On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials....
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Інститут математики НАН України
2011
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| Цитувати: | On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1468572019-02-12T01:24:32Z On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials Koornwinder, T.H. A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey-Wilson to Wilson polynomials and from q-Racah to Racah polynomials are given in a more conceptual way. 2011 Article On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 6 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 33C45 DOI:10.3842/SIGMA.2011.040 http://dspace.nbuv.gov.ua/handle/123456789/146857 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey-Wilson to Wilson polynomials and from q-Racah to Racah polynomials are given in a more conceptual way. |
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Article |
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Koornwinder, T.H. |
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Koornwinder, T.H. On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Koornwinder, T.H. |
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Koornwinder, T.H. |
| title |
On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials |
| title_short |
On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials |
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On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials |
| title_fullStr |
On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials |
| title_full_unstemmed |
On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials |
| title_sort |
on the limit from q-racah polynomials to big q-jacobi polynomials |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/146857 |
| citation_txt |
On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials / T.H. Koornwinder // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 6 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT koornwinderth onthelimitfromqracahpolynomialstobigqjacobipolynomials |
| first_indexed |
2025-07-11T00:47:21Z |
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2025-07-11T00:47:21Z |
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1837309472061521920 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 040, 8 pages
On the Limit from q-Racah Polynomials
to Big q-Jacobi Polynomials?
Tom H. KOORNWINDER
Korteweg-de Vries Institute, University of Amsterdam,
P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
E-mail: T.H.Koornwinder@uva.nl
URL: http://www.science.uva.nl/~thk/
Received March 01, 2011; Published online April 21, 2011
doi:10.3842/SIGMA.2011.040
Abstract. A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given
which can be considered as a limit formula for orthogonal polynomials. This is extended
to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little
q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey–Wilson to Wilson
polynomials and from q-Racah to Racah polynomials are given in a more conceptual way.
Key words: Askey scheme; q-Askey scheme; q-Racah polynomials; big q-Jacobi polynomials;
multi-parameter limit
2010 Mathematics Subject Classification: 33D45; 33C45
Dedicated to Willard Miller on the occasion of his retirement
1 Introduction
The q-Askey scheme (see [3, p. 413]) consists of families of q-hypergeometric orthogonal polyno-
mials connected by arrows denoting limit transitions. Askey–Wilson polynomials and q-Racah
polynomials are on the top level. All other families in the scheme can be reached from these
two families by (possibly successive) limit transitions. In particular, the scheme gives an arrow
from the q-Racah polynomials to the big q-Jacobi polynomials. The explicit limit corresponding
to this arrow is given in [3, (14.2.15)]. However, while the q-Racah polynomials approach this
limit, they no longer form a (finite) system of orthogonal polynomials. It is the first aim of
the present paper to give another limit from q-Racah to big q-Jacobi where the orthogonality
property remains present while the limit is approached. I was motivated to look for such a limit
by seeing a reference to [3, (14.2.15)] in Vinet & Zhedanov [6, end of § 5].
The q-Askey scheme is the q-analogue of the Askey scheme (see [3, p. 184]), which was first
presented in [1]. The arrows in the Askey scheme represent limit transitions within that scheme,
but there are also many limit transitions from families in the q-Askey scheme to families in the
Askey scheme. The paper continues with the discussion of two such limits for q ↑ 1: from Askey–
Wilson to Wilson and from q-Racah to Racah. Different from their presentation in [3], these
limits are given here such that a polynomial of degree n remains present in the limit transition.
The final section of this paper returns to the limit from q-Racah to big q-Jacobi and treats it
as part of a multi-parameter limit (with 3 parameters). Thus the author’s work in [5] to combine
the limits in the Askey scheme (for q = 1) into multi-parameter limits, is extended to a small
part of the (q-)Askey scheme.
?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special
Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html
mailto:T.H.Koornwinder@uva.nl
http://www.science.uva.nl/~thk/
http://dx.doi.org/10.3842/SIGMA.2011.040
http://www.emis.de/journals/SIGMA/S4.html
2 T.H. Koornwinder
The book Koekoek, Lesky & Swarttouw [3] is the successor of the report Koekoek & Swart-
touw [4], which can be alternatively used as a reference whenever the present paper refers to
some formula in [3, Chapters 9 and 14]. For notation of q-hypergeometric series used in this
paper the reader is referred to [2]. Throughout it will be assumed that 0 < q < 1, that N is
a positive integer and that n ∈ {0, 1, . . . , N} if N is present.
2 The limit formula
Big q-Jacobi polynomials, see [3, (14.5.1)], are defined as follows:
Pn(x; a, b, c; q) := 3φ2
(
q−n, qn+1ab, x
qa, qc
; q, q
)
.
A special value for x = qc can be obtained by application of [2, (II.6)]:
Pn(qc; a, b, c; q) = (−1)nqn(n+1)/2an
(qb; q)n
qa; q)n
.
Another q-hypergeometric representation can be obtained by using [2, (III.12)]:
Pn(x; a, b, c; q)
Pn(qc; a, b, c; q)
= 3φ2
(
q−n, qn+1ab, qcx−1
qb, qc
; q, a−1x
)
.
q-Racah polynomials, see [3, (14.2.1)], are defined as follows:
Rn
(
q−y + qy−Nδ;α, β, q−N−1, δ | q
)
:= 4φ3
(
q−n, qn+1αβ, q−y, qy−Nδ
qα, qβδ, q−N
; q, q
)
(2.1)
(n = 0, 1, . . . , N).
They are indeed polynomials of degree n in x:
Rn
(
x;α, β, q−N−1, δ | q
)
=
n∑
k=0
(q−n, qn+1αβ; q)k q
k
(qα, qβδ, q; q)k
k−1∏
j=0
1− qjx+ q2j−Nδ
1− qj−N
.
Now observe that
Rn
(
x
qN+1a
; b, a, q−N−1,
c
a
| q
)
=
n∑
k=0
(q−n, qn+1ab; q)k q
k
(qb, qc, q; q)k
k−1∏
j=0
1− qj−N−1a−1x+ q2j−Na−1c
1− qj−N
−→
n∑
k=0
(q−n, qn+1ab; q)k
(qb, qc, q; q)k
(
a−1x
)k k−1∏
j=0
(
1− qj+1cx−1
)
as N →∞.
Thus we have proved our main result:
Theorem 1. There is the following limit formula from q-Racah polynomials to big q-Jacobi
polynomials:
lim
N→∞
Rn
(
x
qN+1a
; b, a, q−N−1,
c
a
| q
)
=
Pn(x; a, b, c; q)
Pn(qc; a, b, c; q)
. (2.2)
Limit from q-Racah to Big q-Jacobi 3
Remark 1. Assume
0 < qa < 1, 0 ≤ qb < 1, c < 0. (2.3)
Then the polynomials
x 7→ Rn
(
x
qN+1a
; b, a, q−N−1,
c
a
| q
)
are orthogonal with respect to positive weights (see [3, (14.2.2)]) on the points
qN+1−ya+ qy+1c (y = 0, 1, . . . , N),
which, for certain M depending on N can be written as the union of the increasing sequence of
nonpositive points
qc+ qN+1a, q2c+ qNa, . . . , qMc+ qN−M+2a
and the decreasing sequence of nonnegative points
qa+ qN+1c, q2a+ qNc, . . . , qN−m+1a+ qM+1c.
Formally, in the limit for N → ∞ this tends to the union of the sequence of negative points
{qk+1c}k=0,1,... and the sequence of positive points {qk+1a}k=0,1,.... But indeed, we know that
under the constraints (2.3) the big q-Jacobi polynomials are orthogonal with respect to positive
weights on this set of points (see [3, (14.5.2)]). Thus the limit formula (2.2) is under the
constraints (2.3) on the parameters really a limit formula for orthogonal polynomials.
Remark 2. The limit formula [3, (14.2.15)], which reads
Pn
(
q−y; a, b, c; q
)
= lim
δ→0
Rn
(
q−y + cδqy+1; a, b, c, δ | q
)
, (2.4)
cannot be considered as a limit formula for orthogonal polynomials. Indeed, for the q-Racah
polynomials on the right-hand side it is required that qa or qbδ or qc is equal to q−N for some
positive integer N (see [3, (14.2.1)]). Since δ → 0 and a, b, c remain fixed in (2.4), we must
have qa or qc equal to q−N . But then we arrive at a limit from q-Racah polynomials to q-Hahn
polynomials (see [3, (14.2.16) or (14.2.18)]) rather than big q-Jacobi polynomials.
Remark 3. For c = 0 (2.2) specializes to a limit formula from q-Hahn polynomials to little
q-Jacobi polynomials. For the left-hand side of (2.2) use that
Rn
(
x
qN+1a
; b, a, q−N−1, 0 | q
)
= Qn
(
x
qN+1a
; b, a,N ; q
)
, (2.5)
see [3, (14.2.16)], where the Qn are q-Hahn polynomials [3, (14.6.1)]. For the right-hand side
of (2.2) use that
Pn(x; a, b, 0; q)
Pn(0; a, b, 0; q)
= pn
(
x
qa
; b, a; q
)
,
see [3, p. 442, Remarks, first formula], where the pn are little q-Jacobi polynomials [3, (14.12.1)].
Thus for c = 0 (2.2) specializes to the limit formula
lim
N→∞
Qn
(
x
qN+1a
; b, a,N ; q
)
= pn
(
x
qa
; b, a; q
)
, (2.6)
which is also given in [3, (14.6.13)].
4 T.H. Koornwinder
3 Limit from Askey–Wilson to Wilson
Consider Askey–Wilson polynomials (see [3, (14.1.1)]), putting x = cos θ:
pn(x; a, b, c, d | q) := a−n(ab, ac, ad; q)n 4φ3
(
q−n, abcdqn−1, aeiθ, ae−iθ
ab, ac, ad
; q, q
)
= a−n(ab, ac, ad; q)n
n∑
k=0
(q−n, abcdqn−1; q)k q
k
(ab, ac, ad, q; q)k
k∏
j=0
(
1− 2qjax+ q2ja2
)
. (3.1)
Also consider Wilson polynomials (see [3, (9.1.1)]), putting x = y2:
Wn(x; a, b, c, d) := (a+ b, a+ c, a+ d)n 4F3
(
−n, n+ a+ b+ c+ d− 1, a+ iy, a− iy
a+ b, a+ c, a+ d
; 1
)
= (a+ b, a+ c, a+ d)n
n∑
k=0
(−n, n+ a+ b+ c+ d− 1)k
(a+ b, a+ c, a+ d)k k!
k∏
j=0
(
(a+ j)2 + x
)
. (3.2)
Rescale (3.1) as
(1− q)−3npn
(
1− 1
2(1− q)
2x; qa, qb, qc, qd | q
)
=
(qa+b, qa+c, qa+d; q)n
qna(1− q)3n
×
n∑
k=0
(q−n, qn+a+b+c+d−1; q)k q
k(1− q)2
(qa+b, qa+c, qa+d, q; q)k
k∏
j=0
(
(1− qa+j)2
(1− q)2
+ qa+jx
)
. (3.3)
From (3.3) and (3.2) we conclude that
lim
q↑1
(1− q)−3npn
(
1− 1
2(1− q)
2x; qa, qb, qc, qd | q
)
=Wn(x; a, b, c, d). (3.4)
Remark 4. In [3, (14.1.21)] the following limit from Askey–Wilson polynomials to Wilson
polynomials is given:
lim
q↑1
v(1− q)−3nvpn
(
1
2(q
iy + q−iy); qa, qb, qc, qd | q
)
=Wn
(
y2; a, b, c, d
)
. (3.5)
This limit follows immediately by comparing the (q-)hypergeometric expressions in (3.1)
and (3.2). However, the limit (3.5) has the draw-back that the rescaled Askey–Wilson poly-
nomial on the left no longer depends polynomially on y. Note that the limit (3.4) can be written
more generally, by the same proof, as
lim
q↑1
(1− q)−3npn
(
1− 1
2(1− q)
2x+ o
(
(1− q)2
)
; qa, qb, qc, qd | q
)
=Wn(x; a, b, c, d). (3.6)
Then (3.5) is a special case of (3.6), since
1
2(q
iy + q−iy) = 1− 1
2(1− q)
2y2 + o
(
(1− q)2
)
.
4 Limit from q-Racah to Racah
In (2.1) we introduced q-Racah polynomials. These are orthogonal with respect to positive
weights if 0 < qα < 1, 0 < qβ < 1 and δ < qNα, as can be read off from [3, (14.2.2)] and
also from the requirement that An−1Cn > 0 for n = 1, 2, . . . , N in the normalized recurrence
relation [3, (14.2.4)]. In order to keep positive weights in the limit from q-Racah polynomials
Limit from q-Racah to Big q-Jacobi 5
to big q-Jacobi polynomials we needed δ < 0, see Remark 1, or δ = 0 in a degenerate case, see
Remark 3. However, for the limit from q-Racah polynomials to Racah polynomials we will need
0 < δ < qNα. We can rewrite (2.1) as
Rn
(
x;α, β, q−N−1, δ | q
)
=
n∑
k=0
(q−n, qn+1αβ; q)k q
k
(qα, qβδ, q−N , q; q)k
k−1∏
j=0
(
1− xqj + qδ−N+2j
)
. (4.1)
Also consider Racah polynomials (see [3, (9.2.1)]), putting x = y(y + δ −N):
Rn(x;α, β,−N − 1, δ) := 4F3
(
−n, n+ α+ β + 1,−y, y + δ −N
α+ 1, β + δ + 1,−N
; 1
)
=
n∑
k=0
(−n, n+ α+ β + 1)k
(α+ 1, β + δ + 1,−N)k k!
k∏
j=0
(−x+ j(δ −N + j)). (4.2)
These are orthogonal with respect to positive weights if α, β > −1 and δ > N+α, see [3, (9.2.2)]
or [3, (9.2.4)]. Rescale (4.1) as
Rn
(
(1− q)2x+ 1 + qδ−N ; qα, qβ, q−N−1, qδ | q
)
=
n∑
k=0
(q−n, qn+α+β+1; q)k (1− q)2kqk
(qα+1, qβ+δ+1, q−N , q; q)k
k−1∏
j=0
(
(1− qj)(1− qδ−N+j)
(1− q)2
− xqj
)
. (4.3)
From (4.3) and (4.2) we conclude that
lim
q↑1
Rn
(
1 + qδ−N + (1− q)2x; qα, qβ, q−N−1, qδ | q
)
= Rn(x;α, β,−N − 1, δ). (4.4)
The orthogonal polynomials involved in this limit have positive weights if α, β > −1 and δ >
N + α.
Remark 5. In [3, (14.2.24)] the following limit from q-Racah polynomials to Racah polynomials
is given:
lim
q↑1
Rn(q
−y + qy+δ−N ; qα, qβ, q−N−1, qδ | q) = Rn(y(y + δ −N);α, β,−N − 1, δ). (4.5)
This limit follows immediately by comparing the (q-)hypergeometric expressions in (2.1)
and (4.2). Just as for (3.5), the limit (4.5) has the draw-back that we no longer have poly-
nomials in y on the left-hand side of (4.5). Note that the limit (4.4) can be written more
generally, by the same proof, as
lim
q↑1
Rn
(
1 + qδ−N + (1− q)2x+ o((1− q)2); qα, qβ, q−N−1, qδ | q
)
= Rn(x;α, β,−N − 1, δ). (4.6)
Then (4.5) is a special case of (4.6) since
y(y + δ −N) = −(1− qδ−N+y)((1− q−y)
(1− q)2
+ o
(
(1− q)2
)
and
1 + qδ−N + (1− q)2x = y(y + δ −N) for x =
(1− qδ−N+y)(1− q−y)
(1− q)2
.
Also note that the polynomials x 7→ Rn(1 + qδ−N + (1− q)2x) on the left-hand side of (4.4) are
orthogonal with respect to weights on the points−(1−q)−2(1−qδ−N+y)(1−q−y) (y = 0, 1, . . . , N)
by [3, (14.2.)]. In the limit for q ↑ 1 this becomes an orthogonality on the points y(y + δ −N)
(y = 0, 1, . . . , N), as is indeed the case for Racah polynomials, see [3, (9.2.2)].
6 T.H. Koornwinder
q-Racah
�
�
�
@
@
@R
big q-Jacobi
?
q-Hahn
�
�
�
�= ?
little q-Jacobi
@
@
@R
Hahn
�
�
�
Jacobi
Figure 1. Part of (q-)Askey scheme.
5 A piece of (q-)Askey scheme below q-Racah
We earlier saw the limits (2.2) (q-Racah → big q-Jacobi), (2.5) (q-Racah → q-Hahn), (1) (big
q-Jacobi → little q-Jacobi) and (2.6) (q-Hahn → little q-Jacobi). To these we can add limits
from q-Hahn to Hahn (see [3, (14.6.18)])
lim
q↑1
Qn(1 + (1− q)x;α, β,N ; q) = Qn(x;α, β,N), (5.1)
from little q-Jacobi to Jacobi (see [3, (14.12.15)])
lim
q↑1
pn
(
x; qα, qβ; q
)
=
P
(α,β)
n (1− 2x)
P
(α,β)
n (1)
,
and from Hahn to Jacobi (see [3, (9.5.14)])
lim
N→∞
Qn(Nx;α, β,N) =
P
(α,β)
n (1− 2x)
P
(α,β)
n (1)
.
Note that in (5.1) the left-hand side of [3, (14.6.18)] was changed in order to keep polynomials
in x while taking the limit. The validity of (5.1) is easily seen from [3, (14.6.1), (9.5.1)].
Fig. 1 combines these seven limits as a subgraph of the (q-)Askey scheme (see the graphs
given in the beginning of Chapters 9 and 14 in [3]).
In [5] I combined the limits in the Askey scheme (i.e., for q = 1) into a small number of
multi-parameter limits. This was done by renormalizing the Racah and Askey–Wilson polyno-
mials on the top level of the scheme as families of orthogonal polynomials depending on four
positive parameters such that these extend continuously for nonnegative parameter values, while
(renormalized) families lower in the scheme are reached if one or more of the parameters become
zero. At the end of [5] the obvious open problem was mentioned to extend this work to the
q-Askey scheme including the limits for q ↑ 1. Below I will work this out for the small part of
the (q-)Askey scheme in Fig. 1.
Fix α, β > −1 and renormalize the q-Racah polynomials as
pn(x) = pn
(
x; c,N−1, 1− q
)
:=
qn(β+1) (qα+1,−qβ+1c, q−N ; q)n
(q−N − 1)n(qn+α+β+1; q)n
×Rn
(
1− q−Nc+ q−β−1(q−N − 1)x; qα, qβ, q−N−1,−c | q
)
. (5.2)
Limit from q-Racah to Big q-Jacobi 7
q-Racah
�
�
�
N →∞
?
c ↓ 0
@
@
@R
q ↑ 1
big q-Jacobi
?
@
@
@R
q-Hahn
�
�
�
@
@
@R
Hahn
�
�
� ?
little q-Jacobi
@
@
@R
Jacobi
?
Hahn
�
�
�
Jacobi
Figure 2. Part of (q-)Askey scheme with multi-parameter limits.
By the chosen coefficient on the right these are monic polynomials of degree n, see [3, (14.2.4)].
For the parameters in the arguments of pn we require
c > 0, 0 < 1− q < 1, N−1 ∈
{
1, 12 ,
1
3 ,
1
4 , . . .
}
. (5.3)
We will see that the polynomials pn(x; c,N
−1, 1−q) remain continuous in (c,N−1, 1−q) if these
three coordinates are also allowed to become zero.
For the demonstration we will use the same tool as in [5]. We will see that the coefficients
in the three-term recurrence relation for the orthogonal polynomials (5.2) depend continuously
on (c,N−1, 1− q) for values of these coordinates as in (5.3) or equal to zero.
It follows from [3, (14.2.4)] that pn given by (5.2) satisfies the recurrence relation
xpn(x) = pn+1(x) + (An + Cn)pn(x) +An−1Cnpn−1(x) (5.4)
with
An = qβ+1
(
1 + qn+β+1c
) (1− qn+α+1)(1− qn+α+β+1)
(1− q2n+α+β+1)(1− q2n+α+β+2)
qn−N − 1
q−N − 1
and
Cn = qβ+2(c+ qn+α)
(1− qn)(1− qn+β)
(1− q2n+α+β)(1− q2n+α+β+1)
q−N − qn+α+β
q−N − 1
.
Clearly, An and Cn are continuous in (c,N−1, 1− q) for (N−1, 1− q) 6= (0, 0). In order to prove
their continuity at (N−1, 1 − q) = (0, 0) we only have to consider the continuity there of the
factors
qn−N − 1
q−N − 1
= 1− 1− qn
1− q
1− q
1− qN
and
q−N − qn+α+β
q−N − 1
= 1 + qN
1− qn+α+β
1− q
1− q
1− qN
.
Their continuity follows from the limit
lim
q↑1; N→∞
1− q
1− qN
= 0,
8 T.H. Koornwinder
which holds because
1− q
1− qN
=
1
1 + q + · · ·+ qN−1
≤ 1
1 + q0 + · · ·+ qN0−1
0
=
1− q0
1− qN0
0
if q0 ≤ q < 1, N ≥ N0.
We can identify the cases where one or more of the parameters c, N−1, 1 − q in (5.2) are
zero, with families situated below the q-Racah box in Fig. 1. This can be done by taking limits
in (5.2) or by taking limits in the recurrence relation (5.4). Thus we see:
pn(x; c, 0, 1− q) = const · Pn
(
x− qβ+1c; qβ, qα,−qβ+1c; q
)
(big q-Jacobi),
pn
(
x; 0, N−1, 1− q
)
= const ·Qn
(
1 + q−β−1(q−N − 1)x; qα, qβ, N ; q
)
(q-Hahn),
pn
(
x; c,N−1, 0
)
= const ·Qn(Nx;α, β,N) (Hahn),
pn(x; 0, 0, 1− q) = const · pn
(
x; qα, qβ; q
)
(little q-Jacobi),
pn(x; c, 0, 0) = const · P (α,β)
n (1− 2x) (Jacobi).
The various limits are collected in Fig. 2.
References
[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials,
Mem. Amer. Math. Soc. (1985), no. 319.
[2] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Appli-
cations, Vol. 96, Cambridge University Press, Cambridge, 2004.
[3] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues,
Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
[4] Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-
analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology,
1998, http://aw.twi.tudelft.nl/~koekoek/askey/.
[5] Koornwinder T.H., The Askey scheme as a four-manifold with corners, Ramanujan J. 20 (2009), 409–439,
arXiv:0909.2822.
[6] Vinet L., Zhedanov A., A limit q = −1 for big q-Jacobi polynomials, Trans. Amer. Math. Soc., to appear,
arXiv:1011.1429.
http://dx.doi.org/10.1017/CBO9780511526251
http://dx.doi.org/10.1017/CBO9780511526251
http://dx.doi.org/10.1007/978-3-642-05014-5
http://aw.twi.tudelft.nl/~koekoek/askey/
http://dx.doi.org/10.1007/s11139-009-9208-7
http://arxiv.org/abs/0909.2822
http://arxiv.org/abs/1011.1429
1 Introduction
2 The limit formula
3 Limit from Askey-Wilson to Wilson
4 Limit from q-Racah to Racah
5 A piece of (q-)Askey scheme below q-Racah
References
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