Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:

Journal of Computational and Applied Mathematics 99 (1998) 327–335

Transverse limits in the Askey tableau1
A. Ronveauxa , A. Zarzob; 2 , I. Areac; 3 , E. Godoyc;∗;3
a
Mathematical Physics, Facultes Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium
Instituto Carlos I de Fsica Teorica y Computacional, Facultad de Ciencias, Universidad de Granada,
18071-Granada, Spain
c
Departamento de Matematica Aplicada, Escuela Tecnica Superior de Ingenieros Industriales y Minas,
Universidad de Vigo, 36200-Vigo, Spain
b

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

Abstract
Limit relations between classical discrete (Charlier, Meixner, Kravchuk, Hahn) to classical continuous (Jacobi, Laguerre,
Hermite) orthogonal polynomials (to be called transverse limits) are nicely presented in the Askey tableau and can be
described by relations of type lim!→0 Pn (x(s; !)) = Qn (s), where Pn (s) and Qn (s) stand for the discrete and continuous
family, respectively. Deeper
P∞information on these limiting processes can be obtained by expanding the discrete polynomial

family as Pn (x(s; !)) = k=0 ! k Rk (s; n). In this paper a method for computing the coecients Rk (s; n) is designed the
c 1998 Elsevier Science
main tool being the consideration of some closely related connection and inversion problems.
B.V. All rights reserved.
AMS classication: 33C25; 39A10
Keywords: Orthogonal polynomials; Connection problem; Limit relation

1. Introduction
The so-called classical continuous (Jacobi, Laguerre, Hermite) and discrete (Charlier, Meixner,
Kravchuk, Hahn) orthogonal polynomials appear at dierent levels of the Askey tableau [8, 9] of
hypergeometric polynomials. An interesting aspect of this tableau are the limit transitions between
∗

Corresponding author. E-mail: egodoy@dma.uvigo.es.
This work has been partially supported by NATO grant CRG-960213.
2
A.Z. also wishes to acknowledge nantial support from DGES (Ministerio de Educacion y Cultura of Spain) under
contract PG95-1205, from Junta de Andaluca (Grant FQM-0207) and from INTAS (Grant INTAS-93-219-Ext). Permanent
address: Depto. Matematica Aplicada, E.T.S. Ingenieros Industriales, Universidad Politecnica de Madrid, c= Jose Gutierrez
Abascal 2, 28006 Madrid, Spain.

3
The work of these authors has been partially nantial supported by Xunta de Galicia-Universidade de Vigo under
grant 64502I703.
1

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 6 7 - 8

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A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

some of these classical families [8, 13], which are relevant in several problems appearing in Mathematical Physics, Quantum Mechanics, Statistics or Coding Theory (see e.g. [4, 11, 12, 16, 20, 21]
and the references therein).
These limit relations are of the form
lim Pn (x(s; !)) = Qn (s)

(1)

!→0

where ! is the limiting parameter, {Pn (s)} and {Qn (s)} are the classical families connected by the
limit and x = x(s; !) is an ane mapping from x to s. Of course, both polynomial families could
depend on some extra parameters which are not involved in the limiting process.
From (1) the polynomial Pn (x(s; !)) can be always expanded as
Pn (x(s; !)) =

∞
X

! k Rk (s; n):

(2)

k=0

On this expansion the only information provided by the limit relation (1) is R0 (s; n) = Qn (s). Knowledge of coecients Rk (s; n) in the latter expression will give deeper information on the limit (1).
Recently [6], we have devised a recursive method which allows us the explicit computation of
these Rk (s; n)-coecients when both polynomials, Pn (x(s; !)) and Qn (s), involved in the limit relation

(1), belong simultaneously to a classical continuous or discrete family, i.e., when considering vertical
limits (so-called because of their appearance in the Askey tableau). The main tool of the method
developed in [6] is the consideration of the connection problem
Pn (x(s; !)) =

n
X

Cm (n; !)Qm (s)

(3)

m=0

(see [6, Tables 1 and 2]) linking both families involved in (1), which can be solved recurrently by
using the “Navima algorithm” [1, 5, 14, 15] developed by the authors.
It turns out that the above method for computing the coecients Rk (s; n) in (2) cannot be applied
in case of transverse limits, i.e., when the polynomials Pn (x(s; !)) and Qm (s) belong to a discrete and
a continuous family, respectively. This is so because, in this transverse situation, the corresponding
connection problem (3) does not fullll the conditions for the “Navima algorithm” to be applicable.

In particular, it requires that Pn (x(s; !)) and Qm (s) in (3) satisfy dierential or dierence equations
of the same type (see [1, 5] for details) and this is not the case when considering transverse limits,
where Pn (x(s; !)) is solution of a dierence equation, but Qm (s) satises a dierential one.
Main aim of this contribution is to present an alternative method (see Section 3) for solving
this kind of connection problems (out of the scope of the “Navima algorithm”) giving rise to a
constructive algorithm for computing the rest term coecients Rk (s; n) in the transverse situation.
For the sake of completeness, let us mention that partial results on these problems have appeared
already in the literature. For instance in [11, p. 45, Eq. (2.6.3)] it is noticed that the rest term in
the expansion of Hahn polynomials, hn(; ) (s; N ) in Jacobi polynomials, Pn(; ) (s) can be written as


1 (; ) N
(1 + s); N
h
Nn n
2





− Pn(; ) (s) = O

1
N



(N → ∞)

A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

329

but also in a more precise form [11, p. 46, Eq. (2.6.5)] when N →∞:
e
1 (; ) N
+1
hn
(1 + s) −

;N
en
2
2
N

!



− Pn(; ) (s) = O

1
e2
N



e = N + ( + )=2.
where N

On the other hand, Sharapudinov [16] has established a uniform asymptotic form of the Kravchuk
polynomial Kn(p) (s; N ) as n = O(N 1=3 ) → ∞, by using a discrete analogue of Leibniz’s theorem as well
as some properties of Hermite polynomials. This paper [16] is concluded by indicating an application
to coding theory. Moreover, the same author [17] has derived suitably transformed Meixner polynomials in terms of Laguerre polynomials. The asymptotic involves not only letting the degree of the
orthogonal polynomial tend to ∞, but also allowing variation in the parameters of the Meixner polynomials. A very explicit uniform error bound containing all Laguerre polynomials Lk() (s), 06k6n,
was provided.
Our approach to the problem here considered gives more information, in the sense that it provides
explicit expressions for the rest terms (Rk (s; n) in Eq. (2)) and not only error bounds as in the
references cited above. If we write the transverse limit in the form given by Eq. (1), then we can
give explicit formulae for the kth rest term Rk (s; n) in the expansion

Pn (x(s; !)) − Qn (s) = !R1 (s; n) +

X

! k Rk (s; n)

(4)

k¿1

which can be deduced from the transverse connection problem
Pn (x(s; !)) =

n
X

Cm (n; !)Qm (s)

(5)

m=0

collecting the ! k terms in Cm (n; !). In particular, the rest term of rst order R1 (s; n) is a polynomial
in s of degree less than or equal to n − 1. Computation of rest terms in the vertical limit cases [6]
shows that the corresponding polynomial R1 (s; n) is always a combination of only two polynomials
Qr (s), Qj (s) where r can be n − 1 or n − 2 and j is r, r − 1 or r − 2. The present computations
will show that in transverse limit problems this interesting property is still true in some cases.
The outline of the paper is as follows: in Section 2 we present the transverse limits and we x
the notation to be considered within this paper. Section 3 is devoted to the description of the method

we use for the computation of the rest terms Rk (s; n) in (4). As illustration of the method we give
in Section 4 the explicit expression of the rest term of rst order (R1 (s; n) in (4)) for some specic
limits. In doing so, the Mathematica symbolic language [22] has been systematically used in each
computational step. Finally, some concluding remarks are pointed out.

2. Transverse limits and connection problems
The specic transverse limits (1) to be treated together with the corresponding connection problems (5) are listed below. From now on monic polynomials will be considered and notations are
the ones commonly used in the literature (see e.g. [2, 8, 11, 19]).

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A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

• Monic Meixner (Mn(
; ) (x)) → Monic Laguerre (Ln(
) (x)): Starting from the dierence equation satised by the Meixner polynomials Mn(
; ) (x), it is easily shown [11] that the change of variable x = x(s; w) = s=w with w = 1 − , gives at the limit w → 0 the dierential equation satised
by the Laguerre polynomial Ln(
−1) (s). So the following limit appears
lim w

n

w→0

Mn(
+1; 1−w)

 

s
= Ln(
) (s);
w

(6)

where Mn(
; ) (s) (
¿0; 0¡¡1) denotes the monic nth degree Meixner polynomial and Ln(
) (s)
(
¿−1) stands for the monic Laguerre polynomial of degree n.
The corresponding connection problem (5) is, in this case
w n Mn(
+1; 1−w)

 

n
X
s
=
Cm (n; w)Lm(
) (s):
w
m=0

(7)

• Monic Kravchuk (kn(p) (x; N )) → Monic Hermite (Hn (x)): The transverse limit is given by
lim w n

w→0



1
2p(1 − p)

n=2

kn(p)





1
p
sp
2p(1 − p); 2 = Hn (s);
+
w2 w
w

(8)

and the connection problem (5) read as follows:
w

n



1
2p(1 − p)

n=2

kn(p)





n
X
1
sp
p
2p(1 − p); 2 =
+
Cm (n; w)Hm (s):
w2 w
w
m=0

(9)

Here, kn(p) (s; N ) (0¡p¡1, N ∈ Z+ ) denotes the monic nth degree Kravchuk polynomial and Hn (s)
stands for the monic Hermite polynomial of degree n.
• Monic Charlier (cn() (x)) → Monic Hermite (Hn (x)): The limit property is now
2
lim (−w) n cn(1=(2w ))
w→0





s
1
−
= Hn (s);
2w2 w

(10)

and the connection problem (5) becomes
2

(−w) n cn(1=(2w ))





n
X
s
1
=
−
Cm (n; w)Hm (s)
2w2 w
m=0

(11)

cn() (x) (¿0) being the monic Charlier polynomial of degee n and Hn (s) the monic nth degree
Hermite polynomial.
• Monic Hahn (hn(; ) (x; N )) → Monic Jacobi (Pn(; ) (x)): The transverse limit is in this case
lim (2w) n hn(; )

w→0





s+1 1
= Pn(; ) (s);
;
2w w

(12)

and the connection problem (5)
(2w) n hn(; )





n
X
s+1 1
;
=
Cm (n; w)Pm(; ) (s):
2w w
m=0

(13)

Concerning notations, hn(; ) (s; N ) (; ¿−1, N ∈ Z+ ) stands for the monic Hahn polynomial of
degree n and Pn(; ) (s) denotes the monic nth degree Jacobi polynomial (; ¿−1).

A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

331

• Monic Gram (Gn (x; N )) → Monic Legendre (Pn (x)): This transverse limit is a particular case of
the previous one. It reads as follows:
lim w n Gn

w→0





s 1
;
= Pn (s);
w w

(14)

where Gn (s; N ) is the monic Gram polynomials of degree n [7, p. 352], which will be introduced
in Section 4, and Pn (s) = Pn(0; 0) (s) is the monic nth degree Legendre polynomial.
The connection problem (5) can be written as
n

w Gn





n
X
s 1
;
=
Cm (n; w)Pm (s):
w w
m=0

(15)

3. The method: computing rest terms via connection problems
As pointed out in the Introduction, the idea of our approach for the computation of rest terms
Rk (s; n) in Eq. (4) is to obtain the connection coecients, Cm (n; w), appearing in Eq. (5). Then, by
expanding these coecients in power series of the limiting parameter w it is possible to collect all
contributions to the power w k giving the searched rest term of order k.
Computation of the Cm (n; w)-coecients in (5) can be done by solving four coupled connection
problems.
Firstly, denoting by A[0] = 1, A[m] = A(A − 1) · · · (A − m + 1) the falling factorial polynomials
[11, 23], we obtain the coecients Dm (n; w) in
Pn (x(s; w)) =

n
X

Dm (n; w)(x(s; w))[m]

(16)

m=0

which can be computed from the hypergeometrical representation [8, 11] of the classical discrete
polynomial Pn (x(s; w)).
Next, we use the well known expansion (see, e.g., [18, p. 20])
(x(s; w))[m] =

m
X

Sk (m)(x(s; w)) k

(17)

k=0

where Sk (m) are the Stirling numbers of rst kind. Moreover, the coecients Bt (k; w) in
(x(s; w)) k =

k
X

Bt (k; w)s t :

(18)

t=0

are also computed, x = x(s; w) being the ane mapping appearing in each specic transverse limit.
Finally, we use the corresponding inversion formula for each classical continuous orthogonal
polynomial, Qr (s),
t

s =

t
X

Ir (t)Qr (s);

r=0

where the coecients Ir (t) are known (see, e.g., [3, 10, 23]).

(19)

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A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

From these four coupled connection problems and after some calculations, the following expression
for the coecients Cm (n; w) in (5) is obtained:
Cm (n; w) =

n
X

Im (l)

l=m

m
X
i=l

Bl (i; w)

n
X

Dj (n; w)Si (j)

j=i

!!

(20)

:

4. Results: explicit formulae for the rest term of rst order
As illustration of the method we have just described, we give here explicit formulae for the rest
term of order one (R1 (s; n) in Eq. (4)) for the transverse limits described in Section 2. In doing so,
we have computed the connection coecients appearing in Eqs. (7), (9), (11), (13) and (15) using
the expression (20). Then, we have collected the contributions to the rst power of w in the power
series expansion of these coecients in terms of this limiting parameter. The results obtained are as
follows
• Coecient R1 (s; n) in the limit Monic Meixner → Monic Laguerre:






n−2
X
n
(−1) j (
) 
(
)
Lj (s)
(n + 2
+ 1)Ln−1
(s) − (−1) n (
+ 1)(n − 1)!
:
R1 (s; n) =

2
j!
j=0

Here Lj(
) (s) stands for the monic Laguerre polynomial of degree j.
• Coecient R1 (s; n) in the limit Monic Kravchuk → Monic Hermite:
√
 

2(2p − 1) n
n−2
; n¿3;
R1 (s; n) = p
(s)
+
H
(s)
H
n−3
n−1
6
2 p − p2 2

(21)

(22)

Notice that, in this case, R1 (s; n) is a linear combination of two monic Hermite polynomials
(Hn−3 (s) and Hn−1 (s)) which means that this rest term is a quasi-orthogonal polynomial of order
two [2].
On the other hand, if p = 1=2 then R1 (s; n) = 0 and in the expansion Pn (x(s; !)) − Qn (s) =
!2 R2 (s; n) + · · ·, we obtain
R2 (s; n) =

 

n
3



n−3
Hn−4 (s) + Hn−2 (s) ;
8

n¿2:

(23)

• Coecient R1 (s; n) in the limit Monic Charlier → Monic Hermite:
 

n
R1 (s; n) =
2



n−2
Hn−3 (s) + Hn−1 (s) ;
6

n¿3:

(24)

As in the latter example, here R1 (s; n) is a linear combination of two Hermite polynomials which
means that this coecient is again a quasi-orthogonal polynomial of order two [2].
• Coecient R1 (s; n) in the limit Monic Gram → Monic Legendre: When considering the Hahn–
Jacobi case, the polynomial R1 (s; n) involves Jacobi polynomials of degree 0 to n − 1. Coecients
in this expansion are rather complicated expressions in terms of the three parameters appearing
in the limit process. Because of this reason we restrict ourselves to the simpler case given by the

A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

333

limit relation between the so-called Gram polynomials, which play a very important role in least
squares approximation over discrete sets [7], and classical Legendre polynomials.
Monic Gram polynomials have the hypergeometric representation


(n!) 2 (−2M )n
−n; n + 1; −M − s
1
3 F2
1; −2M
(2n)!

Gn (s; M ) =



n
(−n)k (n + 1)k (−M − s)k
(n!) 2 (−2M )n X
;
(2n)!
(k!) 2 (−2M )k
k=0

=

where 06n62M and (A)k = A(A + 1) · · · (A + k − 1), (A)0 = 1 denotes the Pochhammer symbol
[18, p. 149]. These polynomials are solution of the second order dierence equation
(M + 1 − s)(M + s)
3Gn (s; M ) − 2s
Gn (s; M ) + n(n + 1)Gn (s; M ) = 0;
P

they are orthogonal on [−M; M ] with respect to the weight (s) = 1, i.e., M
s=−M Gn (s; M )Gm (s; M )
= kn n; m . So, they are related with the monic Hahn polynomials [11] hn(; ) (x; N ) by means of the
following formula:
0)
Gn (s; M ) = h(0;
(s + M ; 2M + 1):
n

(25)

In this case the expression for the rest term R1 (s; n) is

R1 (s; n) =

 p−1
X



F2k+1 (n)P2k+1 (s);

−

if n = 2p + 1;

k=0
p−1


X



F2k (n)P2k (s);
−

(26)
if n = 2p;

k=0

where

F2k+1 (n) =

F2k (n) =

p(

p(

Qk+1

j=1 [2(k

Qk

j=0 [2(k

with the convention

+ j) + 1])(
Qp−1
j=0

j=0

Q−1

j=0

j=0

[2(k + j + 1)])

[2(n − j) − 1]

+ j) + 1])(
Qp−1

Qp−k−2

Qp−k−2
j=0

;

[2(k + j + 1)])

[2(n − j) − 1]

A = 1.

5. Concluding remarks
It is already interesting to comment the dierences in these expansions (21), (22), (24) and (26)
and to explore possible extensions.
(i) The rest term R1 (s; n) contains all limit polynomials R0 (s; j), 06j6n − 1, in the Meixner–
Laguerre and Hahn–Jacobi cases, but only two polynomials Hn−1 (s) and Hn−3 (s) in both situations

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A. Ronveaux et al. / Journal of Computational and Applied Mathematics 99 (1998) 327–335

Kravchuk–Hermite and Charlier–Hermite. In these two latter cases the expansions are the same up
to a numerical factor.
(ii) Clearly, the full rest term in (4) can be written as
Pn (x(s; !)) − Qn (s) =

n
X

! k Rk (s; n)

k=1

and all Rk (s; n) (k¿1) could be obtained using our method as it is described in Section 3. It should
be interesting to investigate the number of polynomials Qi (s) contained in R2 (s; n); R3 (s; n); : : : ; when
R1 (s; n) contains only two terms.
(iii) As pointed out in the introduction, the method we have presented in this work gives explicit
expressions for the rst rest term coecient (R1 (s; n) in (4)) and so previous results giving error
bounds are improved.
(iv) The four coupled connection problems we have considered (including direct and inverse
representations) for the computation of the rest terms can be also used fruitfully for transverse
connection. Results in this direction have been already obtained and will be given elsewhere.
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