Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Lorentzian Groups
Not too long ago, only a few partial results in the
line of Milnor’s study of definite metrics were
known for indefinite metrics (Barnet 1989, Nomizu
1979), and they were Lorentzian.
Guediri (2003) and others have made special
study of Lorentzian two-step groups, partly because
of their relevance to general relativity, where they
can be used to provide interesting and important
(counter)examples. Special features of Lorentzian
geometry frequently enable them to obtain much
more complete and explicit results than are possible
in general.
For example, Guediri (2003) was able to provide
a complete and explicit integration of the geodesic
equations for Lorentzian 2-step groups. This
includes the case of a degenerate center, which
only required extremely careful handling through a
number of cases. He also paid special attention to
the existence of closed timelike geodesics, reflecting
the relativistic concerns.
As usual, N denotes a connected and s imply
connected 2-step nilpotent Lie group. For the rest
of this section, we assume that the left-invariant
metric tensor is Lorentzian. Whenever a lattice is
mentioned, we also assume that the group is
rational.
Proposition 9 If the center is degenerate, then no
timelike geodesic can be translated by a central
element.
Thus, there can be no closed timelike geodesics
parallel to the center in any nilmanifold obtained
from such an N.
Theorem 12 If the center is Lorentzian, then nN
contains no timelike or null closed geodesics for any
lattice .
To handle degenerate centers, three refined
notions for nonsingular are used: almost, weakly,
and strongly nonsingular. The precise definitions
involve an adapted Witt decomposition (as in the
general pseudo-Riemannian case, but a rather
different one here) and are quite technical, as is
typical. We refer to Guediri (2003) for details.
Theorem 13 If N is weakly nonsingular, then no
timelike geodesic c an be translated by an element
of N.
Corollary 6 If N is flat, then no timelike geodesic
can be translated by a non-identity ele ment.
Corollary 7 If N is flat, then nN contains no
closed timelike geodesics for any lattice .
Corollary 8 If N is weakly nonsingular, then nN
contains no closed timelike geo desic.
Corollary 9 If N ¼ H
2kþ1
is a Lorentzian Heisen-
berg group with degenerate center, then nN
contains no closed timelike geo desic.
Guediri also has the only non-Riemannian results
so far about the phenomenon Eberlein called ‘‘in
resonance.’’ Roughly speaking, this occurs when the
eigenvalues of the map j have rational ratios. (The
Lorentzian case actually requires a slightl y more
complicated condition when the center is
degenerate.)
Theorem 14 If N is almost nonsingul ar, then N is
in resonance if and only if every geodesic of N is
translated by some element of N.
See also: Classical Groups and Homogeneous Spaces;
Einstein Equations: Exact Solutions; Lorentzian
Geometry.
Further Reading
Barnet F (1989) On Lie groups that admit left-invariant Lorentz
metrics of constant sectional curvature. Illinois Journal of
Mathematics 33: 631–642.
Berndt J, Tricerri F, and Vanhecke L (1995) Generalized
Heisenberg Groups and Damek-Ricci Harmonic Spaces,
LNM 1598. Berlin: Springer.
Ciatti P (2000) Scalar products on Clifford modules and pseudo-
H-type Lie algebras. Annali di Matematica Pura ed Applicata
178: 1–31.
Cordero LA and Parker PE (1999) Pseudoriemannian 2-step
nilpotent Lie groups, Santiago-Wichita. Preprint DGS/CP4,
(arXiv:math.DG/9905188).
Eberlein P (1994) Geometry of 2-step nilpotent groups with a left-
invariant metric. Annales Scientifique de l’E
´
cole Normale
Supe´rieure 27: 611–660.
Eberlein P (2004) Left-invariant geometry of Lie groups. Cubo 6:
427–510. (See also http://www.math.unc.edu/faculty/pbe.)
Guediri M (2003) Lorentz geometry of 2-step nilpotent Lie
groups. Geometriae Dedicata 100: 11–51.
Guediri M (2004) The timelike cut locus and conjugate points in
Lorentz 2-step nilpotent Lie groups. Manuscripta Mathema-
tica 114: 9–35.
Hervig S (2004) Einstein metrics: homogeneous solvmanifolds,
generalised Heisenberg groups and black holes. Journal of
Geometry and Physics 52: 298–312.
Jang C, Parker PE, and Park K (2005) PseudoH-type 2-step
nilpotent Lie groups. Houston Journal of Mathematics 31:
765–786 (arXiv:math.DG/0307368).
Kaplan A (1981) Riemannian nilmanifolds attached to Clifford
modules. Geometriae Dedicata 11: 127–136.
Kaplan A (1983) On the geometry of Lie groups of Heisenberg
type. Bulletin of the London Mathematical Society 15: 35–42.
Karidi R (1994) Geometry of balls in nilpotent Lie groups. Duke
Mathematical Journal 74: 301–317.
Milnor J (1976) Curvatures of left-invariant metrics on Lie
groups. Advances in Mathematics 21: 293–329.
102 Pseudo-Riemannian Nilpotent Lie Groups
Nomizu K (1979) Left-invariant Lorentz metrics on Lie groups.
Osaka Mathematical Journal 16: 143–150.
O’Neill B (1983) Semi-Riemannian Geometry. New York:
Academic Press.
Pauls SD (2001) The large scale geometry of nilpotent Lie groups.
Communications in Analysis and Geometry 9: 951–982.
Wilson EN (1982) Isometry groups on homogeneous manifolds.
Geometriae Dedicata 12: 337–346.
Wolf JA (1964) Curvature in nilpotent Lie groups. Proceedings of
the American Mathematical Society 15: 271–274.
Pseudo-Riemannian Nilpotent Lie Groups 103
Q
q-Special Functions
T H Koornwinder, University o
f Amsterdam, Amsterdam, The Netherlands
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In this article we give a brief introduction to q-special
functions, that is, q-analogs of the classical special
functions. Here q is a deformation parameter, usually
0 < q < 1, where q = 1 is the classical case. The
deformation is such that the calculus simultaneously
deforms to a q-calculus involving q-derivatives and
q-integrals. The main topics to be treated are
q-hypergeometric series, with some selected evalu-
ation and transformation formulas, and some
q-hypergeometric orthogonal polynomials, most nota-
bly the Askey–Wilson polynomials. In several vari-
ables, we discuss Macdonald polynomials associated
with root systems, with most emphasis on the A
n
case.
The rather new theory of elliptic hypergeometric series
gets some attention. While much of the theory of
q-special functions keeps q fixed, some of the deeper
aspects with number-theoretic and combinatorial
flavor emphasize expansion in q. Finally, we indicate
applications and interpretations in quantum groups,
Chevalley groups, affine Lie algebras, combinatorics,
and statistical mechanics.
Conventions
q 2 Cn{1} in general, but 0 < q < 1 in all infinite
sums and products.
n, m, N will be non-negative integers unless men-
tioned otherwise.
q-Hypergeometric Series
Definitions
For a , q 2 C the q-shifted factorial (a; q)
k
is defined
as a product of k factors:
ða; qÞ
k
:¼ð1 aÞð1 aqÞð1 aq
k1
Þ
ðk 2 Z
>0
Þ; ða; qÞ
0
:¼ 1 ½1
If jqj < 1 this definition remains meaningful for
k = 1 as a convergent infinite product:
ða; qÞ
1
:¼
Y
1
j¼0
ð1 aq
j
Þ½2
We also write (a
1
, ..., a
r
; q)
k
for the product of r
q-shifted factorials:
ða
1
; ...; a
r
; qÞ
k
:¼ða
1
; qÞ
k
...ða
r
; qÞ
k
ðk 2 Z
0
or k ¼1Þ ½3
A q-hypergeometric series is a power series (for the
moment still formal) in one complex variable z with
power series coefficients which depend, apart from q,
on r complex upper parameters a
1
, ..., a
r
and s
complex lower parameters b
1
, ..., b
s
as follows:
r
s
a
1
; ...; a
r
b
1
; ...; b
s
; q; z
"#
¼
r
s
ða
1
;...; a
r
; b
1
; ...; b
s
; q; zÞ
:¼
X
1
k¼0
ða
1
; ...; a
r
; qÞ
k
ðb
1
;...; b
s
; qÞ
k
ðq; qÞ
k
ð1Þ
k
q
ð1=2Þkðk1Þ
srþ1
z
k
ðr; s 2 Z
0
Þ½4
Clearly the above expression is symmetric in
a
1
,..., a
r
and symmetric in b
1
, ..., b
s
. On the right-
hand side of [4], we have that
ðk þ 1Þth term
kth term
¼
ð1 a
1
q
k
Þð1 a
r
q
k
Þðq
k
Þ
srþ1
z
ð1 b
1
q
k
Þð1 b
s
q
k
Þð1 q
kþ1
Þ
½5
is rational in q
k
. Conversely, any rational function in
q
k
can be written in the form of the right-hand side
of [5]. Hence, any series
P
1
k = 0
c
k
with c
0
= 1 and
c
kþ1
=c
k
rational in q
k
is of the form of a
q-hypergeometric series [4].
In order to avoid singularities in the terms of [4],
we assume that b
1
, ..., b
s
6¼ 1, q
1
, q
2
, .... If, for
some i, a
i
= q
n
, then all terms in the series [4] with
k > n will vanish. If none of the a
i
is equal to q
n
and if jqj < 1, then the radius of convergen ce of the
power series [4] equals 1 if r < s þ 1, 1 if r = s þ 1,
and 0 if r > s þ 1.
We can view the q-shifted factorial as a q-analog
of the shifted factorial (or Pochhammer symbol) by
the limit formula
lim
q!1
ðq
a
; qÞ
k
ð1 qÞ
k
¼ðaÞ
k
:¼ aða þ 1Þða þ k 1Þ½6
Hence the q-binomial coefficient
n
k
q
:¼
ðq; qÞ
n
ðq; qÞ
k
ðq; qÞ
nk
ðn; k 2 Z; n k 0Þ
½7
tends to the binomial coefficient for q ! 1:
lim
q!1
n
k
q
¼
n
k
½8
and a suitably renormalized q-hypergeometric series
tends (at least formally) to a hypergeometric series
as q " 1:
lim
q"1
rþr
0
sþs
0
q
a
1
; ...; q
a
r
; c
1
; ...; c
r
0
q
b
1
; ...; q
b
s
; d
1
; ...; d
s
0
; q; ðq 1Þ
1þsr
z
"#
¼
r
F
s
a
1
; ...; a
r
b
1
; ...; b
s
;
ðc
1
1Þðc
r
0
1Þz
ðd
1
1Þðd
s
0
1Þ
!
½9
At least formally, there are limit relations between
q-hypergeometric series with neighboring r, s:
lim
a
r
!1
r
s
a
1
; ...; a
r
b
1
; ...; b
s
; q;
z
a
r
¼
r1
s
a
1
; ...; a
r1
b
1
; ...; b
s
; q; z
½10
lim
b
s
!1
r
s
a
1
; ...; a
r
b
1
; ...; b
s
; q;b
s
z
¼
r
s1
a
1
; ...; a
r
b
1
; ...; b
s1
; q; z
½11
A terminating q- hype rge omet ric serie s
P
n
k = 0
c
k
z
k
rewritten as z
n
P
n
k = 0
c
nk
z
k
yields
another terminating q-hypergeometric series, for
instance:
sþ1
s
q
n
; a
1
; ...; a
s
b
1
; ...; b
s
; q; z
¼ð1Þ
n
q
ð1=2Þnðnþ1Þ
ða
1
; ...; a
n
; qÞ
n
ðb
1
; ...; b
s
; qÞ
n
z
n
sþ1
s
q
n
; q
nþ1
b
1
1
; ...; q
nþ1
b
1
s
q
nþ1
a
1
1
; ...; q
nþ1
a
1
s
;
"
q;
q
nþ1
b
1
b
s
a
1
a
s
z
½12
Often, in physics and quantum groups related
literature, the following notation is used for
q-number, q-factorial, and q-Pochhammer
symbol:
½a
q
:¼
q
ð1=2Þa
q
ð1=2Þa
q
1=2
q
1=2
½k
q
!:¼
Y
k
j¼1
½j
q
ð½a
q
Þ
k
:¼
Y
k1
j¼0
½a þ j
q
ðk 2 Z
0
Þ½13
For q !1, these symbols tend to their classical
counterparts without the need for renormalization.
They are expressed in terms of the standard notation
[1] as follows:
½k
q
! ¼ q
ð1=4Þkðk1Þ
ðq; qÞ
k
ð1 qÞ
k
ð½a
q
Þ
k
¼ q
ð1=2Þkða1Þ
q
ð1=4Þkðk1Þ
ðq
a
; qÞ
k
ð1 qÞ
k
½14
Special Cases
For s = r 1, formula [4] simplifies to
r
r1
a
1
; ...; a
r
b
1
; ...; b
r1
; q; z
¼
X
1
k¼0
ða
1
; ...; a
r
; qÞ
k
ðb
1
; ...; b
r1
; qÞ
k
ðq; qÞ
k
z
k
½15
which has radius of convergence 1 in the nontermi-
nating case. The case r = 2of[15] is the q -analog of
the Gauss hypergeometric series.
q-Binomial series
1
0
ða; ; q; zÞ¼
X
1
k¼0
ða; qÞ
k
z
k
ðq; qÞ
k
¼
ðaz; qÞ
1
ðz; qÞ
1
ðif series is not terminating, then jzj < 1Þ½16
q-Exponential series
e
q
ðzÞ:¼
1
0
ð0; ; q; zÞ
¼
X
1
k¼0
z
k
ðq; qÞ
k
¼
1
ðz; qÞ
1
ðjzj < 1Þ½17
E
q
ðzÞ:¼
0
0
ð; ; q; zÞ¼
X
1
k¼0
q
ð1=2Þkðk1Þ
z
k
ðq; qÞ
k
¼ðz; qÞ
1
¼ e
q
ðzÞ
1
ðz 2 CÞ½18
"
q
ðzÞ:¼
1
1
ð0; q
1=2
; q
1=2
; zÞ
¼
X
1
k¼0
q
ð1=4Þkðk1Þ
ðq; qÞ
k
z
k
ðz 2 CÞ½19
106 q-Special Functions
Jackson’s q-Bessel functions
J
ð1Þ
ðx; qÞ:¼
ðq
þ1
; qÞ
1
ðq; qÞ
1
1
2
x
2
1
0; 0
q
þ1
; q;
1
4
x
2
ð0 < x < 2Þ½20
J
ð2Þ
ðx; qÞ :¼
ðq
þ1
; qÞ
1
ðq; qÞ
1
1
2
x
0
1
q
þ1
; q;
1
4
q
þ1
x
2
¼
1
4
x; q
1
J
ð1Þ
ðx; qÞðx > 0 Þ½21
J
ð3Þ
ðx; qÞ:¼
ðq
þ1
; qÞ
1
ðq; q Þ
1
1
2
x
1
1
0
q
þ1
; q;
1
4
qx
2
ðx > 0Þ½22
See [90] for the orthogonality relation for J
(3)
(x; q).
If exp
q
(z) denotes one of the three q-exponentials
[17]–[19], then (1=2)( exp
q
(ix) þ exp
q
(ix)) is a
q-analog of the cosine and (1=2)i( exp
q
(ix)
exp
q
(ix)) is a q-analog of the sine. The three
q-cosines are essentially the case = 1=2 of the
corresponding q-Bessel functions [20]–[22], and the
three q-sines are essenti ally the case = 1=2ofx
times the corresponding q-Bessel functions.
q-Derivative and q-Integral
The q-derivative of a function f given on a subset of
R or C is defined by
ðD
q
f ÞðxÞ :¼
f ðxÞf ðqxÞ
ð1 qÞx
ðx 6¼ 0; q 6¼ 1Þ½23
where x and qx should be in the domain of f.By
continuity, we set (D
q
f )(0) := f
0
(0), provided f
0
(0)
exists. If f is differentiable on an open interval
I, then
lim
q"1
ðD
q
f ÞðxÞ¼f
0
ðxÞðx 2 IÞ½24
For a 2 Rn{0} and a function f given on (0, a]or
[a, 0), we define the q-integral by
Z
a
0
f ðxÞd
q
x : ¼ að1 qÞ
X
1
k¼0
f ðaq
k
Þq
k
¼
X
1
k¼0
f ðaq
k
Þðaq
k
aq
kþ1
Þ½25
provided the infinite sum converges absolutely (e.g.,
if f is bounded). If F(a) is given by the left-hand side
of [25], then D
q
F = f . The right-hand side of [25] is
an infinite Riemann sum. For q " 1 it converges, at
least formally, to
R
a
0
f (x)dx.
For nonzero a, b 2 R we define
Z
b
a
f ðxÞd
q
x :¼
Z
b
0
f ðxÞd
q
x
Z
a
0
f ðxÞd
q
x ½26
For a q-integral over (0, 1), we have to specify a
q-lattice {aq
k
}
k2Z
for some a > 0 (up to multi-
plication by an integer power of q):
Z
a:1
0
f ðxÞd
q
x :¼ að1 qÞ
X
1
k¼1
f ðaq
k
Þq
k
¼ lim
n!1
Z
q
n
a
0
f ðxÞd
q
x ½27
The q-Gamma and q-Beta Functions
The q-gamma function is defined by
q
ðzÞ :¼
ðq; qÞ
1
ð1 qÞ
1z
ðq
z
; qÞ
1
ðz 6¼ 0; 1; 2; ...Þ½28
¼
Z
ð1qÞ
1
0
t
z1
E
q
ðð1 qÞqtÞd
q
t ð<z > 0Þ½29
Then
q
ðz þ 1Þ¼
1 q
z
1 q
q
ðzÞ½30
q
ðn þ 1Þ¼
ðq; qÞ
n
ð1 qÞ
n
½31
lim
q"1
q
ðzÞ¼ðzÞ½32
The q-beta function is defined by
B
q
ða; bÞ : ¼
q
ðaÞ
q
ðbÞ
q
ða þ bÞ
¼
ð1 qÞðq; q
aþb
; qÞ
1
ðq
a
; q
b
; qÞ
1
ða; b 6¼ 0; 1; 2; ...Þ½33
¼
Z
1
0
t
b1
ðqt; qÞ
1
ðq
a
t ; qÞ
1
d
q
t
ð<b > 0; a 6¼ 0; 1; 2; ...Þ½34
The q-Gauss Hypergeometric Series
q-Analog of Euler’s integral representation
2
1
ðq
a
; q
b
; q
c
; q; zÞ
¼
q
ðcÞ
q
ðaÞ
q
ðc bÞ
Z
1
0
t
b1
ðtq; qÞ
1
ðtq
cb
; qÞ
1
ðtzq
a
; qÞ
1
tz; qÞ
1
d
q
t ð<b > 0; jzj < 1Þ½35
q-Special Functions 107
By substitution of [25], formula [35] becomes a
transformation formula:
2
1
ða; b; c; q; zÞ
¼
ðaz; qÞ
1
ðz; qÞ
1
ðb; qÞ
1
ðc; qÞ
1
2
1
ðc=b; z ; az; q; bÞ½36
Note the mixing of argument z and parameters
a, b, c on the right-hand side.
Evaluation formulas in special points
2
1
a; b; c; q; c=ðabÞðÞ
¼
ðc=a; c=b; qÞ
1
ðc; c=ðabÞ; qÞ
1
ðjc=ðabÞj < 1Þ½37
2
1
ðq
n
; b; c; q; cq
n
=bÞ¼
ðc=b; qÞ
n
ðc; qÞ
n
½38
2
1
ðq
n
; b; c; q; qÞ¼
ðc=b; qÞ
n
b
n
ðc; qÞ
n
½39
Two general transformation formulas
2
1
a; b
c
; q; z
¼
ðaz; qÞ
1
ðz; qÞ
1
2
2
a; c=b
c; az
; q; bz
½40
¼
ðabz=c; qÞ
1
ðz; qÞ
1
2
1
c=a; c=b
c
; q;
abz
c
½41
Transformation formulas in the terminating case
2
1
q
n
; b
c
; q; z
¼
ðc=b; qÞ
n
ðc; qÞ
n
3
2
q
n
; b; q
n
bc
1
z
q
1n
bc
1
; 0
; q; q
"#
½42
¼ðq
n
bc
1
z; qÞ
n
3
2
q
n
; cb
1
; 0
c; qcb
1
z
1
; q; q
½43
¼
ðc=b; qÞ
n
ðc; qÞ
n
b
n
3
1
q
n
; b; qz
1
q
1n
bc
1
; q;
z
c
½44
Second order q-difference equation
zðq
c
q
aþbþ1
zÞðD
2
q
uÞðzÞ
þ
1 q
c
1 q
q
b
1 q
a
1 q
þq
a
1 q
bþ1
1 q
z
ðD
q
uÞðzÞ
1 q
a
1 q
1 q
b
1 q
uðzÞ¼0 ½45
Some special solutions of [45] are:
u
1
ðzÞ :¼
2
1
ðq
a
; q
b
; q
c
; q; zÞ½46
u
2
ðzÞ :¼ z
1c
2
1
ðq
1þac
; q
1þbc
; q
2c
; q; zÞ½47
u
3
ðzÞ :¼ z
a
2
1
ðq
a
; q
acþ1
; q
abþ1
; q; q
abþcþ1
z
1
Þ½48
They are related by:
u
1
ðzÞþ
ðq
a
; q
1c
; q
cb
; qÞ
1
ðq
c1
; q
acþ1
; q
1b
; qÞ
1
ðq
b1
z; q
2b
z
1
; qÞ
1
ðq
bc
z; q
cbþ1
z
1
; qÞ
1
u
2
ðzÞ
¼
ðq
1c
; q
abþ1
; qÞ
1
ðq
1b
; q
acþ1
; qÞ
1
ðq
aþbc
z; q
cabþ1
z
1
; qÞ
1
z
a
ðq
bc
z; q
cbþ1
z
1
; qÞ
1
u
3
ðzÞ½49
Summation and Transformation Formulas
for
r
r1
Series
An
r
r1
series [15] is called ‘‘ balanced’’ if b
1
...b
r1
=
qa
1
...a
r
and z = q, and the series is called ‘ ‘very well-
poised’’ if qa
1
= a
2
b
1
= a
3
b
2
= = a
r
b
r1
and qa
1=2
1
=
a
2
= a
3
. The following more compact notation is
used for very well-poised series:
r
W
r1
ða
1
; a
4
; a
5
; ...; a
r
; q; zÞ
:¼
r
r1
a
1
; qa
1=2
1
; qa
1=2
1
; a
4
; ...; a
r
a
1=2
1
; a
1=2
1
; qa
1
=a
4
; ...; qa
1
=a
r
; q; z
2
4
3
5
½50
Below only a few of the most important identities
are given. See Gasper and Rahman (2004) for many
more. An important tool for obtaining complicated
identities from more simple ones is Bailey’s Lemma,
which can moreover be iterated (Bailey chain), see
Andrews (1986, ch.3).
The q-Saalschu¨ tz sum for a terminating balanced
3
2
3
2
a; b; q
n
c; q
1n
abc
1
; q; q
¼
ðc=a; c=b; qÞ
n
ðc; c=ðabÞ; qÞ
n
½51
Jackson’s sum for a terminating balanced
8
W
7
8
W
7
ða; b; c; d; q
nþ1
a
2
=ðbcdÞ; q
n
; q; qÞ
¼
ðqa; qa=ðbcÞ; qa=ðbdÞ; qa=ðcdÞ; qÞ
n
ðqa=b; qa=c; qa=d; qa=ðbcdÞ; qÞ
n
½52
108 q-Special Functions
Watson’s transformation of a terminating
8
W
7
into a
terminating balanced
4
3
8
W
7
a; b; c; d; e; q
n
; q;
q
nþ2
a
2
bcde
¼
ðqa; qa=ðdeÞ; qÞ
n
ðqa=d; qa=e; qÞ
n
4
3
q
n
; d; e; qa=ðbcÞ
qa=b; qa =c; q
n
de=a
; q; q
½53
Sears’ transformation of a terminating balanced
4
3
4
3
q
n
; a; b; c
d; e; f
; q; q
¼
ðe=a; f =a; qÞ
n
ðe; f ; qÞ
n
a
n
4
3
q
n
; a; d=b; d=c
d; q
1n
a=e; q
1n
a=f
; q; q
½54
By iteration and by symmetries in the upper and in
the lower parameters, many other versions of this
identity can be found. An elegant comprehensive
formulation of all these versions is as follows.
Let x
1
x
2
x
3
x
4
x
5
x
6
= q
1n
. Then the following
expression is symmetric in x
1
, x
2
, x
3
, x
4
, x
5
, x
6
:
q
ð1=2Þnðn1Þ
ðx
1
x
2
x
3
x
4
; x
1
x
2
x
3
x
5
; x
1
x
2
x
3
x
6
; qÞ
n
ðx
1
x
2
x
3
Þ
n
4
3
q
n
; x
2
x
3
; x
1
x
3
; x
1
x
2
x
1
x
2
x
3
x
4
; x
1
x
2
x
3
x
5
; x
1
x
2
x
3
x
6
; q; q
"#
½55
Similar formulations involving symmetry groups can
be given for other transformations, see Van der Jeugt
and Srinivasa Rao (1999).
Bailey’s transformation of a term inating
balanced
10
W
9
10
W
9
a; b; c; d; e; f ;
q
nþ2
a
3
bcdef
; q
n
; q; q
¼
ðqa; qa=ðef Þ; ðqaÞ
2
=ðbcdeÞ; ðqaÞ
2
=ðbcdf Þ; qÞ
n
ðqa=e; qa=f ; ðqaÞ
2
=ðbcdef Þ; ðqaÞ
2
=ðbcdÞ; qÞ
n
10
W
9
qa
2
bcd
;
qa
cd
;
qa
bd
;
qa
bc
; e; f ;
q
nþ2
a
3
bcdef
; q
n
; q; q
½56
Rogers–Ramanujan Identities
0
1
ð; 0; q; qÞ¼
X
1
k¼0
q
k
2
ðq; qÞ
k
¼
1
ðq; q
4
; q
5
Þ
1
½57
0
1
ð; 0; q; q
2
Þ¼
X
1
k¼0
q
kðkþ1Þ
ðq; qÞ
k
¼
1
ðq
2
; q
3
; q
5
Þ
1
½58
Bilateral Series
Definition [1] can be extended by
ða; qÞ
k
:¼
ða; qÞ
1
ðaq
k
; qÞ
1
ðk 2 ZÞ½59
Define a bilateral q-hypergeometric series by the
Laurent series
r
s
a
1
; ...; a
r
b
1
; ...; b
s
; q; z
"#
¼
r
s
ða
1
; ...; a
r
; b
1
; ...; b
s
; q; zÞ
:¼
X
1
k¼1
ða
1
; ...; a
r
; qÞ
k
ðb
1
; ...; b
s
; qÞ
k
ð1Þ
k
q
ð1=2Þkðk1Þ
sr
z
k
ða
1
; ...; a
r
; b
1
; ...; b
s
6¼ 0; s rÞ½60
The Laurent series is convergent if jb
1
...b
s
=(a
1
...a
r
)j <
jzj and moreover, for s = r, jzj< 1.
Ramanujan’s
1
1
summation formula
1
1
ðb; c; q; zÞ
¼
ðq; c=b; bz; q=ðbzÞ; qÞ
1
ðc; q=b; z; c=ðbzÞ; qÞ
1
ðjc=bj < jz < 1Þ½61
This has as a limit case
0
1
ð; c; q; zÞ¼
ðq; z; q=z; qÞ
1
ðc; c=z; qÞ
1
ðjzj > jcjÞ ½ 62
and as a further specialization the Jacobi triple
product identity
X
1
k¼1
ð1Þ
k
q
ð1=2Þkðk1Þ
z
k
¼ðq; z; q=z; qÞ
1
ðz 6¼ 0Þ½63
which can be rewritten as a product formula for a
theta function:
4
ðx; qÞ :¼
X
1
k¼1
ð1Þ
k
q
k
2
e
2ikx
¼
Y
1
k¼1
ð1 q
2k
Þ
1 2q
k1
cosð2xÞþq
4k2
½64
q-Hypergeometric Orthogonal
Polynomials
Here we discuss families of orthogonal polyno-
mials {p
n
(x)} which are expressible as terminating
q-hyper geometric series (0 < q < 1) and for
which either (1) P
n
(x):= p
n
(x)or(2)P
n
(x):= p
n
q-Special Functions 109
((1=2)(x þ x
1
)) are eigenfunctions of a second-
order q-difference operator, that is,
AðxÞP
n
ðqxÞþBðxÞP
n
ðxÞþCðxÞP
n
ðq
1
xÞ
¼
n
P
n
ðxÞ½65
where A(x), B(x), and C(x) are independent of n,
and where the
n
are the eigenvalues. The generic
cases are the four-parameter classes of ‘‘Askey–
Wilson polynomials’’ (continuous weight function)
and q-Racah polynomials (discrete weights
on finitely many points). They are of type (2) (quad-
ratic q-lattice). All other cases can be obtained from
the generic cases by specia lization or limit transition.
In particular, one thus obtains the generic three-
parameter classes of type (1) (linear q-lattice). These
are the big q-Jacobi polynomials (orthogonality by
q-integral) and the q-Hahn polynomials (discrete
weights on finitely many points).
Askey–Wilson Polynomials
Definition as q-hypergeometric series
p
n
ðcos Þ¼p
n
ðcos ; a; b; c; d jqÞ
:¼
ðab; ac; ad; qÞ
n
a
n
4
3
q
n
; q
n1
abcd; ae
i
; ae
i
ab; ac; ad
; q; q
"#
½66
This is symmetric in a, b, c, d.
Orthogonality relation Assume that a, b, c, d are
four reals, or two reals and one pair of complex
conjugates, or two pairs of complex conjugates.
Also assume that jabj, jacj, jadj, jbcj, jbdj, jcdj < 1.
Then
Z
1
1
p
n
ðxÞp
m
ðxÞwðxÞdx
þ
X
k
p
n
ðx
k
Þp
m
ðx
k
Þ!
k
¼ h
n
n;m
½67
where
2 sin wðcos Þ¼
ðe
2i
; qÞ
1
ðae
i
; be
i
; ce
i
; de
i
; qÞ
1
2
½68
h
0
¼
ðabcd; qÞ
1
ðq; ab; ac; ad; bc; bd; cd; qÞ
1
h
n
h
0
¼
1 abcdq
n1
1 abcdq
2n1
ðq; ab; ac; ad; bc; bd; cd; qÞ
n
ðabcd; qÞ
n
½69
and the x
k
are the points (1=2)(eq
k
þ e
1
q
k
)with
e any of the a, b, c, d of absolute value >1; the sum
is over the k 2 Z
0
with jeq
k
j > 1. The !
k
are
certain weights which can be given explicitly. The
sum in [67] does not occur if moreover
jaj, jbj , jcj, jdj < 1.
A more uniform way of writing the orthogonality
relation [67] is by the contour integral
1
2i
I
C
p
n
1
2
ðz þ z
1
Þ
p
m
1
2
ðz þ z
1
Þ
ðz
2
; z
2
; qÞ
1
ðaz; az
1
; bz; bz
1
; cz; cz
1
; dz; dz
1
; qÞ
1
dz
z
¼ 2h
n
n;m
½70
where C is the unit circle traversed in positive
direction with suitable deform ations to separate the
sequences of poles converging to zero from the
sequences of poles diverging to 1.
The case n = m = 0of[70] or [67] is known as the
Askey–Wilson integral.
q-Difference equation
AðzÞP
n
ðqzÞ AðzÞþAðz
1
Þ
P
n
ðzÞþAðz
1
ÞP
n
ðq
1
zÞ
¼ðq
n
1Þð1 q
n1
abcdÞP
n
ðzÞ½71
where P
n
(z)=p
n
(
1
2
(z þz
1
)) and A(z)= (1 az)
(1 bz)(1 cz)(1dz)=((1 z
2
)(1 qz
2
))
Special cases These include the continuous
q-Jacobi polynomials (two parameters), the contin-
uous q-ultraspherical polynomials (symmetric one-
parameter case of continuous q-Jacobi), the
Al-Salam-Chihara polynomials (Askey–Wilson with
c = d = 0), and the continuous q-Hermite polyno-
mials (Askey–Wilson with a = b = c = d = 0).
Continuous q-Ultraspherical Polynomials
Definitions as finite Fourier series and as special
Askey–Wilson polynomial
C
n
ðcos ; jqÞ
:¼
X
n
k¼0
ð; qÞ
k
ð; qÞ
nk
ðq; qÞ
k
ðq; qÞ
nk
e
iðn2kÞ
½72
¼
ð; qÞ
n
ðq; qÞ
n
p
n
ðcos ;
1=2
; q
1=2
1=2
;
1=2
;
q
1=2
1=2
j qÞ½73
110 q-Special Functions
Orthogonality relation (1 <<1)
1
2
Z
0
C
n
ðcos ; ; qÞ C
m
ðcos ; ; qÞ
ðe
2i
; qÞ
1
ðe
2i
; qÞ
1
2
d
¼
ð; q; qÞ
1
ð
2
; q; qÞ
1
1
1 q
n
ð
2
; qÞ
n
ðq; qÞ
n
n;m
½74
q-Difference equation
AðzÞP
n
ðqzÞ AðzÞþAðz
1
Þ
P
n
ðzÞþAðz
1
ÞP
n
ðq
1
zÞ
¼ðq
n
1Þð1 q
n
2
ÞP
n
ðzÞ½75
where P
n
(z)= C
n
(
1
2
(z þz
1
); jq)andA(z)= (1 z
2
)
(1 qz
2
)=((1 z
2
)(1 qz
2
)).
Generating function
ðe
i
z;e
i
z; qÞ
1
ðe
i
z; e
i
z; qÞ
1
¼
X
1
n¼0
C
n
ðcos ; jqÞz
n
ðjzj < 1; 0 ; 1 <<1Þ½76
Special case: the continuous q-Hermite polynomials
H
n
ðxjqÞ¼ðq; qÞ
n
C
n
ðx; 0 jqÞ½77
Special cases: the Chebyshev polynomials
C
n
ðcos ; qjqÞ¼U
n
ðcos Þ :¼
sinððn þ 1ÞÞ
sin
½78
lim
"1
ðq; qÞ
n
ð; qÞ
n
C
n
ðcos ; jqÞ¼T
n
ðcos Þ
:¼ cosðnÞðn > 0Þ½79
q-Racah Polynomials
Definition as q-hypergeometric series
(n = 0, 1, ..., N)
R
n
ðq
y
þ q
yþ1
; ; ; ; jqÞ
:¼
4
3
q
n
;q
nþ1
; q
y
;q
yþ1
q; q ; q
; q; q
"#
ð; or ¼ q
N1
Þ½80
Orthogonality relation
X
N
y¼0
R
n
ðq
y
þ q
yþ1
ÞR
m
ðq
y
þ q
yþ1
Þ!
y
¼ h
n
n;m
½81
where !
y
and h
n
can be explicitly given.
Big q-Jacobi Polynomials
Definition as q-hypergeometric series
P
n
ðxÞ¼P
n
ðx; a; b; c; qÞ
:¼
3
2
q
n
; q
nþ1
ab; x
qa; qc
; q; q
"#
½82
Orthogonality relation
Z
qa
qc
P
n
ðxÞP
m
ðxÞ
ða
1
x; c
1
x; qÞ
1
ðx; bc
1
x; qÞ
1
d
q
x ¼ h
n
n;m
;
ð0 < a < q
1
; 0 < b < q
1
; c < 0Þ½83
where h
n
can be explicitly given.
q-Difference equation
AðxÞP
n
ðqxÞðAðxÞþCðxÞÞ P
n
ðxÞþCðxÞP
n
ðq
1
xÞ
¼ðq
n
1Þð1 abq
nþ1
ÞP
n
ðxÞ½84
where A(x)= aq(x 1)(bx c)=x
2
and C(x)=(x qa)
(x qc)=x
2
Limit case: Jacobi polynomials P
(, )
n
(x)
lim
q"1
P
n
ðx; q
; q
; q
1
d; qÞ
¼
n!
ð þ 1Þ
n
P
ð;Þ
n
2x þ d 1
d þ 1
½85
Special case: the little q-Jacobi polynomials
p
n
ðx; a; b; qÞ¼ðbÞ
n
q
ð1=2Þnðnþ1Þ
ðqb; qÞ
n
ðqa; q Þ
n
P
n
ðqbx; b; a; 0; qÞ½86
¼
2
1
ðq
n
; q
nþ1
ab; qa; q; qxÞ½87
which satisfy orthogonality relation (for 0 < a < q
1
and b < q
1
)
Z
1
0
p
n
ðx; a; b; qÞp
m
ðx; a; b; qÞ
ðqx; qÞ
1
ðqbx; qÞ
1
x
log
q
a
d
q
x
¼
ðq; qab ; qÞ
1
ðqa; qb ; qÞ
1
ð1 qÞðqaÞ
n
1 abq
2nþ1
ðq; qb; qÞ
n
ðqa; qab; qÞ
n
n;m
½88
Limit case: Jackson’s third q-Bessel function (see [22])
lim
N !1
p
Nn
ðq
Nþk
; q
; b; qÞ¼
ðq; q Þ
1
ðq
þ1
; qÞ
1
q
ðnþkÞ
J
ð3Þ
ð2q
ð1=2ÞðnþkÞ
; qÞð>1Þ½89
q-Special Functions 111