1 Introduction

Due to the popularity of the q-calculus, numerous q-analogues of classical probability distributions have emerged, both for discrete and absolutely continuous cases. For example, there are q-binomial, q-Poisson, q-exponential, q-Erlang, and other q-distributions. These distributions play a significant role not only in the q-calculus itself, but also in various applications, primarily in theoretical physics. See, for example, [1, 6, 10, 14]. Comprehensive information concerning q-distributions is presented in [6] and, in this article, we follow the terminology and exposition of this monograph. Throughout the paper, \(q\in (0,1)\) is taken to be fixed. The q-integral defined by Jackson for \(0<a<b\) as

$$\begin{aligned} \int _0^a f(t) d_qt = a(1-q)\sum _{j=0}^{\infty } f(aq^j)q^j, \quad \int _a^b f(t) d_qt=\int _0^b f(t) d_qt - \int _0^a f(t) d_qt \end{aligned}$$

will be used along with the improper q-integral on \([0,+\infty )\) defined as

$$\begin{aligned} \int _0^\infty f(t) d_qt = (1-q)\sum _{j=-\infty }^{\infty } f(q^j)q^j. \end{aligned}$$

See [11, Sec. 19].

Definition 1.1

[6] Let P be a probability distribution with a distribution function F satisfying \(F(0)=0.\) A function f(t),  \(t>0,\) is a q-density of P if

$$\begin{aligned} F(x)=\int _0^x f(t) d_qt, \quad x>0. \end{aligned}$$
(1.1)

Correspondingly, the nth-order q-moment of P is

$$\begin{aligned} m_q(n; P):=m_q(n;f):=\int _0^\infty t^nf(t) d_qt, \quad n\in {{\mathbb {N}}}_0. \end{aligned}$$
(1.2)

Clearly,

$$\begin{aligned} \displaystyle m_q(n;f)=(1-q)\sum _{j\in {\mathbb {Z}}} f(q^{-j})q^{-j(n+1)} \quad n\in {{\mathbb {N}}}_0. \end{aligned}$$
(1.3)

It has to be mentioned here that if P has a q-density f, then f is the q-derivative of the distribution function F,  that is,

$$\begin{aligned} f(t)=D_qF(t):=\frac{F(t)-F(qt)}{t(1-q)}, \quad t>0. \end{aligned}$$

It is known ([11, Theorem 20.1]) that if \(F(0)= 0\), and it is continuous at 0, and then F can be represented in the form (1.1) and, therefore, possesses a q-density. In this paper, only probability distributions satisfying these conditions and possessing finite q-moments of all orders will be considered. The set of such distributions will be denoted by \({{\mathcal {A}}}.\) The q-moment problem in terms of q-densities has been studied in [17].

Notice that q-moments \(m_q(n;f)\) may also be obtained as the moments of a discrete distribution concentrated on \(\{q^j\}_{j\in {{\mathbb {Z}}}},\) whose probability mass function is given as

$$\begin{aligned} {{\mathbf {P}}}\{X=q^j\}=f(q^j)q^j(1-q),\quad j\in {\mathbb {Z}}. \end{aligned}$$

The moment problem for such discrete distributions was investigated in [2] by C. Berg, who explicitly found infinite families of distributions all possessing the same moments of all orders. These families can also be viewed as discrete Stieltjes classes, although the name “Stieltjes class” was suggested by Stoyanov ([18]) a few years after [2] had been published. The classical moment problem has deep connections with q-orthogonal polynomials, basic hypergeometric series, and theory of special functions. See, for example, [7, 8, 13] and the bibliography therein. The moment problems associated with various sequences of q-orthogonal polynomials have been investigated extensively by a large number of authors since an indeterminate moment problem implies non-uniqueness of the weight functions defining the respective inner product, like, for example, for the q-Laguerre, the discrete q-Hermite II, and Stieltjes-Wigert polynomials ([12, 3.21 and 3.27] and [2, Sect. 3 and 4]).

In the context of the q-calculus, the q-moments and strict q-moments were defined by G. Carnovale and T. Koornwinder in the form slightly different from (1.2). See [5, formula (1.2)] and [4, formula (2.3)]. In these works, the notion of the q-convolution on the line was studied, and it was found that the condition of q-moment determinacy is crucial for the commutativity of the q-convolution. Pertinent to this problem, certain conditions concerning the rate of growth of the q-moments were considered.

In this article, we stick with the notions and terminology of [6], where q-moments are defined for q-densities of continuous probability distributions by (1.2). Since the q-moments depend only on the values of a q-density on the sequence \(\{q^j\}_{j\in {{\mathbb {Z}}}},\) it is reasonable, therefore, to consider the following equivalence relation for functions on \((0, \infty ),\)

$$\begin{aligned} f\sim g \Leftrightarrow f(q^j)=g(q^j), \quad j\in {{\mathbb {Z}}}. \end{aligned}$$

Definition 1.2

[17] A distribution \(P\in {{\mathcal {A}}}\) with a q-density f is q-moment determinate if \(m_q(n;f)=m_q(n;f_1)\) for all \(n\in {\mathbb {N}}_0\) implies that \(f \sim f_1.\) Otherwise, P is q-moment indeterminate.

It should be pointed out that every absolutely continuous distribution possessing finite moments of all orders can be examined from two different perspectives: those of moment determinacy and q-moment determinacy.

In [17], some conditions have been provided both for q-moment determinacy and indeterminacy in terms of the values \(f(q^{-j}).\) More precisely, it has been proved that

  1. (i)

    if

    $$\begin{aligned} f(q^{-j})=o(q^{j(j+1)/2}), \quad j\rightarrow \infty , \end{aligned}$$
    (1.4)

    then P is q-moment determinate;

  2. (ii)

    if

    $$\begin{aligned} f(q^{-j}) \geqslant Cq^{j(j+1)/2}, \quad j\geqslant 0, \end{aligned}$$
    (1.5)

    then P is q-moment indeterminate.

Statement (i) implies immediately that if a q-density f has a bounded support, then the distribution P is q-moment determinate.

In the sequel, the following q-analogue of the exponential function

$$\begin{aligned} e_q(t)=\prod _{j=0}^\infty \left( 1-t(1-q)q^j\right) ^{-1} \end{aligned}$$
(1.6)

is used as well as the q-exponential distribution with parameter \(\lambda >0,\) whose q-density is

$$\begin{aligned} f(t)=\lambda e_q(-\lambda t), \quad t>0. \end{aligned}$$
(1.7)

For ample information on \(e_q(t)\) and the related distributions, we refer to [6, Section 1] and [11, Section 9].

In this work, new results on q-moment (in)determinacy are presented, both in terms of q-moments and q-density itself. Alternatively, it can be stated that some ‘checkable’ conditions for q-moment (in)determinacy are given. For the classical moment problem, an extensive review of such conditions can be found in [15]. The exact relation between the two moment problems is yet to be described. To this end, the present work is an attempt to connect these two aspects.

2 Statement of Results

Throughout the paper, the letter C with or without an index denotes a positive constant whose exact value does not need to be specified. We start with the assertion providing a condition for q-moment determinacy in terms of q-moments, in distinction to (1.4), where the values of the q-density are used.

Theorem 2.1

Let \(P\in \mathcal {A}\) have a q-density f and a sequence of q-moments \(\{m_q(n;f)\}_{n=0}^\infty \). If

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\ln m_q(n;f)}{n^2} = A < \frac{\ln (1/q)}{2}, \end{aligned}$$
(2.1)

then P is q-moment determinate.

To establish conditions for q-moment indeterminacy, we have to impose some restrictions on the behavior of a q-density, as in the next statement. In the theorem below, it is assumed that the sequence \(\{f(q^{-j})\}_{j\in {\mathbb {N}}_0}\) is log-concave.

Theorem 2.2

Let \(P\in \mathcal {A}\) have a q-density f and a sequence of q-moments \(\{m_q(n;f)\}_{n=0}^\infty \). If

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\ln m_q(n;f)}{n^2} =A >\frac{\ln (1/q)}{2} \end{aligned}$$
(2.2)

and

$$\begin{aligned} 0< f(q^{-j-1}) f(q^{-j+1}) \leqslant [f(q^{-j})]^2, \quad j\geqslant 0, \end{aligned}$$
(2.3)

then P is q-moment indeterminate.

Remark 2.1

It has to be pointed out that in both Theorems 2.1 and 2.2, the inequalities for A have to be strict. If \(A=\ln (1/q)/2,\) even if the q-density is subject to (2.3), the q-moment determinacy cannot be established without additional information. This is illustrated in Example 3.1 with the help of the q-gamma distribution whose q-density is equivalent to (3.3).

The next result provides a condition for the q-moment indeterminacy in situations not covered by the outcomes of Theorems 2.1 and 2.2. More precisely, an analogue of condition (1.5) for a subsequence \(\{q^{-mj}\}_{j=0}^\infty \) of \(\{q^{-j}\}_{j=0}^\infty \) is established.

Theorem 2.3

Let f be a q-density of a probability distribution \(P\in {{\mathcal {A}}}\). If, for a fixed positive integer m, 

$$\begin{aligned} f(q^{-mj}) \geqslant C q^{mj(j+1)/2}, \quad j \geqslant 0, \end{aligned}$$
(2.4)

then the distribution P is q-moment indeterminate.

At this stage, the results related to an interrelation between the notions of moment- and q-moment determinacy will be presented. The next statement shows that the properties of a probability distribution with respect to moment- and q-moment determinacy may be diverse. Notice that the conditions of the next theorem are given in terms of the classical moments, while the conclusion is concerned with the q-moment problem.

Theorem 2.4

Let \(P\in {{\mathcal {A}}}\) and \(\{\mu _n\}_{n=1}^{\infty }\) be a sequence of its moments such that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\ln \mu _n}{n^2} = a. \end{aligned}$$

If \(q<e^{-2a},\) then P is q-moment determinate. In particular, if \(q\in (0,1)\) and

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\ln \mu _n}{n^2}=0, \end{aligned}$$

then P is q-moment determinate.

The preceding results lead to the following

Theorem 2.5

There exist probability distributions which are simultaneously moment indeterminate and q-moment determinate. Moreover, there exist probability distributions which are moment indeterminate and q-moment determinate for each \(q\in (0,1).\)

3 Proofs of the Results

In this section, the notation \(M(r;f):=\max _{|z|=r} |f(z)|\), where f(z) is a function analytic in \(\{z:|z|=r\}\), will be used.

Proof of Theorem 2.1

Assume that there exists a q-density \(g\not \sim f\) such that \(m_q(n;f)=m_q(n;g)\) for all \(n\in {{\mathbb {N}}}_0,\) that is,

$$\begin{aligned} \sum _{j\in {{\mathbb {Z}}}} f(q^{-j}) q^{-nj}=\sum _{j\in {{\mathbb {Z}}}} g(q^{-j}) q^{-nj}, \quad n\in {{\mathbb {N}}}. \end{aligned}$$

The existence of the q-moments implies that the Laurent series \(\sum _{j\in {{\mathbb {Z}}}} f(q^{-j})z^j\) and \(\sum _{j\in {{\mathbb {Z}}}} g(q^{-j})z^j\) converge in \({{\mathbb {C}}}^*={{\mathbb {C}}}{\setminus } \{0\}\) to \(\phi _1(z)\) and \(\phi _2(z),\) respectively, both of which are analytic in \({{\mathbb {C}}}^*.\) Now, we apply the following lemma proved in [17, Lemma 2.6]:

Lemma 3.1

Let \(\phi (z)=\sum _{j\in {{\mathbb {Z}}}} c_j z^j\) satisfy \(\phi (q^{-n})=0\) for all \(n\in {{\mathbb {N}}}.\) Then, for some \(C=C(q)>0, \) one has

$$\begin{aligned} M(q^{-n}; \phi ) \geqslant C q^{-n(n-1)/2} \quad n\in {{\mathbb {N}}}. \end{aligned}$$
(3.1)

By Lemma 3.1, for \(\phi =\phi _1-\phi _2,\) (3.1) holds. On the other hand,

$$\begin{aligned}&M(q^{-n};\phi ) \leqslant M(q^{-n};\phi _1) + M(q^{-n};\phi _2) = \frac{m_q(n-1;f)}{1-q} \\&\quad + \frac{m_q(n-1;g)}{1-q} = \frac{2 m_q(n-1;f)}{1-q}, \quad n\in {{\mathbb {N}}}, \end{aligned}$$

whence

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\ln M(q^{-n};\phi )}{n^2} \leqslant \limsup _{n\rightarrow \infty }\frac{\ln m_q(n;f)}{n^2} < \frac{\ln (1/q)}{2}, \end{aligned}$$
(3.2)

due to the assumption (2.1). Meanwhile, (3.1) yields

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\ln M(q^{-n};\phi )}{n^2} \geqslant \frac{\ln (1/q)}{2} \end{aligned}$$

which contradicts (3.2). \(\square \)

Proof of Theorem 2.2

To prove the statement, it suffices to show that, under the conditions (2.2) and (2.3), the density f satisfies the condition (1.5). Consider

$$\begin{aligned} \psi (z)=\sum _{j\in {{\mathbb {Z}}}} f(q^{-j})z^j = \sum _{j=-\infty }^{-1} f(q^{-j})z^j+\sum _{j=0}^\infty f(q^{-j})z^j=:\psi _1(z)+\psi _2(z). \end{aligned}$$

Here, \(\psi _1\) is a function analytic at \(\infty \) with \(\psi _1(\infty )=0,\) whence \(M(r;\psi _1)\rightarrow 0\) as \(r\rightarrow \infty .\) Consequently,

$$\begin{aligned} \limsup _{r\rightarrow \infty } \frac{\ln M(r;\psi )}{\ln ^2 r} = \limsup _{r\rightarrow \infty } \frac{\ln M(r;\psi _2)}{\ln ^2r}. \end{aligned}$$

For a q-density f,  one has \(m_q(n;f)=(1-q)\psi (q^{-(n+1)})=(1-q)M(q^{-(n+1)};\psi )\) for all \(n\in {{\mathbb {N}}}.\) Therefore,

$$\begin{aligned}&\limsup _{r\rightarrow \infty } \frac{\ln M(r;\psi _2)}{\ln ^2 r} = \limsup _{r\rightarrow \infty } \frac{\ln M(r;\psi )}{\ln ^2 r} = \limsup _{n\rightarrow \infty } \frac{\ln M(q^{-(n+1)};\psi )}{n^2\ln ^2(1/q)} \\&\quad =\limsup _{n\rightarrow \infty } \frac{\ln m_q(n;f)-\ln (1-q)}{n^2 \ln ^2(1/q)} = \frac{A}{\ln ^2(1/q)}< \infty . \end{aligned}$$

To complete the proof, we refer to the result below by V. Boicuk and A. Eremenko.

Theorem 3.2

[3, Theorem 3] Let \(f(z)=\sum _{j=0}^\infty c_jz^j\) be an entire function such that \(|c_{j-1}c_{j+1}|\leqslant |c_j|^2\) and

$$\begin{aligned} \lim _{r\rightarrow \infty } \frac{\ln M(r;f)}{\ln ^2r} = \beta < \infty . \end{aligned}$$

Then,

$$\begin{aligned} \liminf _{j\rightarrow \infty }\frac{\ln |c_j|}{j^2}\geqslant -\frac{1}{4\beta }. \end{aligned}$$

Applying this result to \(\psi _2\) with \(c_j=f(q^{-j})\) and \(\beta =A/\ln ^2(1/q)\) yields

$$\begin{aligned} f(q^{-j}) \geqslant \exp \left\{ -\frac{j^2}{4\beta }+o(j^2)\right\} =\exp \left\{ -\frac{\ln ^2(1/q)}{4A}j^2+o(j^2)\right\} ,\quad j\rightarrow \infty , \end{aligned}$$

implying that, for \(A>\ln (1/q)/2, \) one has

$$\begin{aligned} f(q^{-j})\ge C\exp \left\{ -\frac{j(j+1)}{2}\ln (1/q)\right\} = Cq^{j(j+1)/2} \end{aligned}$$

for some \(C>0\) and j large enough. Since the sequence \(\{f(q^{-j})\}_{j\geqslant 0}\) does not vanish, it follows that this condition is satisfied for all \(j\in {\mathbb {N}}_0\). Thus, by (1.5) the distribution P is q-moment indeterminate. \(\square \)

Example 3.1

Using the q-exponential function given in (1.6), consider the q-gamma function of the form

$$\begin{aligned} \gamma _q(\alpha ):=\int _0^\infty t^{\alpha -1} e_q(-t) d_q t, \quad \alpha > 0, \end{aligned}$$

as defined, for example, in [9, formula (4.1) with \(A=(1-q)^{-1}\)]. By a q-gamma distribution with a shape parameter \(\alpha >0\) and scaling parameter \(\lambda >0,\) it is meant a distribution whose q-density is equivalent to

$$\begin{aligned} f(t) = C\, t^{\alpha -1} e_q(-\lambda t), \quad t>0. \end{aligned}$$
(3.3)

Clearly, if \(\alpha =1,\) one recovers (1.7), while if \(\alpha =n\in {{\mathbb {N}}},\) one has the q-density of the n-stage q-Erlang distribution. See [6, Sec. 24] and [14]. Notice that f(t) satisfies condition (2.3) whatever \(\lambda >0\) is since

$$\begin{aligned} 0<\frac{f(q^{-j-1}) f(q^{-j+1})}{[f(q^{-j})]^2 } = \frac{1+\lambda (1-q) q^{-j+1}}{1+\lambda (1-q) q^{-j-1}} < 1. \end{aligned}$$

As the q-moments of the q-gamma distribution are

$$\begin{aligned} m_q(n;f)=C \int _0^\infty t^{n+\alpha -1} e_q(-\lambda t) d_q t, \end{aligned}$$

one derives the recurrence relation

$$\begin{aligned} m_q(n;f) = \frac{1- q^{n+\alpha -1}}{\lambda (1-q)}\, q^{-(n+\alpha -1)}\, m_q(n-1;f), \end{aligned}$$

and, hence,

$$\begin{aligned} m_q(n;f)=\frac{(q^\alpha ;q)_n}{[\lambda (1-q)]^n}\, q^{-\frac{n(n-1)}{2}-n\alpha }, \end{aligned}$$
(3.4)

where \((a;q)_j\) stands for the q-shifted factorial defined by

$$\begin{aligned} (a;q)_0:=1, \quad (a;q)_j=\prod _{s=0}^{j-1}(1-aq^s), \quad a\in {{\mathbb {C}}}. \end{aligned}$$

It is worth pointing out that, when \(\lambda = 1,\) the q-moments (3.4) coincide with the moment sequence associated with the q-Laguerre polynomials ([16, formula (2.13)], [2, formula (4.1)] and [8, formula (2.0.9)]), and that this moment sequence corresponds to an indeterminate Stieltjes moment problem. Explicit examples of distributions with these moments are provided in [2, Proposition 4.1] and [8, Propositions 2.1 and 2.2].

As for the q-moment determinacy, it can be easily obtained from (3.4) that, for all \(\lambda , \alpha >0,\)

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\ln m_q(n;f)}{n^2} = \frac{\ln (1/q)}{2}, \end{aligned}$$

that is, Theorems 2.1 and 2.2 are not applicable. Meanwhile, estimating, for \(j\geqslant 0,\)

$$\begin{aligned} e_q(-\lambda q^{-j})&= \prod _{s=0}^\infty \frac{1}{1+\lambda (1-q)q^{s-j}} = \prod _{s=1}^j \frac{q^s}{q^s+\lambda (1-q)} \prod _{s=0}^\infty \frac{1}{1+\lambda (1-q)q^{s}} \\&= \frac{e_q(-\lambda )}{\left( -\frac{q}{\lambda (1-q)};q\right) _j} \frac{q^{j(j+1)/2}}{[\lambda (1-q)]^j}{\sim } \frac{e_q(-\lambda )}{\left( -\frac{q}{\lambda (1-q)};q\right) _\infty }\frac{q^{j(j+1)/2}}{[\lambda (1-q)]^j},\quad j{\rightarrow +}\infty , \end{aligned}$$

one derives

$$\begin{aligned} f(q^{-j}) \sim C q^{j(j+1)/2}\left[ \frac{q^{1-\alpha }}{\lambda (1-q)}\right] ^j,\quad j\rightarrow +\infty . \end{aligned}$$

By virtue of formulae (1.4) and (1.5), one concludes that a q-gamma distribution is q-moment determinate if and only if \(\lambda > q^{1-\alpha }/(1-q).\)

In particular, the q-exponential distribution with q-density (1.7) is q-moment determinate if and only if \(\lambda > 1/(1-q).\)

Proof of Theorem 2.3

Recall the well-known Euler identity [11, Section 9]:

$$\begin{aligned} \prod _{j=0}^{\infty }(1+q^j t)=\sum _{j=0}^{\infty } \frac{q^{j(j-1)/2}}{(q;q)_j}\, t^j, \quad t\in {{\mathbb {C}}}. \end{aligned}$$

Consider the entire function \(\phi _m(z)=\prod _{j=1}^\infty (1-q^{mj}z)\), for which it is clear that \(\phi _m(q^{-m(n+1)})=0\) for all \(n\in {{\mathbb {N}}}_0.\) By virtue of the Euler identity

$$\begin{aligned} \sum _{j=0}^\infty \frac{(-1)^k q^{mk(k+1)/2}}{(q^m;q^m)_k}\, q^{-m(n+1)k}=0, \quad n\in {{\mathbb {N}}}_0. \end{aligned}$$
(3.5)

Define the sequence \(\{c_j\}_{j\in {\mathbb {Z}}}\) in such a way that

$$\begin{aligned} c_{-j}=\left\{ \begin{array}{cl} f(q^{-j})+\alpha \frac{(-1)^{j/m}q^{j(j/m+1)/2}}{(q^m;q^m)_{j/m}}&{}\quad \text {if}\quad j\in m{\mathbb {N}}_0,\\ &{}\\ f(q^{-j})&{}\quad \text {otherwise},\end{array}\right. \end{aligned}$$

where \(\alpha >0\) is selected so that \(c_j\geqslant 0\) for all \(j\in {{\mathbb {Z}}}.\) Such a selection is possible due to condition (2.4). Furthermore, using (3.5) and the fact that f is a q-density, one obtains that

$$\begin{aligned} (1-q)\sum _{j\in {\mathbb {Z}}}c_{j}q^{j}=(1-q)\sum _{j\in {\mathbb {Z}}}f(q^j)q^j+ \alpha (1-q)\sum _{k=0}^\infty \frac{(-1)^k q^{mk(k+1)/2}}{(q^m;q^m)_k}q^{-mk}=1. \end{aligned}$$

Setting \(G(0)=0\) and \(G(q^j)=(1-q)\displaystyle \sum _{l=j}^\infty c_lq^l,\;j\in {\mathbb {Z}},\) and taking G(x) to be non-decreasing on each interval \([q^{j+1},q^j],\) one obtains a distribution function G,  whose q-density g satisfies \(g(q^j)=c_j\) for all \(j\in {\mathbb {Z}}.\) Obviously, \(g \not \sim f\). Meanwhile, applying (3.5), one obtains:

$$\begin{aligned} m_q(n;g)&=(1-q)\sum _{j\in {{\mathbb {Z}}}} g(q^{-j}) q^{-j(n+1)} \\&=(1-q)\sum _{j=0}^{\infty } f(q^{-j}) q^{-j(n+1)}+\alpha (1-q)\phi _m\left( q^{-m(n+1)}\right) \\&=(1-q)\sum _{j\in {{\mathbb {Z}}}} f(q^{-j}) q^{-j(n+1)}=m_q(n;f). \end{aligned}$$

Thus, P is q-moment indeterminate.\(\square \)

Note that the result cannot be derived from Theorem 2.2, although

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\ln m_q(n;f)}{n^2} \geqslant \frac{m}{2}\, \ln (1/q). \end{aligned}$$

Proof of Theorem 2.4

Let F be the distribution function and f be the q-density of P,  respectively. Then one has:

$$\begin{aligned} \mu _n&= \int _{0}^\infty t^n dF(t) = \sum _{j\in {\mathbb {Z}}} \int _{q^{j+1}}^{q^{j}} t^n\, dF(t) \geqslant q^n\sum _{j\in {\mathbb {Z}}} q^{jn} \left[ F(q^{j-1})-F(q^{j})\right] \\&=q^n\sum _{j\in {\mathbb {Z}}} q^{(j+1)n} (1-q)f(q^j)=q^n m_q(n;p) \end{aligned}$$

by virtue of (1.3). Hence,

$$\begin{aligned} a=\limsup _{n\rightarrow \infty } \frac{\ln m_q(n;X)}{n^2} \leqslant \limsup _{n\rightarrow \infty } \frac{n\ln (1/q)+\ln \mu _n }{n^2}< \frac{\ln (1/q)}{2} \end{aligned}$$

since \(q<e^{-2a}\) according to the assumption of the theorem. By Theorem 2.1, P is q-moment determinate.\(\square \)

Example 3.2

Let f(t) be a density of a generalized gamma distribution with parameters \(\alpha ,\) \(\beta ,\) \(\gamma >0,\) that is,

$$\begin{aligned} f(t)=\frac{\gamma \beta ^{-\alpha /\gamma }}{\Gamma (\alpha /\gamma )}\, t^{\alpha -1} \exp \left( -t^{\gamma }/\beta \right) , \quad t>0. \end{aligned}$$

It is known—see [19, Section 11.4]—that the moments of the distribution are

$$\begin{aligned} \mu _n=\frac{\beta ^{n/\gamma }}{\Gamma (\alpha /\gamma )} \Gamma \left( \frac{n+\alpha }{\gamma }\right) \end{aligned}$$

and that it is moment indeterminate if and only if \(\gamma \in (0, 1/2)\). Since

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\ln \mu _n}{n^2}=0, \end{aligned}$$

we conclude by Theorem 2.4 that the generalized gamma distribution is q-moment determinate for all \(q\in (0,1)\) regardless of its parameter values.

Proof of Theorem 2.5

The assertion of the theorem is an immediate consequence of either Examples 3.1 or 3.2. The details are below.

(a) Example 3.1 shows that the q-exponential distribution possessing q-density (1.7) is q-moment determinate whenever \(\lambda >1/(1-q)\). Meanwhile, P is moment indeterminate for all \(\lambda >0.\) This can be established with the help of the Krein condition [19, Section 11, p.101]. Indeed, the density \(\rho \) of P can be estimated as

$$\begin{aligned} \rho (t)&=\frac{dF(t)}{dt}\, =\lambda (1-q)e_q(-\lambda t) \sum _{j=0}^\infty \frac{q^j}{1+\lambda (1-q)q^j t} \ge \frac{\lambda (1-q) e_q(-\lambda t)}{1+\lambda (1-q)t}. \end{aligned}$$

The bounds for \(e_q(-\lambda t)\) are derived from the next inequality obtained in [20, formula (2.6)], which holds for some positive constants \(C_1=C_1(q),\) \(C_2=C_2(q)\) and t large enough:

$$\begin{aligned} C_1 \exp \left\{ \frac{\ln ^2 t}{2\ln (1/q)}+\frac{\ln t}{2}\right\} \leqslant \prod _{j=0}^{\infty }(1+q^j t) \leqslant C_2 \exp \left\{ \frac{\ln ^2 t}{2\ln (1/q)}+\frac{\ln t}{2}\right\} . \end{aligned}$$

Thus,

$$\begin{aligned} \frac{1}{\rho (t^2)} \leqslant C \exp \left\{ \frac{2\ln ^2 t}{\ln (1/q)}[1+\omega (t)]\right\} ,\quad \text {where}\quad \omega (t)=o(1)\quad \text {as}\quad t\rightarrow \infty . \end{aligned}$$

Consequently, the Krein integral \(\displaystyle \int _{t_0}^{\infty } \frac{-\ln \rho (t^2)}{1+t^2} \, dt \) converges and, hence, the distribution P is moment indeterminate for all \(\lambda > 0\) and \(q\in (0,1).\)

(b) Example 3.2 also proves this theorem since, when \(\gamma \in (0,1/2),\) a generalized gamma distribution is moment indeterminate while being q-moment determinate for any given \(q\in (0,1).\) \(\square \)