The favoured solutions of the moment problem

Ferenc Oravecz

1999

Abstract. The sequence of Catalan numbers is known to be the moments of the Wigner law. In the paper a characterization of this sequence is given in terms of some determinants and the moment sequence of the q-Gaussian law is characterized in a similar fashion.

0. Introduction

According to the Hamburger moment problem, a sequence {a_{n}}_{n=0}^{∞} is the moment sequence of a positive measure μ, if and only if the determinants

Δ_{n}^{(μ)} = | \begin{matrix}a_{0} & a_{1} & … & a_{n} \\ a_{1} & a_{2} & … & a_{n+1} \\ . & . & . & . \\ a_{n} & a_{n+1} & … & a_{2n} \end{matrix} |

are non-negative for every n ∈ N (see e.g. [ST]). In this case μ is a probability measure if and only if Δ_{0} = a_{0} = 1. Bellow we will study the following question: for which probability distributions will all the Δ_{n} be 1? The answer will give a new, highly non-linear characterization of the Catalan numbers. In the end of the paper we will also give the sequence of determinants {Δ_{n}}_{n=0}^{∞} corresponding to the q-Gaussian distribution (see [LM]).

1. Results

1.1. Theorem. Let {a_{n}}_{n=0}^{∞} be a sequence of real numbers. Then the sequence {a_{n}}_{n=0}^{∞} is identical to the sequence of the Catalan numbers if and only if

Δ_{n} = | \begin{matrix}a_{0} & 0 & a_{1} & 0 & a_{2} & … \\ 0 & a_{1} & 0 & a_{2} & 0 & … \\ a_{1} & 0 & a_{2} & 0 & a_{3} & … \\ 0 & a_{2} & 0 & a_{3} & 0 & … \\ . & . & . & . & ... & . \\ … & … & … & … & … & a_{n} \end{matrix} | = 1  (1.1)

for every n ∈ N.

Proof. As from the relation

Δ_{n+1} = a_{n+1}Δ_{n} + f(a_{0}, a_{1}, …, a_{n})

(where f is a certain function of the numbers a_{0}, a_{1}, …, a_{n}) a_{n+1} is determined uniquely by the numbers a_{0}, a_{1}, …, a_{n}, Δ_{n} and Δ_{n+1}, it is enough to show that the sequence of the Catalan numbers satisfy (1.1) for every n.

It is known (see e.g. [HP]) that the moments of the standard Wigner law

w = \frac{1}{2π} \sqrt{4 − x^{2}}χ_{[−2;2]} (x) dx

is given by

m_{n} = \{ \begin{matrix}0, if n is odd \\ c_{k}, if n = 2k, \end{matrix} ,  (1.2)

where c_{k} is the kth Catalan number.

It is also known (see e.g. [ST]) that for every symmetric measure μ (with even density function) whose all moments exist, for the corresponding determinants Δ_{n}^{(μ)}

\frac{Δ_{n}^{(μ)}}{Δ_{n−1}^{(μ)}} = β_{1}β_{2} … β_{n}  (1.3)

holds, where the numbers β_{i} are the parameters in the continued fraction form of the Cauchy transform of the given measure, that is

G_{μ} (z) = ∫_{−∞}^{∞} \frac{dμ (x)}{z − x} = \frac{1}{z − \frac{β_{1}}{z − \frac{β_{2}}{z − ...}}} .  (1.4)

In [Ch] it is also shown that for the weighted shift S defined by

Sδ_{n} = b_{n+1}δ_{n+1} (∀n ∈ Z^{+})

on l^{2} (Z^{+}) , where {δ_{n}}_{n=0}^{∞} is the usual base in l^{2} (Z^{+}) and {b_{n}}_{n=0}^{∞} is a bounded sequence, the distribution of the ’field’ operator S + S^{∗} (where S^{∗} is the adjoint of S) in δ_{0} is given by the measure ν whose Cauchy transform is given by

G_{ν} (z) = ∫_{−∞}^{∞} \frac{dν (x)}{z − x} = \frac{1}{z − \frac{b_{1}^{2}}{z − \frac{b_{2}^{2}}{z − ...}}} .  (1.5)

(The distribution of S + S^{∗} in δ_{0} is ment in the usual way via the moments given by m_{n} = ⟨δ_{0}, (S + S^{∗})^{n} δ_{0}⟩.)

If we take b_{n} = 1 for every n, S is the usual shift acting as

Sδ_{n} = δ_{n+1} (∀n ∈ Z^{+}) ,

and in this case the distribution of S + S^{∗} in δ_{0} is the standard Wigner law (see e.g. [HP]). From this and (1.2 − 5) , as c_{0} = 1, the statement of the theorem follows. □

The above statement also means, that among the symmetric probability measures the standard Wigner law is the only one, for which the corresponding sequence {Δ_{n}^{(μ)}}_{n=0}^{∞} is identically 1, so it can be called the ’favoured solution’ of the Hamburger moment problem.

Remark 1.2. According to the basic properties of determinants, Theorem 1.1 also shows that the determinants

Δ̃_{n} = | \begin{matrix}c_{0} & c_{1} & … & c_{n} \\ c_{1} & c_{2} & … & c_{n+1} \\ . & . & . & . \\ c_{n} & c_{n+1} & … & c_{2n} \end{matrix} |

equal 1 for every n, too (here c_{n} stands for the nth Catalan number again). Of course it is not true any more, that for determinants of such structure the sequence of the Catalan numbers is the only sequence with this property. But with the additional requirements that the determinants

Δ̂_{n} = | \begin{matrix}c_{1} & c_{2} & … & c_{n} \\ c_{2} & c_{3} & … & c_{n+1} \\ . & . & . & . \\ c_{n} & c_{n+1} & … & c_{2n} \end{matrix} |

should also equal 1 for every n = 1, 2, …, the sequence of the Catalan numbers is unique solution again (due to basic properties of determinants). As the sequence of the Catalan numbers equals the moment sequence of the free Poisson law of parameter 1 (see e.g. [OP])

μ = \frac{\sqrt{4 − (x − 2)^{2}}}{2πx} χ_{[1;4]} (x) dx,

we may say, that the free Poisson law is the ’favoured solution’ of the Stieltjes moment problem (see e.g. [ST]). From the proof of Theorem 1.1 it also follows, that in the continued fraction form of the Cauchy transform

G_{μ} (z) = \frac{1}{z − α_{1} − \frac{β_{1}}{z − α_{2} − \frac{β_{2}}{z − α_{3} − ...}}}

of the free Poisson law of parameter 1 all the coefficients β_{n} equal 1.

In the following theorem the q-Gaussian (or q-normal) distribution is involved. For further details on this family of distributions see e.g. [LM] .

Theorem 1.3. Let q ∈ (−1; 1) . Let {m_{n}^{(q)}}_{n=0}^{∞} be the moment sequence of the q-Gaussian law whose density function is given by

C_{q}\sqrt{1 − \frac{(1 − q) x^{2}}{4}} ∏_{k=1}^{∞} (1 − \frac{(1 − q) q^{2}x^{2}}{(1 + q^{k})^{2}} ) ,

where

C_{k} = \frac{\sqrt{1 − q}}{π} ∏_{k=1}^{∞} (1 − q^{2k}) ∏_{k=1}^{∞} (1 + q^{k}) .

Then the for the determinants

Δ_{n}^{(q)} = | \begin{matrix}m_{0}^{(q)} & m_{1}^{(q)} & … & m_{n}^{(q)} \\ m_{1}^{(q)} & m_{2}^{(q)} & … & m_{n+1}^{(q)} \\ . & . & . & . \\ m_{n}^{(q)} & m_{n+1}^{(q)} & … & m_{2n}^{(q)} \end{matrix} |  (1.6)

Δ_{n}^{(q)} = ∏_{k=1}^{n} [k]_{q}!  (1.7)

holds for every n, where [k]_{q}! is the product of the first k q-natural number (see e.g. [AI]). Conversely, if (1.7) holds for the determinants (1.6) (for every n), than the sequence of numbers appearing in the determinants (1.6) is identical to the moment sequence of the corresponding q-Gaussian distribution.

Proof. The proof is the same as in the case of Theorem 1.1; the only difference is that the corresponding weighted shift operator now acts according to

Sδ_{n} = \sqrt{[n + 1]_{q}}δ_{n+1} (∀n ∈ Z^{+})

(see e.g. [JSW]). □

Remark 1.4. In the case of q = 0 we get back the statement of Theorem 1.1. When q → −1 the above formula gives Δ_{0} = Δ_{1} = 1, Δ_{n} = 0 (for n > 1), as one would expect for the distribution \frac{1}{2}δ_{−1} + \frac{1}{2}δ_{1} of the ’fermionic’ field operator. Finally, the case q → 1 corresponds to the standard Gaussian law (the distribution of the ’bosonic’ field operator), when for the determinants Δ_{n}^{(μ)} one gets

Δ_{n}^{(μ)} = ∏_{k=1}^{n} k!.

References

[AI] R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoires Mathematical Society, Vol. 49, (1984).

[Ch] T.S. Chihara, An introduction to orthogonal Polynomials, Gordon and Breach, (1978).

[HP] F. Hiai and D. Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs. 77. Providence, RI: AMS.

[JSW] P. Jørgensen, L. Schmitt, R. Werner, q-canonical commutation relations and stability of the Cuntz algebra, Pac. J. Math. 165, (1991) 131-151.

[LM] H. van Leeuwen and H. Maassen, A q-deformation of the Gaussian distribution, J. Math. Phys., 36 (9) (1995), 4743-4756.

[OP] F. Oravecz and D. Petz, On the eigenvalue distribution of some symmetric random matrices, Acta sci. Math. 63 (1997), 483-495.

[ST] J.A. Shohat and J.D. Tamarkin, The problem of moments, Math. Surveys No 1, (1942).

Ferenc Oravecz

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda u. 13-15 H-1053, Budapest, Hungary

e-mail: oravecz@math-inst.hu

Supported by OTKA T23447