I nternat. J. Math. & Math. Sci.
Vol. 3 No. 4 (1980) 761-771
761
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
FOR THE MEIXNER-POLLACZEK POLYNOMIALS
ABSTRACT.
__
P. A. LEE
Department of Mathematics
University of Malaya
Kuala Lumpur 22-!
Malaysia
(Received August 24, 1978)
Using the technique of canonical expansion in probability theory,
a bilinear summation formula is derived for the special case of the Meixner-
Pollaczek polynomials
Z
n=0
{%(k)
(x)}
n
%(k)(x)zn/n
n
which are defined by the generating function
z)(x-k)/(l
(I +
z) (x+k)
JzJ
<
I.
These polynomials satisfy the orthogonality condition
(k)(ix)dx (-l)nn.’(k)nBm n’
Pk(X)(k)(ix)
m
n
with respect to the weight function
P l(X) =sech x
Pk(X)
F_
sech (x-
sech
x!
Xl
sech x
2
Xk_l)dXldX 2
dXk_
,
k
2, 3,
762
P.A. LEE
oo ly
KEY WORDS AND PHRASES. Mixnr-Pollaczek polynomials,
COD.
1980 MATHEMATICS SJECT CLASSIFICATION
62H20.
1.
Pr
42A60, S_ondarg 33A70
INTRODUCTION
Let
U
be a Cauchy random variable with the probability density function
(p.d.f.)
_
f(u)
U
Consider the transformation
p(v)
< u < m.
2
+u
sinh V.
The p.d.f, of
< v <
sech v,
V
is
()
(R).
This is the hyperbolic secant distribution considered by Baten [2], and is a
special case of the generalizdd hyperbolic secant distribution treated by
Harkness and Harkness
Let
X
and
in comnon [6], i.e.
where
V.
(i
X
2
I0].
be two random variables having additive random elements
I, 2, 3)
X
X
p(x l, x 2)
V
2
+ V
3
-x
x
X
P(Xl, x 2)
The joint p.d.f.
sech z sech w(x
sech
The marginal p.d.f.’s for
2
2
are mutually independent random variables each
having the p.d.f, given in (I).
easily shown to be
+ V
V
sech
and
X
2
2 sech
z) sech w(x 2
(x
x
2
I)
2
are respectively
of
x
X
and
X
2
is
+ z)dz
<
Xl
-<x2
<
(R)’
< =.
(2)
_. .
_
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
g(xl)
-o
P(xI’ x2)dx2
2xl
cosech
763
Xl,
()
h(x 2)
P(Xl, x2)dx
2x
2
cosech x
2.
The orthogonal polynomials with the above marginals as weight function are
related to the Euler nt,bers and have been discussed by Carlitz in
[4].
Specifically, for the weight function
Pk(X)
P l(X)
the polynomials
Z
nffi0
sech
Xl
sech
x2
)dx
ldx 2
’’’sech (x
k
dXk_ I,
x2
Xl
2, 3,
(4)
sech x,
In(k) (x)}
with generating function
(k)(x)zn/n’.
n
(l +
z)(x-k)/(l- z) (x+k),
(-, )
are the orthogonal polynomials in the interval
[z[
<
(5)
and satisfy the
or thogonality condition
(ix)X n
Pk(X) x(k)
m
where
i
/Z and
m,n
(k)
(ix)dx
(-l)nn
(k) n m,n
denotes the Kronecker delta.
The explicit form of the orthogonal polynomial is given by Carlitz
x(k)
(x)
n
(6)
n
Z
r--O
2rI(x k))in
r
+ k
n
r
[4] as
)
(-l) n(k) n 2 F [-n, (x + k); k; 2].
(7)
The last result follows easily from the following well-known generating function
for the Gaussian hypergeometric function
2
Fl
[7, p. 82]
764
P.A. LEE
x) z] -b
(I +z) b-C[l + (l
I-n,
7.
b; c; x]z
n
n=0
A related system of polynomials has been discussed by Bateman
who
It happens that both the
referred to them as the Mittag-Leffler polynomials.
polynomials discussed by Bateman and Carlitz are but special cases of the system
of orthogonal polynomials first discussed by Meixner Ill] and later
Following the notation of [8, p. 219] (See
independently by Pollaczek [12].
[5, p. 184]), the Meixner-Pollaczek polynomials
also
P(a)(X;n
where
a > 0,
0 <
)
<
(2a,)nn,. ein 2Fl[-n,
< x <
and
are given explicitly by
e -2i
a + ix; 2a;
(8)
(R).
These polynomials are orthogonal with respect to the weight function
o
(a)
(xl )
(2 sin )
2a-
11"
e-(-2)r(
al
+ i)
12
The orthogonality relation is given by
co(a) (x; )p(a)
(x; )p(a)
(x;
n
m
A generating function for
tnp (a)(x; )
n
7.
n--O
p(a) (x;
n
(l
)
)dx
r(2a
/
n’.
n)
cosec
m,n"
is
tell) -a+ix (l
te
-i)-a-ix, it
< I.
(9)
It is clear when comparing (7) with (8) or (5) with (9) that
(-i)nl(nk)(ix)
and thus
1
(k)
n
polynomials.
(x)
nl P(k/2)(x/2;
/2),
n
k
l, 2, ...; n
0, I, 2,
may be regarded as a special case of the Meixner-Pollaczek
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
2.
765
A BILINEAR SUMMATION FORMULA
From the generating function in (5) it is innediately clear that
%(k)
n (x)
satisfies the following so-called Runge-type identity
(kl +k2) (x
n
. I)x(kl) (Xl) (k2) (x2)
n
+
x2"
n-r
r
r=O
k
I’
k
2
2
3
(I0)
n.
and all
It has been shown that the result in (I0) is both necessary and suffic’ient
for the joint p.d.f, in (2) to possess a bilinear expansion (also called a
canonical expansion in statistical literature) of the form [6]
g(Xl)h(x 2)
P(Xl, x 2)
Z
r--O
Pn0n(Xl)n(X 2)
{en(X) } ({n(X) })
where the canonical variables
orthonormal polynomials with weight function
are a complete set of
g(x) (h(x)).
The canonical
correlation is
E
Pn
where
E
[0n(Xl) n(X2)
denotes the expectation operation.
For the joint p.d.f, in (2) with the equal marginal p.d.f.’s given in (3),
we note that the canonical variable in this case is
.-n
8
On(X)
(x)
n
(2) (ix).
/n’.(2) n n
i
The canonical correlation is
0n
E[On(Xl)On(X 2)
(-l) -n
n’. (2) n
n
E{X(2)n [i(Vl
2) n r.
r.
+
V2) ]X(2)n [i(V2
E[X r )(iv)]
+
V3) ]}
P.A. LEE
766
)]
2)]E[I(1)(iV3
E[I(1)(iV2)I(I)(iV
n-s
s
n-r
]2
E[l(1).(iV2)
n
n’. (2) n
(II)
n+
We therefore have the following interesting bilinear sunmation formula for
the Meixner-Pollaczek polynomials
(_l)n
Z
nffiO
[(n
+
1):]
2
A(2)
(ix
n
)(2)(ix2)
n
sinh
Xl
< x
3.
A GENERALIZATION
x2
sinh
<
r(x
<
and
x
2
sech
1)
2
x2
/
/( 2Xl x2)
(12)
< oo.
Consider the following more general scheme of additive random variables as
in [9].
Let
and
be
{i }
{i }
(n
for
i
l, 2,
n
for
i
l, 2,
n
+ n
2
-m)
m,
m
2
{n i}
for
i
.<
where
l, 2,
m < min(n l, n
m
2)
mutually independent random variables each having the p.d.f.
given in (1).
Define
nl-m
U
Z
i=l
n2"m
m
_E i,
V
X
Z
i=l
n-,
W
7.
i=l
.
=U+V
X2=V+W.
It is clear that the joint characteristic function
X
2
is
(l’ 2
E[exp(iIXl
+
i2X2)
(l’ 2
of
X
and
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
E{exp[iiU
+
i2W
2)V]}
+
+ i(
m
_
2
EEe i%]
EeiqJ]
since
2
sech
sech
sech
767
sech v dv
sech()
<
for
i
.<
n
-m,
.< j .<m,
.<k.<
n2-m.
The joint p.d.f, in question is therefore
P(X
x
2)
42
f=of
and the marginal p.d.f.’s for
g(x
-it
X
lXl-it2x2
sech
and
X
2
sec
sech
mI
2
2)d
are respectively
dm
sech
e
+
nl-I
7
h(x 2)
2
n
on using the fact that
I)’.
(n
(n 2
[3, p. 31]
e
F
-im2x2
n2-1
1)
+ ix
sech
n21
d
2
d
2
(13)
768
-
P.A. LEE
e
-ivx
[sech(Sx + y)
2u
]U+ldx
The respective canonical variables are
.-n
x
On(X l)
/n
I
n(X2)
IF[
_
X
+
r(u + 1)
(n 1)
n
(n2)
-n
n’-(n2) n
iv/8) 12 eiVy/8
+
P
n
(ix
(14)
l)
(ix2)
By a repeated application of the Runge-type identity in (I0) analogous to
the derivation leading to the result in (l|), it may be shown that the
canonical correlation in this case is
(m) n
n
l(nl) n(n2
.<
n
On the other hand, note that from (14)
1"-!-4 -o -=o
2
eitlXl-i2x2
m-1
2
2
m
m-1
2
m
,,,( ,): It( 7
where
(S(x)
m
seth
/
m < min(n I,
n2).
+
[ t2)dt2 dr2
2
exp[-
i2(x2
Xl)]d2
ix)l 2 (x 2 -xz)
denotes the Dirac delta rune=ion.
A double convolution operation applied to (13) then yields the following
expression for
p(x I, x 2)
n
P(Xl’ x2)
3(m- l):(n
r
2
+ iv
+n2m-3
I’
-m- l):(n
2
-m- 1)’
Ir( + i(xl
(x2
2
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
nl +n2+m-3
n
l)’.(n
(m
r
m
u)
+ i(x
m
F
+ i(x
2’
2
)’.
l)’.(n 2
+ u)
x
2
2
m
F
769
(15)
du.
Finally, the result in (15) may be rewritten into the following Barnes type
contour integral
P(Xl’ x2)
2
2
l_J_
2wi
’.(nl-m-l) ’.(n2-m-l)
(m-l)
n2-m
nl +n2+m-2
+s F
F
ixl+s
_i
m
+i(x2-xl)+S
F
Inl; )I +ixl-s
-s F
which my be evaluated in terms of a sum of
3F2
-i(x2-xl)-S
F
functions [13, p. 133] or,
perhaps more conveniently, in terms of Meijer’s G-function as follows
[7,
See. 5.3]
P(Xl x2)
2
2
(m
nl +n2-m-2
l)’.(n
m
n
-m
l)’.(n 2
m +
2
3
3,3
G3,
n
m
m
m +
2
2
l):
n2-m
iXl,
2
n
ixl’
i
(x2
-x)
2
Xl)
m
2
2
i(x
The existence of a diagonal expansion then implies the following summation
formula
E
n=O
(-I) n(m) n
(nl)n n2)nn’.
(n
2m(m
l,):(n
k(n
n
(n 2)
(x) kn
(x2)
l)’.(n 2
m
l):
l):(n 2
m
1).
ds
770
P.A. LEE
n
3,3
G3, 3
.
m
m
2
m
n|
m
2
for
.< m
< min(n I,
n2),
< x
n
+ x 1,
<
(x2 Xl)
2
(16)
n
+
m
2
xl’
,
m
2
(x2
2
< x
2
<
Xl
.
It is perhaps interesting to note in passing that a comparison of the two
results in (12) and (16) allows us to deduce the following special case of the
G-function, viz.
3,3
-sec
1/2,
ACKNOWLEDGEMENT.
+ x 1,
+ x
x
2
(x
sec
sec
2
x
The author is grateful to the referee for his helpful comments,
in particular regarding the connection of the system of orthogonal polynomials
discussed by Carlitz with that of the Meixner-Pollaczek polynomials.
REFERENCES
1.
Bateman, H. The Polynomials of Mittag-Leffler, Proc. Nat. Acad. Sci.
(U.S.A. 26 (|940) 49|-496.
2.
Baten, W.D.
The Probability Law for the Sum of n Independent Variables,
each subject to the Law (!/(2h)) sech(x/(2h)), Bull. Amer. Math. Soc.
40 (!934) 284-290.
3.
Buchholz, H. The Confluente Hypergeometric Functions, Springer Verlag,
New York, !969.
4.
Carlitz, L.
5.
Chihara, T.S. An Introduction to Ortho$onal Polynomials, Gordon and Breach,
New York, |978.
6.
Eagleson, G.K. and H.O. Lancaster. The Regression System of Sums with
Random Elements in Common, Aust. J. Statist. 9 (|967) I|9-|25.
7.
Erd6lyi, A., A. Magnus, F. Oberhettinger and F. Tricomi. Higher
Transcendental Functions, Vol. !, McGraw-Hill, New York, |953.
8.
Erdlyi, A., A. Magnus, F.
Bernoulli and Euler Numbers and Orthogonal Polynomials,
Math. J. 26 (1959) 1-15.
Oberhettinger and F. Tricomi.
Transcendental Functions, Vol. 2, McGraw-Hill, New York,
|953.
Duk___e
PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA
9.
771
Hamden, M.A. Canonical Expansion of the Bivariate Binomial Distribution
With Unequal Marginal Indices, Int. Statist. Rev. 40 (1972) 277-280.
10.
Harkness, W.L. and M.L. Harkness. Generalized Hyperbolic Secant
Distributions, J. Amer. Statist. Assoc. 63 (1968) 329-337.
11.
Meixner, J.
12.
Pollaczek, F.
Orthogonale Polynomsysteme mit einem besonderen Gestalt der
erzeugenden Funktion, J. London Math. Soc. 9 (|934) 6-|3.
Sur une famille de polynSmes orthogonaux qui contient les
et de Laguerre comme cas limites, C.R. Acad.
Sci., Paris 230 (1950) |563-1565.
polynmes d’Hermite
13.
Slater, L.J. Generalized Hypergeometric Functions, Cambridge University
Press, London, 1966.