Ismail M., Koelink E. (editors) Theory and Applications of Special Functions
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68 THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
where the coefficients
cn,k
satisfy the recurrence equation
n
Cn+l,l+l=
c
Ck,l
(;)an-k.
k=l
Here
an
=
(-l)nf(n+l)(~),
where
f
is given by
(2.4))
cf. (Berg,
2000,
Prop.
2.4),
so
an
is easily calculated to be
It follows also
by
(Berg,
2000)
that
cn,n
=
a:
=
(1
-
q)-n,
cn,l
=
an-l
=
(n
-
1)!/(1
-
qn).
Defining
dn,k
=
(1
-
q)kcn,k
we have
and
dn+ 1,
In particular
We give the first coefficients
It follows by induction using
(2.13)
that
dn,k
as a function of
q
has a
finite limit for
q
--+
I-.
The image measures
pt
=
~1-~
under
q-,(x)
=
x(1- q)
form a
convolution semigroup
(P~),,~
with
On a generalized Gamma convolution related to the q-calculus
and
n
It follows that
s, (pt)
-+
tn
for
q
-+
I-,
so lim
pt
=
bt
weakly by the
041-
2
-
method of moments. This is also in accordance with
because the q-exponential function
Eq
given in
(1.7)
converges to the
exponential function in the following sense
lim
Eq
(z(1
-
q))
=
exp(z),
q+l-
cf. (Gasper and Rahman,
1990).
Remark
2.2.
Consider a non-zero Bernstein function
f.
In (Carmona
et al., 1994; Carmona et al., 1997)
it
was proved by probabilistic methods
that the sequence
n
!
Sn
=
f(l)
.
.
.
. .
f(n)
is a determinate Stieltjes moment sequence, meaning that
it
is the mo-
ment sequence of a unique probability on
[0,
o~[.
The special case
f
(s)
=
1
-
qS gives the moment sequence (2.10). In (Berg and Duran, 2004) the
above result of (Carmona et al., 1994; Carmona et al., 1997) is obtained
as a special case of the following result:
Let (a,) be a non-vanishing Hausdorff moment sequence. Then (s,)
defined by so
=
1 and s,
=
I/ (al
-
. . . .
a,) for
n
2
1 is a normalized
Stieltjes moment sequence.
In order to find an expression for
Iq
we consider the discrete signed
measure
sqk
(2.14)
k=O
with moments
where we have used
(1.7).
In particular, the signed measure
pq
has mass
1.
70
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
For measures v,
T
on 10,
co[
we define the product convolution v
o
T
as the image of the product measure v
@I
T
under
x, y
H
xy.
The prod-
uct convolution is the ordinary convolution of measures on the locally
compact abelian group 10,
co[
with multiplication as group operation. In
particular we have
From this equation we get the moment equation
hence
sn (p,
o
El)
=
n!/(q;
q)n,
which shows that
p,
o
El
has the same
moments as
I,.
Since the first measure is not known to be non-negative,
we cannot conclude right-away that the two measures are equal, although
I,
is Stieltjes determinate. We shall show that
pqoEl
has a density
i,(x),
which is non-negative. Since
6,
o
El
=
Ell,
for
a
>
0, it is easy to see
that
m
(-l)nqn(n-1)/2
is
(x)
=
x
exp
(-xq-")
(2.15)
n=O
(4;
q)n(q;
q)m
)
but it is not obvious that
i,(x)
2
0.
Proposition
2.3.
The function i4(x) given
by
(2.15) is non-negative
for x
>
0.
Therefore I,
=
pq
o
El
=
iq(x)llo,m[(x) dx.
Proof.
The Laplace transform of the function
i,
is
which is the partial fraction expansion of
l/(-s; q),,
since the residue
of this function at the pole
s
=
q-"
is
We claim that
which shows that
I,
and
i,
have the same Laplace transform, so
i,
is the
density of
I,
and hence non-negative.
On a generalized Gamma convolution related to the q-calculus
71
To see the equation (2.17) we note that the left-hand side minus the
right-hand side of the equation is an entire function cp, and by (2.9) we
get
but by
(1.7)
the sum above equals (qn+';
q)OO,
and we get cp(n)(~)/n!
=
0,
which shows that
cp
is identically zero.
We call the attention to the fact that the identity (2.17) was also
used in the work (Dumas et al., 2002) of Dumas et al., but it is in fact
a special case of Jackson's transformations, see
(111
4) in (Gasper and
Rahman, 1990) with
b
=
1,
a
=
-s,
z
=
q.
0
Let Rq denote the following positive discrete measure
with moments
by (1.6). We claim that pq given by (2.14) and Rq are the inverse of
each other under the product convolution, i.e.
This amounts to proving that
but this follows by Cauchy multiplication of the power series (1.6), (1.7).
Combining Proposition 2.3 with (2.20) we get:
Corollary
2.4.
The following factorization hold
El
=
Iq
0
Rq,
which corresponds to the factorization of the moments of
£1
as
72
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
Remark
2.5.
The factorization of Corollary 2.4 is a special case of a
general factorization in (Bertoin and Yor,
2001):
where
f
is a non-zero Bernstein function (2.2), and
If,
Rf are deter-
mined
by
their moments
sn(Rf)
=
f
(1)
.
.
. .
f
(n).
3.
The entire function
h(x)
:=
I?(%)
(q"; q),
Since the Gamma function has simple poles at z
=
-n,
n
=
0,1,.
.
.
,
where (qr; q), has simple zeros, it is clear that the product h(z)
:=
r(z) (q"; q), is entire. We have
and from this it is easy to see that
Proposition
3.1.
For z
E
cC
we have
I
x" dIq (x)
=
h(z
+
1)
0
(4; 4)m
'
Proof. For Re
z
>
-1
the following calculation holds by (2.15) and (1.7):
i
00
(-
l)nqn(n-1)/2
x" dIq (x)
=
i
x~e-xq-n
dx
,=, (q; q)n(q; q)m
0
0
Since the right-hand side is entire and
I,
is a positive measure, we get by
a classical result (going back to Landau for Dirichlet series, see (Widder,
1941, p.
58))
that the integral in (3.2) must converge for all z
E
C, and
therefore the equation holds for all z
E
C.
0
When discussing measures on 10,
oo[
it is useful to consider this set as
a locally compact group under multiplication. The Haar measure is then
On a generalized Gamma convolution related to the q-calculus
73
dm(x)
=
(112) dx,
and it is useful to consider the density of a measure
with respect to the Haar measure
m.
The Mellin transformation is the
Fourier tranformation of the locally compact abelian group
(10,
oo[,
-),
and when the dual group is realized as the additive group
R,
the Mellin
transformation of a finite measure
p
on
10,
oo[
is defined as
We get from
(3.2)
that
From Proposition
3.1
it follows that
I,
has negative moments of any
order, and from
(3.1)
we get in particular that
is a probability.
The image of
Jq
under the reflection
x
I+
l/x
is denoted
jq.
Theorem
3.2.
The product convolution
Lq
:=
Iq
o
jq
has the density
(1.2)
1
with respect to Lebesgue measure on the half-line.
Proof.
For
z
E
@
we clearly have
and by
(3.2)
we get
By
(1.5)
it follows that for
z
E
C
74
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
so
Lq
=
Xq(x) dx.
0
Remark
3.3.
In (Bertoin et al., 2004) the authors prove Theorem 3.2
by showing that
for
-1
<
Re
z
<
0,
and then they prove the partial fraction expansion of
the meromorphic density
Xq
(x)
Remark
3.4.
The moments of
Jq
are given by
Therefore Carleman's criterion gives no information about determinacy
of
jq.
By the Krein criterion, cf. (Berg, 1995), (Stoyanov, 2000), we
can conclude that
because Iq is determinate. The substitution x
=
lly
in this integral leads
to
but since
we see that (3.6) gives no information about indeterminacy of
jq.
We
do not know if
jq
is determinate or indeterminate, and as a factor of an
indeterminate distribution Lq none of these possibilities can be excluded.
On
a generalized Gamma convolution related to the q-calculus
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RAMANUJAN AND CRANKS
Bruce C. Berndt*
Department of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801
Heng Huat chant
Department of Mathematics
National University of Singapore
2 Science Drive
2
Singapore 11 7543
REPUBLIC OF SINGAPORE
Song Heng Chan
Department of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801
Wen-Chin ~iawt
Department of Mathematics
National Chung Cheng University
Min-Hsiung, Chia- Yi 621 01, Taiwan
REPUBLIC OF CHINA
wcliaw0math.ccu.edu.tw
*Research partially supported by grant MDA904-00-1-0015 from the National Security
Agency.
+Research partially supported by National University of Singapore Academic Research Fund,
Project Number R146000027112.
T~esearch partially supported by grant NSC91-2115-M-194-016 from the National Science
Council of the Republic of China.
O
2005
Springer
Science+Business
Media,
Inc.