Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:

Journal of Computational and Applied Mathematics 99 (1998) 67–75

On some indeterminate moment problems for measures
on a geometric progression
Christian Berg ∗
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen , Denmark
Received 20 October 1997; received in revised form 8 June 1998

Abstract
We consider some indeterminate moment problems which all have a discrete solution concentrated on geometric progressions of the form cq k ; k ∈ Z or ± cq k ; k ∈ Z. It turns out to be possible that such a moment problem has innitely
c 1998 Elsevier Science B.V. All rights reserved.
many solutions concentrated on the geometric progression.
AMS classication: primary 44A60; secondary 33A65; 33D45
Keywords: Indeterminate moment problems; q-orthogonal polynomials; Stieltjes–Wigert polynomials; q-Hermite
polynomials; q-Laguerre polynomials

1. Introduction
If s = (sn )n¿0 is an indeterminate moment sequence, it is well known that the set V of solutions to
the corresponding moment problem contains measures of a very dierent nature: There are measures
 ∈ V with a C ∞ -density, discrete measures and measures which are continuous singular. It is even
so that there are many solutions of each of the three types in the sense that each of these classes

of measures is a dense subset of V , cf. [3].
It is true however that in concrete cases only a few families of solutions are explicitly known.
Recently, Andreas Rung asked the author if the solution associated with the discrete q-Hermite
1
polynomials II is uniquely determined by being restricted to the countable set {± q k+ 2 | k ∈ Z}. We
shall answer this question in the negative. We also treat the same question for the Stieltjes–Wigert
polynomials [21], which are associated with the log-normal distribution, and for the q-Laguerre
polynomials, cf. [17]. The question by Rung was motivated by the study of a q-deformation of
the harmonic oscillator, see [10, 16, 18, 19].
∗

E-mail: berg@math.ku.dk.

c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 6 - 0

68

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

We shall make use of the following elementary result, which is a special case of a theorem of
Naimark, cf. [1, p. 47], characterizing the extreme points of the convex set V as those  ∈ V for
which the polynomials are dense in L1 (). We recall that a point in a convex set is called extreme,
if it is not the midpoint of any segment contained in the convex set.
Proposition 1.1. Let  ⊂ R be an innite countable set and let
=

X

a  ” ;

a ¿0

(1.1)

∈

be a solution to an indeterminate moment problem. Then  is not an extreme point of V if and
only if there exists a non-zero sequence ’ :  → [−1; 1] such that

X

n ’()a = 0

(1.2)

for n¿0:

∈

If the conditions are satised; there are innitely many measures in V concentrated on .
We include the proof, since it shows how to construct solutions on  dierent from (1.1), using
the sequence ’ from (1.2). For s ∈ [−1; 1] we dene
s =

X

a (1 + s’())” ;

(1.3)

∈

which is a family of measures concentrated on  with the same moments as  = 0 , s 6= t for s 6= t
and  = (1 + −1 )=2. This shows that  is not an extreme point of V .
Conversely, if  is not an extreme point, it is of the form  = ( + )=2, where ;  are dierent
measures from V . Since (R\) = 0; we get that  and  are both concentrated on , and dening
({}) − ({})
;  ∈ ;
’() =
2a
we get a non-zero sequence ’ :  → [−1; 1] satisfying (1.2).
2. The log-normal distribution
The log-normal distribution with parameter ¿0 has the following density on ]0; ∞[
−1
d (x) = (22 ) 2 x−1

(log x)2
exp −
22

For p ∈ R one has
sp (d ) =

Z

∞

1

!

:

2 2

xp d (x) dx = e 2 p  ;

(2.1)

(2.2)

0

and in particular the moment sequence is given as
2

sn (d ) = q −n ;

n ¿ 0;

(2.3)
−1 2
q = e 2 .

where 0¡q¡1 is dened by
removes the factor x−1 in (2.1).

Note that d (q 2 x) = (x=q)d (x), so a suitable change of scale

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

69

Stieltjes showed the indeterminacy of (2.3) by remarking that all the densities (s ∈ [−1; 1])


d (x) 1 + s sin



2
log x
2



have the same moments (2.3), cf. [20]. The crucial point in the calculation is the Z-periodicity of
sin(2x).
Chihara [6] and later Leipnik [15] gave the following family of solutions to (2.3) concentrated

on countable sets. For a¿0 dene the measure
∞
2
1 X
a k q k ”aq 2k
c(a) k=−∞

Ha =

(2.4)

concentrated on (a) = {aq 2k | k ∈ Z}, where
∞
X

c(a) =

2

ak qk :

(2.5)

k=−∞

Note that c(a) = (q 2 ; −aq; −q=a; q 2 )∞ by Jacobi’s triple product identity, cf. P
[9].
The moments of Ha can be calculated using the translation invariance of ∞
−∞ :
2

sn (Ha ) =

∞
∞
2
2
q −n X
1 X
a k q k (aq 2k )n =

a k+n q (k+n)
c(a) k=−∞
c(a) k=−∞
2

= q −n :
Note that this calculation is valid for n ∈ Z, so that d and Ha have the same positive and negative
moments. The calculation in [15] involved characteristic functions.
We now put a = q and show that Hq is not an extreme point of V .
The following formula holds
∞
X

qk

2

+k

(−1) k = 0:

(2.6)

k=−∞

In fact, the terms corresponding to k and −k − 1 cancel.
The bounded sequence ’ : (q) → [−1; 1] given by ’(q 2k+1 ) = (−1) k satises (1.2), since for n ∈ Z
2

∞
∞
2
2+k+n
q −n X
1 X
q (2k+1)n (−1) k q k +k = (−1)n
q (k+n) (−1)n+k = 0:
c(q) k=−∞
c(q) k=−∞

(2.7)

Proposition 2.1. The measure Hq is not an extreme point of V . All the measures on (q) of the
form
s =

∞
2
1 X
q k +k (1 + s(−1) k )”q 2k+1 ;
c(q) k=−∞

have the same moments (2.3).

s ∈ [−1; 1]

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C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

Using the translation invariance it follows that Haq 2 = Ha and in particular Hq 2m+1 = Hq for all
m ∈Z. One may ask for which values of a¿0 it is the case that Ha is an extreme point of V . We
cannot answer this question, but shall answer the corresponding question for the strong (or twosided) Stieltjes moment problem. Concerning this problem see [4, 12]. We notice that Ha and d
are solutions of a strong Stieltjes moment problem since
sn (Ha ) = sn (d ) = q −n

2

for n ∈ Z:

(2.8)

Proposition 2.2. Let V˜ be the convex set of solutions to the strong Stieltjes moment problem with
moments (2.8).
Then Ha ; a¿0 is an extreme point of V˜ if and only if a 6= q 2m+1 for all m ∈ Z.
Proof. Since (2.7) holds for all n ∈ Z, it follows by an obvious extension of Proposition 1.1 that
Hq is not an extreme point of V˜ . We shall next prove that if a is a positive number not equal to an
odd power of q, then Ha is an extreme point of V˜ . Let ’ : Z → [−1; 1] annihilate x n for all integers
n with respect to Ha . By showing that ’ is identically zero, we obtain the assertion.
2
Putting fa (k) = a k q k , the assumption on ’ is equivalent to the convolution equation fa ∗ ’ = 0 on
the group Z. By Wiener’s Tauberian Theorem we get ’ ≡ 0 provided that
fˆa () =

∞
X

k=−∞

2

a k q k eik 6= 0

for all  ∈ R:

Using [5, Theorem 1, p. 59], we see that the above function vanishes precisely if a is an odd power
of q and  is an odd multiple of .

3. The discrete q-Hermite polynomials II
According to Koekoek and Swarttouw [13] the monic discrete q-Hermite polynomials II are given
by the recurrence relation
xh˜ n (x; q) = h˜ n+1 (x; q) + q −2n+1 (1 − q n ) h˜ n−1 (x; q)

(3.1)

with the initial conditions h˜ −1 (x; q) = 0, h˜ 0 (x; q) = 1. For more information about these polynomials
see [14]. They appear in another normalization as the analogues of the classical Hermite polynomials
in the study of a q-deformation of the harmonic oscillator, see [10, 16, 18, 19].
By Favard’s theorem it is easy to see that (h˜n (x; q)) are orthogonal with respect to a positive
measure if and only if
q −2n+1 (1 − q n ) ¿ 0

for n¿1;

which is satised if and only if 0¡q¡1. The corresponding orthonormal polynomials (Pn ) with
positive leading coecients satisfy
xPn (x) = bn Pn+1 (x) + bn−1 Pn−1 (x)

(3.2)

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

71

with
1q

bn = q −n− 2

1 − q n+1 ;

(3.3)

!1

(3.4)

2

qn
(q; q)n

Pn (x) =

2

h˜ n (x; q);

where we use the notation from [9] for q-shifted factorials
(z; q)n =

n
Y
k=1

(1 − zq k−1 );

n ∈ N ∪ {∞}; z ∈ C;

(3.5)

and extended to negative n by
(z; q)−n =

(z; q)∞
;
(zq −n ; q)∞

n ∈ N:

(3.6)

The polynomials h˜ n (x; q) correspond to a symmetric Hamburger moment problem with vanishing
odd moments. For c ¿ 0 we consider the family (c ) of symmetric probabilities
c =

∞
X
qk
1
(”cq k + ”−cq k );
2A(c) k=−∞ (−c2 q 2k ; q 2 )∞

(3.7)

where the normalizing constant
A(c) =

∞
X

qk
(−c2 q 2k ; q 2 )∞
k=−∞

can be found from Ramanujan’s

1 1 -sum

(|b=a| ¡ |x| ¡ 1)

∞
X

(a; q)k k (ax; q=ax; q; b=a; q)∞
x =
;
(b; q)k
(x; b=ax; b; q=a; q)∞
k=−∞

(3.8)

cf. [2, 9].
We nd
A(c) =

∞
X
1
(−qc2 ; −q=c2 ; q 2 ; q 2 )∞
2
2
k
:
(−c
;
q
)
q
=
k
(−c 2 ; q 2 )∞ k=−∞
(−c2 ; −q 2 =c2 ; q; q 2 )∞

The even moments of c can also be evaluated by (3.8)
∞
1 X
(cq k )2n
s2n (c ) =
qk
A(c) k=−∞ (−c 2 q 2k ; q 2 )∞

∞
X
c 2n
(−c 2 ; q 2 )k q (2n+1)k
=
A(c)(−c 2 ; q 2 )∞ k=−∞

= c 2n

1−2n 2
2
=c ; q; q 2 )n
(−c 2 q 2n+1 ; −q 1−2n =c 2 ; q; q 2 )∞
2n (−q
=
c
= (q; q 2 )n q −n ;
2n+1
2
2
2
2
2
(q
; −qc ; −q=c ; q )∞
(−qc ; q )n

(3.9)

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C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

so
sn (c ) =

(

0;

n = 2m + 1;

(3.10)

2

(q; q 2 )m q −m ; n = 2m:

This shows that all the measures c have the same moments (3.10). From the moments (sn ) we
can calculate the Hankel determinants and nd
Dn = det(si+j )06i; j6n = q −

Pn

k=1

k2

n
Y
k=1

(1 − q k )n−k+1 ;

(3.11)

and we then get
√
q
Dn−1 Dn+1
−n− 1
2
bn =
1 − q n+1 ;
=q
Dn
which shows that (Pn ) given by Eq. (3.2) are orthonormal with respect to c for any c ¿ 0.
√
We now put c = q and show that √q is non-extreme and that there are dierent measures with
the moments (3.10) concentrated on
 = {± q

k+ 1
2

| k ∈ Z}:

The bounded even sequence ’ :  → [−1; 1] given by ’(± q
Z

x n ’(x) d√q (x) = 0

k+ 1
2 ) = (−1) k

satises

for n¿0:

This is clear for n odd, and for even n we nd by (3.8)
Z

∞
X
1
2n
√
x ’(x) d q (x) = √
q
A( q) k=−∞



k+ 1
2



2n

(−1) k

qk
(−q 2k+1 ; q 2 )∞

∞
X
qn
(−q; q 2 )k (−q 2n+1 ) k ;
= √
A( q)(−q; q 2 )∞ k=−∞

which vanishes because of the factor (q −2n ; q 2 )∞ . Summing up we have
Proposition 3.1. The family of measures (s ∈ [−1; 1])
∞
X
1
qk
(1 + s(−1) k )(”q k+ 12 + ”−q k+ 12 )
√
2A( q) k=−∞ (−q 2k+1 ; q 2 )∞

on  has the moments (3.10).
Taking the image measure of c under the mapping x 7→ x2 , we obtain an indeterminate Stieltjes
moment problem with moment sequence
2

sn = (q; q 2 )n q −n :

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

Replacing q by
(q; c) =

√

q and c by

√

73

c we get the following family of measures on ]0; ∞[

∞
1 X
q k=2
”cq k
˜
(−cq k ; q)∞
A(c)
k=−∞

(3.12)

for c ¿ 0; 0 ¡ q ¡ 1 and
˜ =
A(c)

∞
X

q k=2
:
(−cq k ; q)∞
k=−∞

They all have the moments
1 2
√
sn ((q; c)) = ( q; q)n q − 2 n :

(3.13)

4. The q-Laguerre polynomials
The q-Laguerre polynomials Ln() (x; q) are studied in [17]. We restrict the parameters to 0 ¡ q ¡ 1;
 ¿ −1, and in this case they are associated with an indeterminate Stieltjes moment problem with
the moment sequence
sn (; q) = (1 − q)−n q−n−( 2 ) (q +1 ; q)n :
n+1

(4.1)

See [11] for a calculation of the corresponding Nevanlinna matrix. In [2] and in [17] one nds
the following family ((c; ))c¿0 of probabilities with moment sequence (4.1):
(c; ) =

∞
1 X
q k(+1)
”cq k ;
B(c) k=−∞ (−q k c(1 − q); q)∞

(4.2)

where
B(c) =

∞
X

(−q +1 c(1 − q); −q − =c(1 − q); q; q)∞
q k(+1)
=
:
(−q k c(1 − q); q)∞
(q +1 ; −q=c(1 − q); −c(1 − q); q)∞
k=−∞

The sum above is evaluated using (3.8), and by the same formula it is easy to show that the n’th
moment of (c; ) is given by (4.1). We nally use (3.8) to establish (1.2) for a certain value of c.
For ’(cq k ) = (−1) k we get
Z

∞
X
cn
(−c(1 − q); q)k (−q n++1 ) k
x ’(x) d(c; )(x) =
(−c(1 − q); q)∞ B(c) k=−∞
n

= cn

(q +1 ; q n++1 c(1 − q); q −n− =c(1 − q); q)∞
;
(−q +1 c(1 − q); −q − =c(1 − q); −q n++1 ; q)∞

which vanishes for c = q m− =(1 − q); m = 0; 1; : : : :
We have established

74

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

Proposition 4.1. The moments of the measures (s ∈ [−1; 1])
∞
X
1
q k(+1) =(−q k− ; q)∞ (1 + s(−1) k )”q k− =(1−q)
B(q − =(1 − q)) k=−∞

are given by (4.1).
For a measure  on [0; ∞[ and a¿0 we denote by ha () the image measure of  under the
mapping ha (x) = ax. Note that
h1−q ((c=(1 − q); − 21 )) = (q; c):
The q-Laguerre polynomials are related to the generalized Stieltjes–Wigert polynomials of Chihara,
cf. [7, 8]. They are obtained from the Stieltjes–Wigert polynomials by introducing a parameter
06p¡1 and forming the density function on ]0; ∞[
!(x) = (p; −pq=x; q 2 )∞ d (x);
where d is given by (2.1). In order to nd the moments of ! we use the well-known formula for
the q-exponential function
2
∞
X
qk pk

2

(−pq=x; q )∞ =

k=0

(q 2 ; q 2 )k

x−k ;

and integrating the corresponding series term by term, it is easy by (2.3) to get
2

sn (!) = (p; q 2 )n q −n ;

(4.3)

n¿0:

The family (Ha ) from (2.4) can be extended in the following way to yield a family of discrete
measures with the moments (4.3). Dene
Ha;p =

∞
X
1
pk
”aq 2k ;
2k+1
c(a; p) k=−∞ (−q
a=p; q 2 )∞

a ¿ 0; 0 ¡ p ¡ 1;

(4.4)

where c(a; p) normalizes Ha;p to have mass one. The mass at aq 2k is by (3.8) given as
pk
(−aq=p; q 2 )k (p; −pq=a; q 2 )∞ ;
c(a)
2

which has limit a k q k =c(a) for p → 0. This shows that limp→0 Ha; p = Ha . The moments of Ha; p are
indeed given by (4.3). This follows easily from (3.8) like the proof of (3.10). The family Ha; p is
given in [8] in a slightly dierent normalization. Note that the image measure of c (cf. (3.7)) under
x 7→ x2 is equal to Hc2 ; q .
Putting a = q we shall see that Hq;p is not an extreme point in the set of solutions to the moment
problem with moments (4.3). This follows as above calculating that
Z

x n ’(x) dHq; p (x) = 0

for n¿0;

where as before ’(q 2k+1 ) = (−1) k .

C. Berg / Journal of Computational and Applied Mathematics 99 (1998) 67–75

75

We shall nally see how the families Ha; p and (c; ) are related. If in Ha;p we put p = q 2(+1) ,
1

where ¿−1, replace q 2 by q, put a = c(1 − q)q + 2 and take the image measure under x 7→
1

xq −− 2 =(1 − q), we get the family (c; ) from (4.2).
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