory
86 (3) (1996) 335–357.
[15] S.K. Suslov, The theory of dierence analogues of special functions of hypergeometric type, Russian Math. Surveys
44 (2) (1989) 227–278.