Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
Подождите немного. Документ загружается.
Ordinary Special Functions
W Van Assche, Katholieke Universiteit Leuven,
Leuven, Belgium
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The exponential function, the logarithm, the trigo-
nometric functions, and various other functions are
often used in mathematics and physics. They are
transcendental functions in the sense that they
cannot be obtained by a finite number of operations
as a solution of an algebraic (polynomial) equation.
Typically, they are obtained by a Taylor series
expansion. Many other higher transcendental func-
tions arise in mathematical physics, often as solu-
tions of differential equations. A precise knowledge
of the behavior of such functions, their relation with
other functions, addition, multiplication and com-
position properties, representations as an infinite
series, or as an integral, often shed a lot of light onto
the problem in which they arise. If they are
sufficiently useful to a large audience, then they
usually get a name and they will be called special
functions. In what follows, we describe a few of
these special functions of one variable, but clearly
this is just a tip of the iceberg. Many other special
functions exist and we refer to the classical tables of
Abramowitz and Stegun (1964) and the Bateman
manuscript project (Erde´lyi et al. 1953–55) for more
special functions. Nowadays, there have been
numerous q-extensions of special functions (see
q-Special Functions).
Gamma and Beta Function
The gamma function is defined by
ðzÞ¼
Z
1
0
t
z1
e
t
dt ; <z > 0: ½1
It satisfies the functional equation (z þ 1) = z(z)
and since (1) = 1wehave(n þ 1) = n! for n 2 N.
The gamma function therefore extends the factorial
function for integers to complex numbers. The
functional equation
ðzÞð1 zÞ¼
sin z
½2
allows to continue the gamma function analytically
to <z < 0 and the gamma function becomes an
analytic function in the complex plane, with a
simple pole at 0 and at all the negative integers.
The residue of (z)atz = n is equal to (1)
n
=n!.
Legendre’s duplication formula is
ð2zÞ¼
2
2z1
ffiffiffi
p
ðzÞðz þ 1=2Þ½3
from which one can obtain the special value
(1=2) =
ffiffiffi
p
. Finally, two useful infinite product
representations are
ðzÞ¼lim
n!1
n!n
z
zðz þ 1Þðz þ nÞ
and
1
ðzÞ
¼ ze
z
Y
1
n¼1
ð1 þ z=nÞe
z=n
where is Euler’s constant:
¼ lim
n!1
X
n
k¼1
1
k
log n
!
¼ 0:577 215 664 9 ... ½4
The beta function is a function of two variables
given by
Bðx; yÞ¼
Z
1
0
t
x1
ð1 tÞ
y1
dt
<x > 0; <y > 0 ½5
Clearly it satisfies B( x, y) = B(y, x) and it is related to
the gamma function by
Bðx; yÞ¼
ðxÞðyÞ
ðx þ yÞ
½6
The gamma and beta function are quite useful in
probability theory. One of the most common
probability distributions on the positive real line is
the gamma distribution
PrðX xÞ¼
1
ðÞ
Z
x
0
e
t=
t
1
dt ; x 0
The case = 3=2 is the Maxwell–Boltzmann dis-
tribution. The most common probability distribu-
tion on the interval [0, 1] is the beta distribution
PrðY xÞ¼
1
Bð; Þ
Z
x
0
t
1
ð1 tÞ
1
dt
where 0 x 1.
The psi function is the logarithmic derivative of
the gamma function
ðzÞ¼
0
ðzÞ
ðzÞ
½7
Ordinary Special Functions 637
It is meromorphic with simple poles at 0 and at the
negative integers. Special values are (1) = and
ðn þ 1Þ¼
X
n
k¼1
1
k
where is Euler’s constant. These can be obtained
from the functional equation
ðzÞ¼ ðz þ 1Þ
1
z
Bessel Functions
Bessel’s differential equ ation is
x
2
y
00
þ xy
0
þðx
2
2
Þy ¼ 0 ½8
where derivatives are with respect to x and is a
complex number. This differential equation has a
regular singularity at x = 0 and an irregular singu-
larity at x = 1. The standard method of finding a
solution in the neighborhood of a regular singularity
gives the solution
J
ðxÞ¼ðx=2Þ
X
1
k¼0
ðx
2
=4Þ
k
k!ðk þ þ 1Þ
and J
(x) is another solution (if 6¼ 0). The
function J
is called the ‘‘Bessel function of the first
kind’’ and is the ‘‘order’’ of the Bessel function.
The series x
J
(x) is an entire function of the
variable x. The function
Y
ðxÞ¼
J
ðxÞcosðÞJ
ðxÞ
sinðÞ
is also a solution of Bessel’s differential equation
and is known as the ‘‘Bessel function of the second
kind of order .’’ Two other solutions that are often
used are
H
ð1Þ
ðxÞ¼J
ðxÞþiY
ðxÞ
H
ð2Þ
ðxÞ¼J
ðxÞiY
ðxÞ
which are the first and second ‘‘Hankel functions.’’
Bessel functions appear if one solves the wave
equation in cylindrical or spherical coordinates, using
separation of variables. The Helmholtz equation
r
2
F þ k
2
F = 0 in cylindrical coordinates , , z is
@
2
F
@
2
þ
1
@F
@
þ
1
2
@
2
F
@
2
þ
@
2
F
@z
2
þ k
2
F ¼ 0
and if we look for a solution of the form
f ()g()h(z), then this leads to a differential equation
for f of the form
d
2
f
d
2
þ
1
df
d
þ½k
2
a
2
ð=Þ
2
f ¼ 0
where a and are separation constants. The general
solution is f () = Z
((k
2
a
2
)), where Z
is any of
the Bessel functions given higher or linear combina-
tions of them. In spherical coordinates r, , the
Helmholtz equation is
@
2
F
@r
2
þ
2
r
@F
@r
þ
1
r
2
@
2
F
@
2
þ
cot
r
2
@F
@
þ
1
r
2
sin
2
@
2
F
@
2
þ k
2
F ¼ 0
and for a solution of the form f (r)g()h() one
obtains a differential equation for f of the form
1
r
d
2
ðrf Þ
dr
2
þ½k
2
ð þ 1Þ=r
2
f ¼ 0
with general solution f (r) = Z
þ(1=2)
(kr)=
ffiffi
r
p
.
Bessel functions have very simple differentiation
formulas:
½z
J
ðzÞ
0
¼ z
J
1
ðzÞ
½z
J
ðzÞ
0
¼z
J
þ1
ðzÞ
The first formula can be seen as a lowering
operation, the second as a raising operation. Some
integral representations are
J
ðzÞ¼
ðz=2Þ
ffiffiffi
p
ð þ 1=2Þ
Z
0
sin
2
cosð z cos Þd
or
J
ðzÞ¼
ðz=2Þ
ffiffiffi
p
ð þ 1=2Þ
Z
1
1
ð1 x
2
Þ
1=2
cos zx dx
which hold for <>1=2. For real the Bessel
function J
has infinitely many real zeros, and when
>1, then all the zeros are real. All the zeros are
simple (except possibly at the origin). Each of
the functions J
(z), Y
(z), H
(1)
(z), or H
(2)
(z) satisfies
the recurrence relation
za
1
ðzÞþza
þ1
ðzÞ¼2a
ðzÞ
and the differential–recurrence relation
a
1
ðzÞa
1
ðzÞ¼2a
0
ðzÞ
638 Ordinary Special Functions
Modified Bessel Functions
The modified Bessel equation is
x
2
y
00
þ xy
0
ðx
2
þ
2
Þy ¼ 0 ½9
Clearly J
(ix) is a solution of this equation. The
‘‘modified Bessel function of the first kind’’ is
defined as
I
ðxÞ¼e
i=2
J
ðxe
i=2
Þ; <arg x =2 ½10
so that
I
ðxÞ¼ðx=2Þ
X
1
k¼0
ðx=2Þ
2k
k!ð þ k þ 1Þ
If is not an integer, then I
(x)andI
(x) are two
linearly independent solutions of [9], and when = n
is an integer one has I
n
(x) = I
n
(x). The ‘‘modified
Bessel function of the second kind’’ is defined by
K
ðxÞ¼
2 sin
½I
ðxÞI
ðxÞ
Some special cases of modified Bessel functions are
I
1=2
ðxÞ¼
ffiffiffiffiffiffi
2
x
r
sinh x
I
1=2
ðxÞ¼
ffiffiffiffiffiffi
2
x
r
cosh x
and
K
1=2
ðxÞ¼
ffiffiffiffiffiffi
2x
r
e
x
One has the integral representa tion
K
ðzÞ¼
Z
1
0
e
z cosh x
cosh x dx
and
I
ðzÞ¼
ðz=2Þ
ffiffiffi
p
ð þ 1=2Þ
Z
1
1
ð1 x
2
Þ
1=2
e
zx
dx
whenever <>1=2. The ‘‘Airy functions’’ are
given by
AiðzÞ¼
ffiffiffi
z
p
3
I
1=3
ðÞI
1=3
ðÞ
¼
ffiffiffiffiffiffiffiffi
z=3
p
K
1=3
ðÞ
BiðzÞ¼
ffiffiffiffiffiffiffiffi
z=3
p
I
1=3
ðÞþI
1=3
ðÞ
where = 2z
2=3
=3. They are both a solution of Airy’s
differential equation
y
00
ðzÞzyðzÞ¼0
Hypergeometric Series
A power series
P
1
n = 0
c
n
z
n
is said to be hypergeo-
metric when the ratio c
nþ1
=c
n
is a rational function
of the index n. Most series that one finds in calculus
textbooks are hypergeometric series and some of
them define important special functions. When
c
nþ1
c
n
¼
ðn þ a
1
Þðn þ a
2
Þðn þ a
p
Þ
ðn þ b
1
Þðn þ b
2
Þðn þ b
q
Þðn þ 1Þ
then we write the corresponding series as
p
F
q
a
1
; a
2
; ...; a
p
b
1
; b
2
; ...; b
q
z
¼
X
1
n¼0
ða
1
Þ
n
ða
2
Þ
n
ða
p
Þ
n
ðb
1
Þ
n
ðb
2
Þ
n
ðb
q
Þ
n
z
n
n!
½11
where (a)
n
= a(a þ 1)(a þ 2) (a þ n 1), with
(a)
0
= 1, is the rising factorial or Pochhammer
symbol. When p and q are small, one also uses the
notation
p
F
q
(a
1
, ..., a
p
; b
1
, ..., b
q
; z) where a semi-
colon (;) is used to separate the parameters in the
numerator from the parameters in the denominator
and also to separate the parameters from the
variable z. Some special cases are:
the exponential series
0
F
0
ð; ; zÞ¼
X
1
n¼0
z
n
n!
¼ expðzÞ
the geometric series
1
F
0
ð1; ; zÞ¼
X
1
n¼0
z
n
¼
1
1 z
the binomial series
1
F
0
ð; ; zÞ¼
X
1
n¼0
n
z
n
¼ð1 þ zÞ
the logarithmic function
2
F
1
ð1; 1; 2; zÞ¼
X
1
n¼0
z
n
n þ 1
¼
1
z
logð1 zÞ
the Bessel function
ðz=2Þ
0
F
1
ð; þ 1; z
2
=4Þ¼ð þ 1 ÞJ
ðzÞ
For generic values of the parameters, we see that the
hypergeometric series converges everywhere in the
complex plane when q p, it converges for jzj < 1
when p = q þ 1, and for p > q þ 1 it is only defined at
Ordinary Special Functions 639
z = 0. When one of the numerator parameters is a
negative integer, say a
1
= m, then the series is
terminating and defines a polynomial of degree m.
None of the denominator parameters is allowed to
be a negative integer m, unless there is a
numerator parameter which is a negative integer
k with k < m. For q p, the hypergeometric
series therefore defines an entire function which is
the corresponding hypergeometric function. For
p = q þ 1, the hypergeometric series only converges
in the open unit disk, but sometimes it can be
continued analytically to a larger domain in the
complex plane. The analytic continuation of the
hypergeometric series is then called the hypergeo-
metric function. Take for example the geometric
series, then it is clear that the hypergeometric
series converges in the open unit disk, but the
corresponding hypergeometric function is defined
in the whole complex plane with a simple pole at
z = 1. The logarithmic function log (1 z)hasa
hypergeometric series in the open unit disk, but it
can be continued analytically to the complex plane
with a cut along [1, 1) and a branch point
at z = 1.
Gauss Hypergeometric Function
The most famous hypergeometric function is the
Gauss hypergeometric function defined for jzj < 1
by the hypergeometric series
2
F
1
ða; b; c; zÞ¼
X
1
n¼0
ðaÞ
n
ðbÞ
n
ðcÞ
n
n!
z
n
½12
which is ofte n denoted by F(a, b; c; z). It is a solution
of the hypergeometric equation
zð1 zÞy
00
ðzÞþ½c ða þ b þ 1Þzy
0
ðzÞ
abyðzÞ¼0 ½13
and this solution is regular at z = 0. Obviously,
2
F
1
(a, b; c; z) =
2
F
1
(b, a; c; z). The six functions
2
F
1
(a 1, b; c; z),
2
F
1
(a, b 1; c; z), and
2
F
1
(a, b; c
1; z) are called contiguous to
2
F
1
(a, b; c; z) and there
are 15 linear relations (with coefficients which are
linear functions of z) between
2
F
1
(a, b; c; z) and any
two contiguous functions. Two of these relations are
ð2a c az þbzÞFða; b; c; zÞþðc aÞFða 1; b; c; zÞ
þaðz 1ÞFða þ1; v ; c; zÞ¼0
and
cða ðc bÞzÞFða; b; c; zÞacð1 zÞFða þ 1; b;
c; zÞ
þðc aÞðc bÞzFða; b; c þ 1; zÞ¼0
Euler gave the integral representation
2
F
1
ða; b; c; zÞ
¼
ðcÞ
ðbÞðc bÞ
Z
1
0
x
b1
ð1 xÞ
cb1
ð1 zxÞ
a
dx ½14
for <c > 0 and <b > 0. This allows to find the
analytic continuation from the open unit disk to the
complex plane. A useful result is the Gauss summa-
tion formula
2
F
1
ða; b; c; 1Þ¼
ðcÞðc a bÞ
ðc aÞðc bÞ
<ðc a bÞ > 0
The special case for a terminating series is known as
the Chu–Vandermonde sum
2
F
1
ðn; a; c; 1Þ¼
ðc aÞ
n
ðcÞ
n
Pfaff’s transformation is
2
F
1
ða; b; c; zÞ¼ð1 zÞ
a
2
F
1
a; c b; c;
z
z 1
and Euler’s transformation is
2
F
1
ða; b; c; zÞ¼ð1 zÞ
cab
2
F
1
ðc a; c b; c; zÞ
Confluent Hypergeometric Function
The hypergeometric series
1
F
1
(a; c; z) defines an
entire function in the complex plane and satisfies
the differential equation
zy
00
ðzÞþðc zÞy
0
ðzÞayðzÞ¼0 ½15
This hypergeometric series (and the differential equa-
tion) are formally obtained from
2
F
1
(a, b; c; z=b)by
letting b !1, which gives a confluence of two of the
singularities at z = 1. This is the reason why the
differential equation [15] is known as the confluent
hypergeometric equation. The solution
ða; c; zÞ¼
1
F
1
ða; c; zÞ½16
is called a confluent hypergeometric function, and a
second linearly independent solution of [15] is
z
1c
(c a þ 1, 2 c; z). The function
ða; c; zÞ¼
ð1 cÞ
ða c þ1Þ
ða; c; zÞ
þ
ðc 1Þ
ðaÞ
z
1c
ða c þ1; 2 c; zÞ½17
640 Ordinary Special Functions
is therefore also a solution of eqn [15]. The
following integral representations hold:
ða; c; zÞ¼
ðcÞ
ðaÞðc aÞ
Z
1
0
e
zx
x
a1
ð1 xÞ
ca1
dx
whenever <c > <a > 0, and
ða; c; zÞ¼
1
ðaÞ
Z
1
0
e
zx
x
a1
ð1 þ xÞ
ca1
dx
whenever <a > 0.
The ‘‘Whittaker functions’’ are defined as
M
;
ðzÞ¼e
z=2
z
c=2
ða; c; zÞ
W
;
ðzÞ¼e
z=2
z
c=2
ða; c; zÞ
with = a þ c=2and = (c 1)=2. They are a
solution of the Whittaker equation
y
00
ðzÞþ
1
4
þ
z
þ
1 4
2
4z
2
yðzÞ¼0
The ‘‘parabolic cylinder functions’’ are also con-
fluent hypergeometric functions. They are given by
D
ðzÞ¼2
=2
e
z
2
=4
ð=2; 1=2 ; z
2
=2Þ
¼ 2
ð1Þ=2
e
z
2
=4
zðð1 Þ=2; 3=2; z
2
=2Þ
When is a non-negative integer, one finds Hermite
polynomials
H
n
ðzÞ¼2
n=2
e
z
2
=2
D
n
ð
ffiffiffi
2
p
zÞ
Classical Orthogonal Polynomials
A family of polynomials {p
n
(x), n 2 N}, where p
n
has degree n , is orthogonal on the real line if there is
a positive measure on the real line for which
Z
R
p
n
ðxÞp
m
ðxÞdðxÞ¼h
n
m;n
½18
Usually the measure is absolutely continuous, in
which case d(x) = w(x)dx with w a non-negative
density function on the real line, or is discrete and
supported on a finite or at most countable set. Any
family of orthogonal polynomials satisfies a ‘‘three-
term recurrence relation’’
xp
n
ðxÞ¼A
n
p
nþ1
ðxÞþB
n
p
n
ðxÞþC
n
p
n1
ðxÞ½19
with C
n
A
n1
> 0 for every n 1. For the
monic polynomials P
n
(x) = p
n
(x)=k
n
, with k
n
=
1=(A
0
A
1
A
2
A
n1
) this relation becomes
P
nþ1
ðxÞ¼ðx b
n
ÞP
n
ðxÞa
2
n
P
n1
ðxÞ
with b
n
= B
n
and a
2
n
= A
n1
C
n
. This recurrence
relation gives rise to a tridiagonal matrix
J ¼
b
0
a
1
0000
a
1
b
1
a
2
000
0 a
2
b
2
a
3
00
00a
3
b
3
a
4
0
000a
4
.
.
.
.
.
.
0000
.
.
.
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
which is formally symmetric and which is called the
‘‘Jacobi matrix.’’ The spectral measure of this opera-
tor, acting on ‘
2
(N), is equal to the orthogonality
measure whenever this symmetric operator can be
extended to a self-adjoint operator. If this is not
possibleinauniqueway–asituationwhichcanoccur
for unbounded operators only – then every self-adjoint
extension of J gives rise to a spectral measure which
can be used for the orthogonality conditions [18].In
this case, there are infinitely many positive measures
which can be used in the orthogonality relations and
all these measures have the same moments
m
n
¼
Z
R
x
n
dðxÞ
Some families of orthogonal polynomials have
additional properties which are quite useful in
many practical and physical applications, such as
the following:
The derivatives p
0
n
are again a family of orthogo-
nal polynomials (Hahn property).
The polynomials p
n
satisfy a second-order linear
differential equation of the form
ðxÞy
00
ðxÞþy
0
ðxÞ¼
n
yðxÞ
where is a polynomial of degree at most 2, is a
polynomial of degree 1, both independent of n,
and
n
is a real number (Bochner property).
The polynomials can be obtained by a Rodrigues
formula
wðxÞp
n
ðxÞ¼C
n
d
n
dx
n
wðxÞ
n
ðxÞðÞ
where w is a non-negative function and a
polynomial of degree at most 2 (Hildebrand
property).
There are three families of orthogonal polynomials
on the real line which have these three properties, and
Ordinary Special Functions 641
each of these three properties characterizes these
families. These are the Hermite polynomials, the
Laguerre polynomials, and the Jacobi polynomials. In
a more general situation when the orthogonality
relation is described by a linear functional and the
functional is not required to be positive, one has an
additional family of Bessel polynomials. The densities
w(x) for these families all satisfy a first-order differ-
ential equation [(x)w(x)]
0
= (x)w (x), where is a
polynomial of degree at most 2 and apolynomialof
degree 1. This equation is known as the ‘‘Pearson
equation.’ ’
Hermite Polynomials
Hermite polynomials H
n
(x) are orthogonal with
respect to the normal density w(x) = e
x
2
:
Z
1
1
H
n
ðxÞH
m
ðxÞe
x
2
dx ¼ 2
n
n!
n;m
Observe that the density satisfies w
0
= 2xw so that
= 1 and (x) = 2x. The recurrence relation is
H
nþ1
ðxÞ¼2xH
n
ðxÞ2nH
n1
ðxÞ
and the polynom ials satisf y the second-order differ-
ential equation
y
00
ðxÞ2xy
0
ðxÞþ2nyðxÞ¼0
The functions h
n
(x) = e
x
2
=2
H
n
(x) satisfy the differ-
ential equation
h
00
n
ðxÞþð2n þ 1 x
2
Þh
n
ðxÞ¼0
The derivatives satisfy H
0
n
(x) = 2nH
n1
(x)(lowering
operation) and one also has [e
x
2
H
n
(x)]
0
= e
x
2
H
nþ1
(x) (raising operation). The Rodrigues formula is
e
x
2
H
n
ðxÞ¼ð1Þ
n
d
n
dx
n
e
x
2
The polynomials can be written as a hypergeometric
series
H
n
ðxÞ¼ð2xÞ
n
2
F
0
ðn=2; ðn 1Þ=2; ; 1=x
2
Þ
or alternatively as
H
n
ðxÞ¼n!
X
bn=2c
k¼0
ð1Þ
k
ð2xÞ
n2k
k!ðn 2kÞ!
Their generating function is
X
1
n¼0
H
n
ðxÞ
t
n
n!
¼ expð2xt t
2
Þ
Hermite polynomials are relevant for the analysis of
the quantum harmonic oscillator, and the lowering
and raising operators there correspond to creation
and annihilation.
Laguerre Polynomials
Laguerre polynomials L
n
(x) are for >1orthogo-
nal with respect to the gamma density w(x) = x
e
x
on [0, 1):
Z
1
0
L
n
ðxÞL
m
ðxÞx
e
x
dx ¼
ðn þ Þ
n!
m;n
The Pearson equation is [xw]
0
= ( þ 1 x)w so that
(x) = x and (x) = þ 1 x. The recurrence rela-
tion is
ðn þ 1ÞL
nþ1
ðxÞ
¼ð2n þ þ 1 xÞL
n
ðxÞðn þ ÞL
n1
ðxÞ
and the differential equation is
xy
00
ðxÞþð þ 1 xÞy
0
ðxÞþnyðxÞ¼0
The functions ‘
n
(x) = x
=2
e
x=2
L
n
(x) satisfy
ðx‘
0
n
Þ
0
þ n þ
þ 1
2
x
4
2
4x
‘
n
¼ 0
Differentiation has the effect that
L
n
ðxÞ
0
¼L
þ1
n1
ðxÞ
and
x
e
x
L
n
ðxÞ
0
¼ðn þ 1Þx
1
e
x
L
1
nþ1
ðxÞ
The Rodrigues formula is
x
e
x
L
n
ðxÞ¼
1
n!
d
n
dx
n
½x
nþ
e
x
The hypergeometric exp ression is
n!L
n
ðxÞ¼ð þ 1Þ
n
1
F
1
ðn; þ 1; xÞ
and the generating function is
X
1
n¼0
L
n
ðxÞt
n
¼ð1 tÞ
1
exp
xt
t 1
Laguerre polynomials occur as eigenfunctions of the
hydrogen atom.
642 Ordinary Special Functions
Jacobi Polynomials
Jacobi polynomials P
(, )
n
(x) are orthogonal for the
beta density w(x) = (1 x)
(1 þ x)
on [1, 1]
whenever >1 and >1:
Z
1
1
P
ð;Þ
n
ðxÞP
ð;Þ
m
ðxÞð1 xÞ
ð1 þ xÞ
dx
¼
2
þþ1
2n þ þ þ 1
ðn þ þ1Þðn þ þ 1Þ
ðn þ þ þ 1Þ
n;m
The Pearson equation is [(1 x
2
)w]
0
= [
( þ þ 2)x]w and the differential equation is
ð1 x
2
Þy
00
ðxÞþ½ ð þ þ 2Þxy
0
ðxÞ
þ nðn þ þ þ 1ÞyðxÞ¼0
Differentiation has the effect
P
ð;Þ
n
ðxÞ
hi
0
¼ðn þ þ þ 1Þ=2P
ðþ1;þ1Þ
n1
ðxÞ
and
ð1 xÞ
ð1 þ xÞ
P
ð;Þ
n
ðxÞ
hi
0
¼2ðn þ 1Þð1 xÞ
1
ð1 þ xÞ
1
P
ð1;1Þ
nþ1
ðxÞ
The Rodrigues formula is
ð1 xÞ
ð1 þ xÞ
P
ð;Þ
n
ðxÞ
¼
ð1Þ
n
2
n
n!
d
n
dx
n
ð1 xÞ
nþ
ð1 þ xÞ
nþ
hi
In terms of hypergeometric series, one has
P
ð;Þ
n
ðxÞ¼
ð þ 1Þ
n
n!
2
F
1
n; n þ þ þ 1
þ 1
1 x
2
Observe that one has P
(, )
n
(x) = (1)
n
P
(, )
n
(x).
Special cases of the Jacobi polynomials are as
follows:
The ‘‘Legendre polynomials’’ P
n
(x) = P
(0, 0)
n
(x).
They appear when the Laplacian is separated in
spherical coordinates as functions of the polar
angle , for which x = cos .
The ‘‘Chebyshev polynomials’’ of the first kind
T
n
ðxÞ¼P
ð1=2;1=2Þ
n
ðxÞ=P
ð1=2;1=2Þ
n
ð1Þ
and of the second kind
U
n
ðxÞ¼ðn þ 1ÞP
ð1=2;1=2Þ
n
ðxÞ=P
ð1=2;1=2Þ
n
ð1Þ
These functions are more easily written by using the
change of variable x = cos and then T
n
(cos) =
cos n and U
n
(cos) = sin (n þ 1)= sin .
The ‘‘Gegenbauer polynomials’’ or ultraspherical
polynomials are Jacobi polynomials with equal
parameters:
C
n
ðxÞ¼ð2Þ
n
=ð þ 1=2Þ
n
P
ð1=2;1=2Þ
n
ðxÞ
Gegenbauer polynomials are involved in the
angular or spatial part of the wave function of
physical systems in a central potential in both
position and momentum space, and in the spatial
part of the wave function of hydrogenic systems in
momentum space, as well as in the eigen functions of
several quantum-mechanical potentials, such as the
relativistic harmonic oscillator.
Other Classical Orthogonal Polynomials
Instead of restri cting attention to the differential
operator D = d=dx, one can also use the (forward)
difference operator for which f (x) = f (x þ 1)
f (x), the divided difference operator
for which
f (x) =f ((x))=(x) with a quadratic function
, or certain q-difference operators and look for
orthogonal polynomials that satisfy difference equa-
tions in the variable x. Together with the three-term
recurrence relation (in the degree n), one then has
families of polynomials satisfying a bispectral
problem. For the differen ce operator and the divided
difference operator, this gives several important
families of orthogonal polynomials which all have
a hypergeometric representation. These hypergeo-
metric polynomials are usually listed in a tabl e, and
each level indicates the number of parameters and/or
the order of the hypergeometric function. This table
is known as Askey’s table and is given in Figure 1.
The extension with q-difference operators involves
basic hypergeomet ric series and q-extensions of
classical orthogonal polynomials.
‘‘Charlier polynomials’’ C
n
(x; a) are orthogonal
with respect to the Poisson distribution
X
1
k¼0
C
n
ðk; aÞC
m
ðk; aÞ
a
k
k!
¼ e
a
=a
n
n;m
The recurrence relatio n is
aC
nþ1
ðx; aÞþðx n aÞC
n
ðx; aÞ
þ nC
n1
ðx; aÞ¼0
and the second-order difference equation is
ayðx þ 1Þþðn x aÞyð xÞþxyðx 1Þ¼0
Ordinary Special Functions 643
The forward difference operator has the effect
C
n
(x; a) = n=aC
n1
(x; a) and the backward differ-
ence operator rf (x) = f (x) f (x 1) has the effect
r[a
x
=x!C
n
(x;a)] = a
x
=x!C
nþ1
(x; a). The hypergeo-
metric representation is C
n
(x; a) =
2
F
0
(n x; ;
1=a). Observe that the variable x appears as a
parameter of the hypergeometric series.
‘‘Krawtchouk polynomials’’ K
n
(x; p, N) are ortho-
gonal with respect to the binomial distribution:
X
N
k¼0
K
n
ðk; p ; NÞK
m
ðk; p ; NÞ
N
k
p
k
ð1 pÞ
Nk
¼
ð1Þ
n
n!
ðNÞ
n
p
n
ð1 pÞ
n
n;m
where N is a positive integer and 0 < p < 1. They
are given by K
n
(x; p, N) =
2
F
1
(n, x; N;1=p)
and correspond to Meixner polynomials for which
the parameter is a negative integer.
‘‘Meixner polynomials’’ m
n
(x; , c) are orthogonal
with respect to the negative binomial distribution
(Pascal distribution)
X
1
k¼0
m
n
ðk; ; cÞm
j
ðk; ; cÞ
ðÞ
k
c
k
k!
¼
n!
c
n
ðÞ
n
ð1 cÞ
n;j
where >0 and 0 < c < 1. They are given by
m
n
(x; , c) =
2
F
1
(n, x; ;1 1=c).
‘‘Meixner–Pollaczek polynomials’’ P
n
(x; ) are
orthogonal on (1, 1):
Z
1
1
P
m
ðx; ÞP
n
ðx; Þe
ð2Þx
jð þ ixÞj
2
dx
¼
2ðn þ 2Þ
ð2 sin Þ
2
n!
m;n
where >0 and 0 <<. The appropriate differ-
ence operator has an imaginary shift f (x) =
f (x þ i =2) f (x i=2) and one has P
n
(x; ) =
2 sin P
þ1=2
n1
(x; ). They are given by
P
n
ðx; Þ¼
ð2Þ
n
n!
e
in
2
F
1
n;þ ix
2
1 e
2i
‘‘Hahn and dual Hahn polynomials’’ are orthogo-
nal on a finite set of points. Hahn polynomials are
given by
Q
n
ðx; ; ; NÞ¼
3
F
2
n; n þ þ þ 1; x
þ 1; N
1
and their orthogonality is with respect to a
hypergeometric distribution on {0, 1, ..., N}. The
appropriate difference operator is the (forward)
difference operator . They are related to the 3 j
symbols or Wigner coefficients that arise when
considering angular momenta in two quantum
systems. Dual Hahn polynomials are given by
R
n
ððxÞ; ; ; NÞ¼
3
F
2
n; x; x þ þ þ 1
þ 1; N
1
where (x) = x(x þ þ þ 1). They are obtained
from the Hahn polynomials by interchanging the
roles of n and x. They are orthogonal on the set
{(0), (1), ..., (N)}. The appropriate difference
operator is the divided difference operator which
acts on f as f ((x))=(x).
‘‘Continuous Hahn and dual Hahn polynomials’’
are orthogonal on the real line. The continuous
Hahn polynomials are
p
n
ðx; a; b; c; dÞ
¼ i
n
ða þ cÞ
n
ða þ dÞ
n
n!
3
F
2
n; n þ a þ b þ c þ d 1; a þ ix
a þ c; a þ d
1
and the appropriate difference operator is the
difference operator with imaginary shift. The
continuous dual Hahn polynomials are
S
n
ðx
2
; a; b; cÞ¼ða þ bÞ
n
ða þ cÞ
n
3
F
2
n; a þ ix; a ix
a þ b; a þ c
1
and the appropriate difference operator is the divided
difference operator which acts on f as f (x
2
)=x
2
.
‘‘Wilson polynomials’’ are the most general system
of hypergeometric polynomials satisfying a bispec-
tral problem. All the other classical orthogonal
polynomials can be obtained from them by taking
Hermite
Laguerre Charlier
Meixner-
Pollaczek
Jacobi
Meixner Krawtchouk
Continuous
dual Hahn
Continuous
Hahn
Hahn Dual Hahn
Wilson
Racah
Figure 1 Askey’s table.
644 Ordinary Special Functions
appropriate parameters or as limiting cases. They are
given by
W
n
ðx
2
; a; b; c; dÞ
ða þbÞ
n
ða þcÞ
n
ða þdÞ
n
¼
4
F
3
n; n þa þb þc þd 1; a þix; a ix
a þb; a þc; a þd
1
and for R(a, b, c, d) > 0 (with nonreal parts appear-
ing in conjugate pairs) they are orthogonal on the
positive real line with respect to the weight function
wðxÞ¼
ða þ ixÞðb þ ixÞðc þ ixÞðd þ ixÞ
ð2ixÞ
‘‘Racah polynomials’’ can be obtained from
Wilson polynomials when the parameters are such
that one of a þ b, a þ c,ora þ d is a negative
integer N. They are given by
R
n
ððxÞ; ; ; ; Þ
¼
4
F
1
n; n þ þ þ 1; x; x þ þ þ 1
þ 1;þ þ 1;þ 1
1
where þ 1 = N or þ þ 1 = N or þ 1 = N,
and N is a non-negative integer. They are orthogonal on
the finite set {(0), (1), ..., (N)}, where (x) = x(x þ
þ þ 1). They arise as 6 j symbols in the coupling
of three angular momenta.
See also: Combinatorics: Overview; Compact Groups
and their Representations; Integrable Systems:
Overview; Painleve
´
Equations; q-Special Functions;
Random Matrix Theory in Physics; Separation of
Variables for Differential Equations.
Further Reading
Abramowitz M and Stegun IA (1964) Handbook of Mathematical
Functions, With Formulas, Graphs, and Mathematical Tables,
National Bureau of Standards Applied Mathematics Series,
vol. 55 (reprinted 1984). New York: Dover.
Andrews GE, Askey R, and Roy R (1999) Special Functions,
Encyclopedia of Mathematics and Its Applications, vol. 71.
Cambridge: Cambridge University Press.
Bailey WN (1935) Generalized Hypergeometric Series, Cambridge
Mathematical Tract, vol. 32. Cambridge: Cambridge University
Press.
Erde´lyi A, Magnus W, Oberhettinger F, and Tricomi FG (1953–1955)
Higher Transcendental Functions, Bateman Manuscript Project,
vols.1–3.NewYork:McGraw-Hill.
Gradshteyn IS and Ryzhik IM (1965) Table of Integrals, Series,
and Products, chs 8–9, pp. 904–1080. New York: Academic
Press.
Koekoek R and Swarttouw R (1998) The Askey-Scheme of
Hypergeometric Orthogonal Polynomials and Its q-Analogue.
Reports of the faculty of Technical Mathematics and Infor-
matics, no. 98–17, Delft University of Technology.
Lozier D, Olver F, Clark C, and Boisvert R Digital Library of
Mathematical Functions, http://dlmf.nist.gov.
Nikiforov AF and Uvarov VB (1988) Special Functions of
Mathematics Physics. Basel: Birkha¨user.
Nikiforov AF, Suslov SK, and Uvarov VB (1991) Classical
Orthogonal Polynomials of a Discrete Variable, Springer
Series in Computational Physics. Berlin: Springer.
Szego¨ G (1939) Orthogonal Polynomials, American Mathema-
tical Society Colloquium Publications XXIII, 4th edn., 1975.
Providence, RI: American Mathematical Society.
Watson GN (1922) A Treatise on the Theory of Bessel Functions.
Cambridge: Cambridge University Press.
Ordinary Special Functions 645
P
Painleve´ Equations
N Joshi, University of Sydney, Sydney, NSW,
Australia
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The Painleve´ equations P
I
–P
VI
are six classical
second-order ordinary differential equations that
appear widely in modern physical applications.
Their conventional forms (governing y(x) with
derivatives y
0
= dy=dx, y
00
= d
2
y=dx
2
) are:
P
I
: y
00
¼6y
2
þ x
P
II
: y
00
¼2y
3
þ xy þ
P
III
: y
00
¼
y
02
y
y
0
x
þ
1
x
y
2
þ
þ y
3
þ
y
P
IV
: y
00
¼
y
02
2y
þ
3
2
y
3
þ 4xy
2
þ 2ðx
2
Þy þ
y
P
V
: y
00
¼
1
2y
þ
1
y 1
y
02
y
0
x
þ
ðy 1Þ
2
x
2
y þ
y
þ
y
x
þ
yðy þ 1Þ
y 1
P
VI
: y
00
¼
1
2
1
y
þ
1
y 1
þ
1
y x
y
02
1
x
þ
1
x 1
þ
1
y x
y
0
þ
yðy 1Þð y xÞ
x
2
ðx 1Þ
2
(
þ
x
y
2
þ
ðx 1Þ
ðy 1Þ
2
þ
xðx 1Þ
ðy xÞ
2
)
where , , , are constants. They were identified
and studi ed by Painleve´ and his school in their
search for ordinary differential equati ons (in the
class y
00
= R(x, y, y
0
), where R is rational in y
0
, y and
analytic in x) that define new transcendental func-
tions. Painleve´ focuss ed his search on equations that
possess what is now known as the Painleve´ property:
that all solutions are single-valued around all
movable singularities (a singularity is ‘‘movable’’ if
its location changes with initial conditions).
For the Painleve´ equations, all movabl e singula-
rities are poles. For P
I
and P
II
, all solutions are
meromorphic functions. However, the solutions of
each of the remaining equations have other singula-
rities called ‘‘fixed’’ singularities, with locations that
are determined by the singularities of the coefficient
functions of the equation. P
III
–P
VI
have a fixed
singularity at x = 1.P
III
and P
V
have additional
fixed singularities at x = 0, and P
VI
has them at x = 0
and 1. Although each solution of P
III
–P
VI
is single-
valued around a movable singularity, it may be
multivalued around a fixed singularity.
Painleve´’s school considered canonical classes of
ordinary differential equations equivalent under linear
fractional transformations of y and x.Ofthefifty
canonical classes of equations they found, all except
six were found to be solvable in terms of already
known functions. These six lead to the Painleve´
equations P
I
–P
VI
as their canonical representatives.
A resurgence of interest in the Painleve´ equations
came about from the observation (due to Ablowitz
and Segur) that they arise as similarity reductions
of well-known integrable partial differential equa-
tions (PDEs), or soliton equations, such as the
Korteweg–de Vries equation, the sine-Gordon equa-
tion, and the self-dual Yang–Mills equations.
As this connection suggests, the Painleve´ equations
possess many of the special properties that are
commonly associated with soliton equations. They
have associated linear problems (i.e., Lax pairs) for
which they act as compatibility conditions. There
exist special transformations (called Ba¨cklund trans-
formations) mapping a solution of one equation to a
solution of another Painleve´ equation (or the same
equation with changed parameters). There exist
Hamiltonian forms that are related to existence of
tau-functions, that are analytic everywhere except at
the fixed singularities. They also possess multilinear
forms (or Hirota forms) that are satisfied by tau-
functions. In the following subsections, for concise-
ness, we give examples of these properties for the first
or second Painleve´ equations and briefly indicate
differences, in any, with other Painleve´ equations.