Transverse limits in the Askey tableau
1 A. Ronveauxa, A. Zarzob;2, I. Areac;3, E. Godoyc;∗;3aMathematical Physics, Facultes Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium b
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
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!→0Pn(x(s; !)) =Qn(s), where Pn(s) and Qn(s) stand for the discrete and continuous family, respectively. Deeper information on these limiting processes can be obtained by expanding the discrete polynomial family as Pn(x(s; !)) =P
∞
k=0! kR
k(s;n). In this paper a method for computing the coecients Rk(s;n) is designed the main tool being the consideration of some closely related connection and inversion problems. c1998 Elsevier Science B.V. All rights reserved.
AMS classication: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.
1This work has been partially supported by NATO grant CRG-960213.
2A.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.
3The work of these authors has been partially nantial supported by Xunta de Galicia-Universidade de Vigo under grant 64502I703.
some of these classical families [8, 13], which are relevant in several problems appearing in Math-ematical 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
!→0
Pn(x(s; !)) =Qn(s) (1)
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=0
!kRk(s;n): (2)
On this expansion the only information provided by the limit relation (1) is R0(s;n) =Qn(s).
Knowl-edge 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 theseRk(s;n)-coecients when both polynomials,Pn(x(s; !)) andQn(s), involved in the limit relation
(1), belong simultaneously to a classical continuous or discrete family, i.e., when consideringvertical
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
m=0
Cm(n;!)Qm(s) (3)
(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 oftransverselimits, i.e., when the polynomialsPn(x(s; !)) andQm(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 fullll 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) satises 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, h(; )
n (s;N) in Jacobi polynomials, Pn(; )(s) can be written as
1
Nnh
(; )
n
N
2(1 +s);N
−Pn(; )(s) = O 1
N
but also in a more precise form [11, p. 46, Eq. (2.6.5)] when N→∞:
1 e
Nnh
(; )
n
e N
2(1 +s)−
+ 1
2 ;N
!
−P(; )
n (s) = O
1
e
N2
where Ne=N+ (+)=2.
On the other hand, Sharapudinov [16] has established a uniform asymptotic form of the Kravchuk polynomialK(p)
n (s;N) asn= O(N1=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 polyno-mials in terms of Laguerre polynopolyno-mials. The asymptotic involves not only letting the degree of the
orthogonal polynomial tend to ∞, but also allowing variation in the parameters of the Meixner
poly-nomials. A very explicit uniform error bound containing all Laguerre polynomials L(k)(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¿1
!kR
k(s;n) (4)
which can be deduced from the transverse connection problem
Pn(x(s; !)) = n
X
m=0
Cm(n;!)Qm(s) (5)
collecting the !k terms inC
m(n;!). In particular, the rest term of rst orderR1(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 specic
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 specic transverse limits (1) to be treated together with the corresponding connection
• Monic Meixner (M(; )
n (s). So the following limit appears
lim
(¿−1) stands for the monic Laguerre polynomial of degree n.
The corresponding connection problem (5) is, in this case
wnM(+1;1−w)
and the connection problem (5) read as follows:
wn
stands for the monic Hermite polynomial of degree n.
• Monic Charlier (c()
and the connection problem (5) becomes
(−w)nc(1n=(2w2))
and the connection problem (5)
• 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
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
wnG
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 wk 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.
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]=
are also computed, x=x(s;w) being the ane mapping appearing in each specic transverse limit.
From these four coupled connection problems and after some calculations, the following expression
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 transverselimits 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
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
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–
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
Gn(s;M) =
[18, p. 149]. These polynomials are solution of the second order dierence equation
(M+ 1−s)(M+s)3Gn(s;M)−2sGn(s;M) +n(n+ 1)Gn(s;M) = 0;
with the convention Q−1
j=0A= 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–
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=1
!kR
k(s;n)
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 polynomialsQi(s) contained inR2(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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