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Citation for published version
Clarkson, Peter (2013) Recurrence coefficients for discrete orthonormal polynomials and the
Painlevé equations. Journal of Physics A: Mathematical and Theoretical, 46 (18). ISSN 1751-8113.
DOI
https://doi.org/10.1088/1751-8113/46/18/185205
Link to record in KAR
https://kar.kent.ac.uk/33120/
Document Version
UNSPECIFIED
Recurrence coefficients for discrete orthonormal
polynomials and the Painlev´e equations
Peter A. Clarkson
School of Mathematics, Statistics and Actuarial Science, University of Kent,
Canterbury, CT2 7NF, UK
E-mail: P.A.Clarkson@kent.ac.uk
Abstract. We investigate semi-classical generalizations of the Charlier and Meixner
polynomials, which are discrete orthogonal polynomials that satisfy three-term
recurrence relations. It is shown that the coefficients in these recurrence relations
can be expressed in terms of Wronskians of modified Bessel functions and confluent
hypergeometric functions, respectively for the generalized Charlier and generalized
Meixner polynomials. These Wronskians arise in the description of special function
solutions of the third and fifth Painlev´e equations.
PACS numbers: 02.30.Gp, 02.30.Hq, 02.30.Ik
AMS classification scheme numbers: 34M55, 33E17, 33C47
Submitted to: J. Phys. A: Math. Theor.
Discrete orthonormal polynomials and the Painlev´e equations 2
1. Introduction
In this paper we are concerned with the coefficients in the three-term recurrence
relations for semi-classical orthonormal polynomials, specifically for generalizations of
the Charlier and Meixner polynomials which are discrete orthonormal polynomials. It
is shown that these recurrence coefficients for the generalized Charlier polynomials and
generalized Meixner polynomials can respectively be expressed in terms of Wronskians
that arise in the description of special function solutions of the third Painlev´e equation
(P
III
)
d
2
w
dz
2
=
1
w
dw
dz
2
−
1
z
dw
dz
+
Aw
2
+ B
z
+ Cw
3
+
D
w
, (1.1)
where A, B, C and D are arbitrary constants, and the fifth Painlev´e equation (P
V
)
d
2
w
dz
2
=
1
2w
+
1
w −1
dw
dz
2
−
1
z
dw
dz
+
(w −1)
2
z
2
Aw +
B
w
+
Cw
z
+
Dw(w + 1)
w −1
. (1.2)
Wronskians for special function solutions of P
III
are expressed in terms of modified
Bessel functions and for P
V
in terms of confluent hypergeometric functions.
The relationship between semi-classical orthogonal polynomials and integrable
equations dates back to the work of Shohat [51] in 1939 and later Freud [26] in 1976.
However it was not until the work of Fokas, Its and Kapaev [21, 22] in the early 1990s
that these equations were identified as discrete Painlev´e equations. The relationship
between semi-classical orthogonal polynomials and the (continuous) Painlev´e equations
was demonstrated by Magnus [37] in 1995. A motivation for this work is that recently
it has been shown that recurrence coefficients for several semi-classical orthogonal
polynomials can be expressed in terms of solutions of Painlev´e equations, see, for
example, [2, 4, 5, 7, 9, 10, 11, 14, 15, 17, 18, 19, 25, 52, 56, 57].
This paper is organized as follows: in §2 we review properties of P
III
(1.1) and
P
V
(1.2), including special function solutions and the Hamiltonian structure of these
equations; in §3 we review properties of orthogonal polynomials and discrete orthogonal
polynomials; in §4 we derive expressions for the recurrence coefficients for the generalized
Charlier polynomials in terms of Wronskians that arise in the description of special
function solutions of P
III
; in §5 we derive expressions for the recurrence coefficients for
the generalized Meixner polynomials in terms of Wronskians that arise in the description
of special function solutions of P
V
; and in §6 we discuss our results.
2. Painlev´e equations
The six Painlev´e equations (P
I
–P
VI
) were first discovered by Painlev´e, Gambier and
their colleagues in an investigation of which second order ordinary differential equations
Discrete orthonormal polynomials and the Painlev´e equations 3
of the form
d
2
w
dz
2
= F
dw
dz
, w, z
, (2.1)
where F is rational in dw/dz and w and analytic in z, have the property that their
solutions have no movable branch points. They showed that there were fifty canonical
equations of the form (2.1) with this property, know known as the Painlev´e property.
Further Painlev´e, Gambier and their colleagues showed that of these fifty equations,
forty-four are either integrable in terms of previously known functions (such as elliptic
functions or are equivalent to linear equations) or reducible to one of six new nonlinear
ordinary differential equations, which define new transcendental functions (see Ince [31]).
The Painlev´e equations can be thought of as nonlinear analogues of the classical special
functions [13, 20, 28, 33], and arise in a wide variety of applications, for example random
matrices, see [23, 50] and the references therein.
The Painlev´e equations P
II
–P
VI
possess hierarchies of solutions expressible in terms
of classical special functions, cf. [13, 28, 40] and the references therein. For P
III
(1.1)
these are expressed in terms of Bessel functions [42, 43, 48], which we discuss in §2.1,
and for P
V
(1.2) in terms of confluent hypergeometric functions (equivalently, Kummer
functions or Whittaker functions) [47, 40, 58], which we discuss in §2.3.
Each of the Painlev´e equations P
I
–P
VI
can be written as a (non-autonomous)
Hamiltonian system. For P
III
(1.1) and P
V
(1.2) these have the form
z
dq
dz
=
∂H
J
∂p
, z
dp
dz
= −
∂H
J
∂q
, J = III, V
for a suitable Hamiltonian function H
J
(q, p, z) [34, 45, 47, 48], which we discuss in
§2.2 and §2.4. Further, the function σ(z) ≡ H
J
(q, p, z) satisfies a second-order, second-
degree equation, which is often called the Jimbo-Miwa-Okamoto equation or Painlev´e σ-
equation, whose solution is expressible in terms of the solution of the associated Painlev´e
equation [34, 46, 47, 48]. Hence there are special function solutions of these equations,
which are also discussed in §2.2 and §2.4.
2.1. Special functions solutions of the third Painlev´e equation.
In the generic case when CD 6= 0 in P
III
(1.1), then we set C = 1 and D = −1, without
loss of generality, so in the sequel we consider the equation
d
2
w
dz
2
=
1
w
dw
dz
2
−
1
z
dw
dz
+
Aw
2
+ B
z
+ w
3
−
1
w
. (2.2)
Special function solutions of (2.2) are expressed in terms of Bessel functions, see
[42, 43, 48].
Theorem 2.1. Equation (2.2) has solutions expressible in terms of Bessel functions if
and only if
ε
1
A + ε
2
B = 4n + 2, (2.3)
with n ∈ Z and ε
1
= ±1, ε
2
= ±1 independently.
Discrete orthonormal polynomials and the Painlev´e equations 4
Proof. See Gromak [27], Mansfield and Webster [38] and Umemura and Watanabe [55];
also [28, §35].
An alternative form of P
III
, due to Okamoto [45, 46, 48], is obtained by making the
transformation w(z) = u(t)/
√
t, with t =
1
4
z
2
, in (2.2) giving
d
2
u
dt
2
=
1
u
du
dt
2
−
1
t
du
dt
+
Au
2
2t
2
+
B
2t
+
u
3
t
2
−
1
u
, (2.4)
which is known as P
III
′
. Equation (2.4) has solutions expressible in terms of solutions
of the Riccati equation
t
du
dt
= ε
1
u
2
+ νu + ε
2
t, (2.5)
if and only if A and B satisfy (2.3). To solve (2.5), we make the transformation
u(t) = −ε
1
t
d
dt
ln ψ
ν
(t),
then ψ
ν
(t) satisfies
t
d
2
ψ
ν
dt
2
+ (1 − ν)
dψ
ν
dt
+ ε
1
ε
2
ψ
ν
= 0, (2.6)
which has solution
ψ
ν
(t) =
t
ν/2
n
C
1
J
ν
(2
√
t) + C
2
Y
ν
(2
√
t)
o
, if ε
1
= 1, ε
2
= 1,
t
−ν/2
n
C
1
J
ν
(2
√
t) + C
2
Y
ν
(2
√
t)
o
, if ε
1
= −1, ε
2
= −1,
t
ν/2
n
C
1
I
ν
(2
√
t) + C
2
K
ν
(2
√
t)
o
, if ε
1
= 1, ε
2
= −1,
t
−ν/2
n
C
1
I
ν
(2
√
t) + C
2
K
ν
(2
√
t)
o
, if ε
1
= −1, ε
2
= 1,
(2.7)
with C
1
and C
2
arbitrary constants, and where J
ν
(z), Y
ν
(z), I
ν
(z) and K
ν
(z) are Bessel
functions.
2.2. Hamiltonian structure for the third Painlev´e equation.
The Hamiltonian associated with P
III
′
(2.4) is
H
III
′
(q, p, t) = q
2
p
2
−
q
2
+ 2θ
0
q − t
p + (θ
0
+ θ
∞
)q, (2.8)
with θ
0
and θ
∞
parameters, where p and q satisfy
t
dq
dt
= 2q
2
p − q
2
− 2θ
0
q + t,
t
dp
dt
= −2qp
2
+ 2qp + 2θ
0
p − (θ
0
+ θ
∞
),
see Okamoto [45, 46, 48]. Eliminating p then q = u satisfies P
III
′
(2.4) with (A, B) =
(−4θ
∞
, 4(θ
0
+ 1)).
The second-order, second-degree equation satisfied by the Hamiltonian function is
given in the following theorem.