Meixner-Pollaczek
polynomials and
the
Heisenberg
algebra
Tom
H.
Koornwinder
Centre for Mathematics
and
Computer Science,
P.
0.
Box
4079, 1009
AB
Amsterdam, The Netherlands
(Received
12
July 1988; accepted for publication 9 November 1988)
An
alternative proof is given for the connection between a system
of
continuous
Hahn
polynomials and identities for symmetric elements in the Heisenberg algebra,
which
was first
observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 ( 1986); J.
Math.
Phys. 28,
509 ( 1987)].
The
continuous Hahn polynomials
turn
out
to
be Meixner-Pollaczek
polynomials. Use is made
of
the connection between Laguerre polynomials
and
Meixner-
Pollaczek polynomials, the Rodrigues formula for Laguerre polynomials,
an
operational
formula involving Meixner-Pollaczek polynomials,
and
the Schrodinger model
for
the
irreducible unitary representations
of
the three-dimensional Heisenberg group.
I.
INTRODUCTION
In
two recent papers
1
•
2
Bender, Mead,
and
Pinsky dis-
cussed
the
connection between certain continuous
Hahn
polynomials and symmetrizations
of
elements in the Heisen-
berg algebra. They showed that, if
[q,p]
=
i
and
T
m,n
is the sum
of
all possible terms containing
m
factors
of
p
and
n
factors
of
q,
then
Tn,n
=
const
Sn
(
T
1
•
1
),
(
1.1)
for
some
polynomial
Sn
of
degree
n,
which turns
out
to
be the
orthogonal
polynomial
of
degree
n
on
lR
with respect to
the
weight
function
Xf---*
1/
eh (
1TX/2).
However, the actual
proof
of
this
result is not very clear from these two papers.
In
the
present paper we give an alternative
proof
of
( 1.1). First, in Sec. II, we observe a transformation connect-
ing
certain
continuous
Hahn
polynomials, in particular, the
above
polynomials
Sn
to certain Meixner-Pollaczek polyno-
mials. Next, in Sec.
III
we use a Mellin transform relating
Laguerre
polynomials
and
Meixner-Pollaczek polynomials
and
the
Rodrigues formula for Laguerre polynomials
in
or-
der
to
derive
an
operational formula involving Meixner-Pol-
laczek
polynomials. Finally, in Sec. IV
we
use this operation-
al
formula
in order to derive formula ( 1.1). Here we
make
use
of
the
Schrodinger
modelfor
the irreducible unitary rep-
resentations
of
the Heisenberg group.
II.
ON
CONTINUOUS HAHN POLYNOMIALS
EXPRESSIBLE
AS
MEIXNER-POLLACZEK
POLYNOMIALS
Continuous
Hahn
polynomials
are defined by
Pn
(x;a,b,c,d):
·n
(a+c)n(a+d)n
=
l
n
X
3
F
2
(-
n,ri
+a+
b
+
c
+dd-
1,a
+ix;
I).
(
2
_
1
)
a+c,a+
If
c
=
a,
d
=
b
and
Re
a,
Re
b
>
0,
then they are orthogonal
on
( -
oo,
oo
)
with respect to the weight function
w(x):
=
r(a
+
ix)r(b
+
ix)r(c
-
ix)r(d
-
ix).
(2.2)
See Refs. 3
and
4,
but
read
a
+
ix
instead
of
a -
ix
in
formula
(3)
of
Ref.
4.
Meixner-Pollaczek
polynomials
are defined by
P~
0
l(x;</J):
=
eimf>
2
F
1
( -
n,a
+
ix;2a;l -
e-
1
i4>).
(2.3)
If
a
>
0
and
0
<
</>
<
1T,
they
are
orthogonal
on
( -
oo
,
oo
)
with respect to the weight function
w(x)
=e<
1
4>-,,.lxjr(a+ix)[2.
(2.4)
See Refs. 5
and
6 and, for standardized notation,
the
Appen-
dix
of
Ref.
7.
For
a
=
c
=
b -
! =
d -
!
>
0
the
weight function
( 2.2) becomes
w(x)
=
2 -
40
+
2
1TI
rc2a
+
2ix)
1
2
•
(2.5)
On
comparing with
(2.4)
we conclude
that
Pn
(x;a,a
+
!,a,a
+
!)
=
const
P
~
1
al
(2x;!
1T).
The
constant can be
computed
by comparing coefficients
of
xn.
We
obtain
Pn
(x;a,a
+
!,a,a
+
!)
= [
(2a)n
(2a
+
Pnln!]
(2.6)
In
terms
ofhypergeometric
functions this formula reads
(
-
n,n
+
4a,a
+
ix
I ) .
3
F
2
1
1
=
2
F
1
( -
n,2a
+
21x;4a;2).
2a,2a
+
2
(2.7)
This
identity can also
be
obtained from Ref.
8,
F (
a,b,n
+
2c,
-
n
·l)
=
F (
2a,2b, - n
·l)
4 3
+b+I
+1'
3 2
b+l2'
'
a
2
,c,c
2
a
+
2
,
c
(2.8)
by letting
b-.
oo.
For
a:=;\
the weight function (2.5) becomes
w(x)
=
2~/ch(21Tx).
In
particular, we find for
the
polynomials
Sn
introduced
in
Sec. I, which were identified with special continuous
Hahn
polynomials in Ref. 2,
that
they
can
be
written as
Meixner-
Pollaczek polynomials:
S
(x)
=
const
p<
112
Jc1xl1T)
n n
2
'2
•
(2.9)
767
J.
Math. Phys.
30
(4), April 1989
0022-2488/89/040767-03$02.50
©
1989 American Institute of Physics
767
Ill.
AN
OPERATIONAL FORMULA INVOLVING
ME,:IXNER-POLLACZEK POLYNOMIALS
Recall
that
we can obtain the Mellin transform pair,
G(..1)
=
i""
F(r)r-
1
-v..dr,
F(
r)
= (21T)-
1
f:
00
G(..1).fA
d..1,
from
the
Fourier transform pair,
g(A.)
=
s:
.,/(t)e-21Tv..1 dt,
f
(t)
=I:"'
g(..1)e2mtt
d..1,
by
making
the substitutions
r=e
2
"
1
,
F(r)=f(t),
G(..1)=21Tg(..1)
(3.1)
(3.2)
in
(3.2).
In particular, Mellin inversion in (3.1) is valid
if
the
function ti-+F( e
2
'1T
1
)
belongs
to
the
class Y
of
rapidly
decreasing C"" function
on
JR..
If
F
1
, F
2
are two such func-
tions
and
G
1
, G
2
their Mellin transforms
then
we have the
Parseval formula
{""
F
1
(r)
F
2
(r)
dr
=J
00
G
1
(..1)
G
2
(i!.)
di!..
(3.3)
Jo
r - ..
21T
Proposition
3.1:
For
a>
0
and
0 <
<P
<
1T
Laguerre poly-
nomials
Xt-+L
!a
-
1
(x)
and
Meixner-Pollaczek polynomials
i!.i-+P
~al
(.i!.;rp) are mapped onto each other by the Mellin
transform
in the following way:
I -in,P
n.e
e-
c112ixc1 + icot.PlxaL
!a-
l(x)x-
1-v..
dx
(2a)n
=
eUa-.<l[.p-
<
1121
1ric2
sin
rp)a-i.<r(a-
i.i!.)P~a
1
(i!.;cp).
(3.4)
Proof
The
left-hand side can
be
rewritten as
e-in,P i
(-n)k
l""e-(l/2)x(l+icot.P)xk+a-V..-ldx
k=O
(2a)kk!
Jo
-in.P n
(-n)k
rca-ii!.+k)
=e
k~o(2a)kk!C!+!icotrp)a-V..+k
=
e-in<Prca
- ii!.)(1 -
e2i.P)a-v.
X
2
F
1
( -
n,a -
i..1;2a;
1 - e
2
;.p)
=
ein.Pr(a
_ i.i!.) (
1
_
ei;.p)a
-
•A.
X
2
F
1
( -
n,a +
i..1;2a;l
-
e-
i;.p),
which
can
be rewritten as
the
right-hand side
of
( 3.4). 0
It
is possible
9
•
10
to give
an
interpretation
of
the above
proposition in
the
context
of
matrix elements
of
discrete
se-
ries representations ofSL(2,lR.).
Corollary 3.2:
For
a>
0 and 0 <
rp
<
1T
Laguerre polyno-
mials can be expressed
by
the
differentiation formula
I -in,P
n.e
cv2)x(l+icot<1»xaL!a-1(x)
(2a)n
=P~ai(-ix
:x
;</J)(e-<112Jx(l+icot.Plxa).
(3.5)
768
J. Math. Phys., Vol. 30, No. 4, April 1989
Proof
In
the left-hand side
of
( 3.4) Mellin transform is
taken
of
a function
that
belongs
to
the
class Y as a function
oft,
where x =
e'.
Hence we can apply Mellin inversion [
cf.
(3.1)] and
we
can write the left-hand side
of
(3.5) as
(217')-1
f_""""
eUa-.1.Jl.P-
(112)1T](2
sin
</J)a-V.
X
r(a
-
i.i!.)P
~
01
(A.;rp)xv.
d..1
= P
~a>(
- ix !
;</J)
x
[e<ia
-.1.l[,P-
(ll2)1r]
(2
sin
rp
)
0
-
i).
xrca-i.i!.)xi)..]'
which equals the right-hand side
of
(3.5).
By substitution
of
the Rodrigues formula
n!e-"xaL~(x)
=
(!rce-"xn+a)
into (3.5) weobtain
(!
)n
(e-xxn+2a-1)
=
(2a)
nein.Pe- (l/2)x(J - icotg,P)x°- I
0
XP~a'(-ix
!
,rp)[e-0/2)x(l+icot,Plxa].
(3.6)
In particular,
for</>
=
~17'
and a = ! we obtain
(i
!rce-xxn>
=
n!e
-
0/2lxp
o121(ix
~
+ J_ i
_!._
17')
[e
- ( J/2Jx].
n dx 2
'2
Hence for arbitrary veC,
· ( d
)n
2·
e"'"
i
dx
(xne - "'")
=
n!P<
112
'(ix~
+
J_i
~)[e-i""].
(3.7)
n
dx
2'2
IV. PROOF OF THE
B~NDER-MEAD-PINSKY
RESULT
Consider the
Heisenberg
group H
1
,
which is
R.
3
equipped
with
the
multiplication rule
c5,-,,,r>
<5
',,,,',r')
=
(5
+
5',7J
+ 7J',r+ r' +
!C5'7J
-57J')).
(4.1)
Let AElR'\ {O} and let
17';,.
denote the unique (up to equiv-
alence) irreducible unitary representation
of
H
1
such
that
'IT;,.
(0,0,r) =
ev..TJ,
reR.
Then,
withµ:=
IA.
j 1
12
and
E:
= sgn(..1),
1T;.
can be realized
on L
2
(R)
by
(17'.i.
<t,TJ,r)f)(x>
=eiµsxe;µ'[ET+(!ls7llj(x+µTJ),
fEL2(R.).
(4.2)
Let X
and
Y be the infinitesimal generators
of
the one-pa-
rameter subgroups
of
elements (5,0,0) and (0,,,,,0), respec-
tively. Let a denote the symmetrization mapping
11
from the
symmetric algebra to the universal enveloping algebra
of
the
Lie algebra
of
H
1
, i.e.,
Tom H. Koornwinder
768
( 4.3)
where
s
runs over all
permutations
of
{l,.
..
,k}. Let
f
be
a
C""
function locally defined
on
R.
Then
and
(1T,dX)f)(x)
=
iµxf(x),
(1T,,_(Y)/)(x)
=µf'(x),
(
1T,i
(cr(X
"Y")
)/)
(x)
=
(_!__)"(_!__)"(eiµ5xeiµ
2
[Er
+
(l/2J511lj(x
+
µ7]))
I
at
a7J
5,71,r=O
=(iµ
~)"((x+
~µ71)"fcx+µ17))i
11
=
0
•
Hence
(
1T,i
(cr(X "Y")
)f
)(x)
=
1-l
l"[i
~r
x((x+
~y)/cx+y>)ly=o.
(4.4)
For
n
=
1
this simplifies
to
(1T,,_(cr(XY))j)(x)
=IA.
l(ix_!__+_!_i)f(x).
(4.5)
ax
2
Let
f
u
(x):
=
e-ivx.
Then
we obtain from
(4.4),
(3.7),
and
(4.5)
that
769
J.
Math. Phys.,
Vol.
30,
No.
4,
April
1989
(1T;.(a(X"Y"))fu) (x)
=
1-l
l"(i
~)"((x+
~
y)"e-iv(x+y))ly=O
=
2 -
"IA.
I
"eivx(i
! r
Cx"e
-
2ivx)
=2-"n!IA.l"P~
112
>(ix
! +
~
i,~
1T)[e-ivx]
=
2 -
"n!IA.
l"P~
112
>
(IA.
1-
1
1T,,_(cr(XY)),!
1T)
[
fv
(x)].
Hence
by integrating
both
sides against suitable functions
of
v,
we obtain
1T,.(a(X"Y"))
=
2 -
"n!IA.
l"P
~ini
(IA.
1-
1
1T,,_
(cr(XY)
j,!1T).
(4.6)
In
view
of
(
2.
9)
and (
4.
3)
this
becomes for
A.
=
1
the result
(
1.1
)
of
Ref.
1.
ACKNOWLEDGMENT
The
author
thanks
G.
Gasper
for a reference to (2.8).
'C. M. Bender,
L.
R. Mead, and
S.
S.
Pinsky, Phys. Rev. Lett. 56, 2445
(1986).
2
C. M. Bender,
L.
R. Mead,
and
S.
S. Pinsky,
J.
Math. Phys. 28,
509
(1987).
3
N. M. Atakishiyev and
S.
K. Suslov,
J.
Phys. A 18,
1583
( 1985).
4
R.
Askey,
J.
Phys. A
18,
Ll017
(1985).
5
J.
Meixner,
J.
London Math. Soc. 9, 6 (1934 ).
6
F. Pollaczek,
C.R.
Acad. Sci. Paris 230, 1563 (1950).
7
R.
Askey
and
J.
Wilson, Mem. Am. Math.
Soc.
54,
319
( 1985).
8
W. N. Bailey, Proc. London Math. Soc. 29, 495 ( 1929).
9
T.
H. Koornwinder,
"Group
theoretic interpretations
of
Askey's scheme
of
hypergeometric orthogonal polynomials," "Orthogonal polynomials
and
their applications,"
Lecture Notes in Mathematics,
Vol. 1329, edited
by
M.
Alfaro
et
al.
(Springer, Berlin, 1988), pp. 46-72.
10
0.
Basu
and
K.
B.
Wolf,
J.
Math. Phys. 23,
189
(1982).
11
S.
Helgason,
Groups and Geometric Analysis
(Academic, New York,
1984).
Tom
H.
Koornwinder
769