Two-Motzkin-Like Numbers and Stieltjes Moment Sequences

Abstract

First, we introduce the two-Motzkin-like number as the weight of vertically constrained Motzkin-like path with no leading vertical steps from (0, 0) to (n, 0) consisting of up steps, down steps, horizontal steps, vertical steps in the down direction and vertical steps in the up direction. Secondly, we provide sufficient conditions under which the two-Motzkin-like numbers (resp. the q-analogue of the two-Motzkin-like numbers) are Stieltjes moment sequences (resp. are q-Stieltjes moment sequences) and therefore infinitely log-convex sequences. As applications, on the one hand, we show that many well-known counting coefficients, including the central trinomial $\left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}$ and pentanomial $\left( {\begin{array}{c}2n\\ 4n\end{array}}\right) _{4}$ numbers of even indices respectively are Stieltjes moment sequences and, therefore, infinitely log-convex sequences in a unified approach. On the other hand, we prove that the sequence of polynomials of square trinomials $\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) ^{2} _{2}q^{k}$ are q-Stieltjes moment sequence of polynomials. Finally, we provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences in more generalized triangular array.

1 Introduction

Let ${\alpha }=(a_{n})_{n\ge 0}$ be a sequence of nonnegative numbers. The sequence is called log-convex (resp. log-concave) if $a_{n}a_{n+2}\ge a_{n+1}^{2}$ (resp. $a_{n}a_{n+2}\le a_{n+1}^{2}$) for all $n\ge 0$. The log-convex and log-concave sequences arise often in combinatorics and have been extensively investigated. We refer the reader to [12, 27, 29] for the log-concavity and [23, 30] for the log-convexity. A basic approach to such problems comes from the theory of total positivity [11,12,13,14, 16, 17, 20].

Let ${\alpha }=(a_{n})_{n\ge 0}$ be an infinite sequence of positive numbers. Define a new sequence ${\mathcal {L}}(a_{n})=(b_{n})_{n\ge 0}$ (resp. $L(a_{n})=(c_{n})_{n\ge 0}$) by $b_{n}=a_{n}a_{n+2}-a_{n+1}^{2}$ (resp. $c_{n}=a_{n}+a_{n+2}-2a_{n+1}$). Then $b_{n}$ and $c_{n}$ are log-convex (resp. convex) if and only if ${\mathcal {L}} ({\alpha })$ (resp. $L({\alpha })$) is nonnegative. Call m-log-convex (resp. m-convex) if ${\mathcal {L}}^{i}( {\alpha })$ (resp. $L^{i}({\alpha })$) is nonnegative for all $1\le i\le m$ and infinitely log-convex (resp. infinitely convex) if ${\mathcal {L}} ^{i}({\alpha })$ (resp. $L^{i}({\alpha })$) is nonnegative for all $i\ge 1$. Chen and Xia [18] showed that some combinatorial sequences, including the Apéry numbers and the Schröder numbers, are 2-log-convex via analytic methods.

We say that ${\alpha }$ is a moment sequence if it admits a representation of the form

$$\begin{aligned} a_{n}=\int _{I}x^{n}\mathrm{{d}}\mu (x) \quad (n=0,1,2,\ldots ), \end{aligned}$$

(1)

where I is some interval of the real line and $d\mu (x)$ is a finite nonnegative Borel measure supported on I. Of special interest are the moment sequences associated with the names of Hamburger (HM for short): I $=(-\infty ,+\infty )$, Stieltjes (SM for short): $I=[0,+\infty )$.

Now, we define the (finite or infinite) Hankel matrix of the sequence ${\alpha }$ by

$$\begin{aligned} H({\alpha })=\left[ a_{i+j}\right] _{i,j\ge 0}=\left[ \begin{array}{ccccc} a_{0}\ \ &{} a_{1}\ \ &{} a_{2}\ \ &{} a_{3}\ \ &{} \cdots \\ a_{1}\ \ &{} a_{2}\ \ &{} a_{3}\ \ &{} a_{4}\ \ &{} \\ a_{2}\ \ &{} a_{3}\ \ &{} a_{4}\ \ &{} a_{5}\ \ &{} \\ a_{3}\ \ &{} a_{4}\ \ &{} a_{5}\ \ &{} a_{6}\ \ &{} \\ \vdots &{} &{} &{} &{} \ddots \end{array} \right] . \end{aligned}$$

Then ${\alpha }$ is a HM if and only if $H({\alpha })$ is positive semi-definite:

$$\begin{aligned} \sum _{i,j\ge 0}a_{i+j}x_{i}x_{j}\ge 0 \end{aligned}$$

(2)

for an arbitrary sequence $(x_{n})_{n\ge 0}$ of real numbers with finite support (i.e. $x_{n}=0$ except for finitely many values of n), or equivalently, all Hankel determinants

$$\begin{aligned} h_{n}({\alpha })=\det \left[ a_{i+j}\right] _{0\le i,j\le n} \end{aligned}$$

are nonnegative and $h_{N}=0$ implies $h_{n}=0$ for $n\ge N$ (see [25, Theorem 1.2] for instance). As example, the sequence of Fibonacci numbers ${\mathcal {F}} =(F_{n})_{n\ge 0},$ defined by $F_{0}=F_{1}=1$ and $F_{n}=F_{n-1}+F_{n-2},$ is HM.

We say that a matrix is totally positive (TP for short) if all its minors are nonnegative. It is well known that a sequence ${\alpha }$ is SM if and only if its Hankel matrix $H({\alpha })$ is TP, or equivalently, both ${\alpha }=(a_{n})_{n\ge 0}$ and its shift $\overline{{\alpha }}=(a_{n+1})_{n\ge 0}$ are HM (see [25, Theorem 1.3] for instance).

The bi$^{s}$nomial coefficients or the ordinary multinomials are a natural extension of binomial coefficients (see, for instance, [9]). These coefficients are defined as follows: let $s\ge 1$ and $n\ge 0$ be two integers, and $k=0,1,\ldots ,sn$, the bi$^{s}$nomial coefficient $\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}$ is the kth coefficient in the expansion,

$$\begin{aligned} (1+x+\cdots +x^{s})^{n}=\sum _{k\in {\mathbb {Z}}}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}x^{k}, \end{aligned}$$

(3)

with $\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}=0$ for $k>sn$ or $k<0$.

Some readily well-known established properties are

• the symmetry relation

  $$\begin{aligned} \left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}=\left( {\begin{array}{c}n\\ sn-k\end{array}}\right) _{s}, \end{aligned}$$

  (4)

• the longitudinal recurrence relation

  $$\begin{aligned} \left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}=\sum _{j=0}^{s}\left( {\begin{array}{c}n-1\\ k-j\end{array}}\right) _{s}, \end{aligned}$$

  (5)

• the absorption identity

  $$\begin{aligned} k\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}=n\sum _{j=1}^{s}j\left( {\begin{array}{c}n-1\\ k-j\end{array}}\right) _{s}. \end{aligned}$$

  (6)

These coefficients, as for usual binomial coefficients, are built as for the Pascal triangle, known as “s-Pascal triangle”. One can find the first values of the s-Pascal triangle in OEIS [26] as A027907 for $s=2$ (Table 1).

Table 1 Triangle of trinomial coefficients: $s=2$

Very recently, Belbachir and Igueroufa [7] generalized the absorption identity (6) and some classical congruence properties for the bi$^{s}$nomial coefficients. They also established under suitable hypothesis the corresponding versions of Kummer’s Theorem and Ram’s Theorem. For other properties of bi$^{s}$nomial coefficients, see, for instance, [6, 9, 15].

The first result dealing with unimodality of bi$^s$nomial coefficients is due to Belbachir and Szalay [10] who proved that any ray crossing Pascal’s triangle provides a unimodal sequence. In [1,2,3,4], respectively, the authors established the strong log-convexity, the unimodality, the preserving log-convexity and log-concavity properties for the bi$^s$nomial coefficients.

A natural extension of the bi$^{s}$nomial coefficients are the q-bi$^{s}$nomial coefficients, which satisfy the following recursion:

$$\begin{aligned} {n\brack k}_{q}^{(s)}=\sum _{j=0}^{s}q^{(n-1)j}{n-1\brack k-j} _{q}^{(s)} \end{aligned}$$

(7)

and which appear as the coefficients of the following product:

$$\begin{aligned} \prod _{j=0}^{n-1}\left( 1+q^{j}z+\cdots +(q^{j}z)^{s}\right) =\sum _{k\ge 0}{n\brack k}_{q}^{(s)}z^{k}. \end{aligned}$$

(8)

For more details, we refer the reader to [5, 6, 8].

Liang et al. [21] gave the sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. These numbers are the 0th column of the infinite lower triangular matrix $[a_{n,k}]_{n,k\ge 0}$, where the $a_{n,k}$ are defined by the recurrence

$$\begin{aligned} a_{n+1,k}=a_{n,k-1}+r_{k}a_{n,k}+s_{k+1}a_{n,k+1} \end{aligned}$$

(9)

subject to the initial conditions that $a_{0,0}=1$ and $a_{n,k}=0$ unless $n\ge k\ge 0$. As applications, they showed that many well known counting coefficients form Stieltjes moment sequences, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the Schröder numbers. Wang and Zhu [28, Theorem 2.3] showed that Stieltjes moment sequences are infinitely log-convex. They introduced the concept of q-Stieltjes moment sequences of polynomials and showed that many well-known polynomials in combinatorics are such sequences. On the other hand, they also proved a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Recently, Liang et al. [22] proved that many well-known counting coefficients associated with the Catalan-like numbers form a Hamburger sequences in certain unified approaches and that Hamburger moment sequences are infinitely convex.

In our paper, we establish the SM, HM and the infinite log-convexity properties of many well-known combinatorial sequences linked in more generalized triangles. In Sect. 2, first, as an extension of the Catalan-like numbers, we define the two-Motzkin-like numbers as the 0th column of the infinite generalized triangular array $A=[a_{n,k}]_{n,k\ge 0}$ of the form

$$\begin{aligned} a_{0,0}=1, a_{n+1,k}=f_{k-2}a_{n,k-2}+p_{k-1}a_{n,k-1}+r_{k}a_{n,k}+s_{k+1}a_{n,k+1}+t_{k+2}a_{n,k+2}, \end{aligned}$$

where $a_{n,k}=0$ unless $2n\ge k\ge 0$. These numbers count the weight of vertically constrained Motzkin-like path with no leading vertical steps from (0, 0) to (n, 0) consisting of up steps, down steps, horizontal steps, vertical steps in the down direction and vertical steps in the up direction. Second, we provide sufficient conditions such that the two-Motzkin-like numbers are Stieltjes moment sequences, and are therefore infinitely log-convex. As applications, we show that the central trinomial, Motzkin and central pentanomial numbers of even indices are Stieltjes moment sequences, and are therefore infinitely log-convex. In Sect. 3, we provide sufficient conditions such that the q-analogue of two-Motzkin-like numbers are q-Stieltjes moment sequences of polynomials. For example, the sequence of polynomials of square trinomials is a q-Stieltjes moment sequence of polynomials. In Sect. 4, we give a criterion for the linear transformations and convolutions preserving Stieltjes moment sequences in more generalized triangles, a parallel result to [28, Theorem 4.2]. As applications, the square trinomial, the bi$^{s}$nomial convolutions and transformations preserve the Stieltjes moment sequences property.

2 Infinitely Log-Convex and Stieltjes Moment Sequences

2.1 Vertically Constrained Motzkin-Like Paths

Following I. Veronika et al.  [19], a vertically constrained Motzkin-like path with no leading vertical steps is a lattice path formed from the set of step vectors ${\mathcal {A}}=\{(1,0),(1,1),(1,-1),(0,1),(0,-1)\}$ with the restriction that vertical steps $\{(0,1),(0,-1)\}$ cannot be consecutive and cannot be leading steps. The ${\mathcal {A}}$ is the union of the well-known Motzkin step vectors ${\mathcal {M}}=\{(1,0)$, (1, 1), $(1,-1)\}$ with the vertical steps $\{(0,1),(0,-1)\}$. The last authors showed that the trinomial number (resp. the Motzkin number) of even index counts the number of vertically constrained Motzkin-like paths with no leading vertical steps from (0, 0) to (n, 0) in the half-plane (resp. in the quarter-plane). They showed also that the number of vertically constrained Motzkin-like paths with no flat and leading vertical steps (i.e. the steps from the set ${\mathcal {B}}=\{(1,1)$, $(1,-1)$, (0, 1), $(0,-1)\}$) from (0, 0) to (n, 0) in the quarter-plane is the number A214938 in OEIS. Now let ${{\mathcal {F}}}=(f_{k})_{k\ge 0}$, ${{\mathcal {P}}}=(p_{k})_{k\ge 0}$, ${\gamma }=(r_{k})_{k\ge 0},$ ${\sigma }=(s_{k})_{k\ge 1}$ and ${\tau }=(t_{k})_{k\ge 2}$ be five sequences of nonnegative numbers. Denote by $a_{n,k}$ the total weight of all vertically constrained Motzkin-like (${\mathcal {A}},{\mathcal {B}},\ldots$) paths from (0, 0) to (n, k) and assume that the last sequences determine the following recurrence relation by considering all ways the vertically constrained $({\mathcal {A}},{\mathcal {B}},\ldots )$ paths can terminate at a lattice point:

$$\begin{aligned} a_{0,0}=1,a_{n+1,k}=f_{k-2}a_{n,k-2}+p_{k-1}a_{n,k-1}+r_{k}a_{n,k}+s_{k+1}a_{n,k+1}+t_{k+2}a_{n,k+2},\nonumber \\ \end{aligned}$$

(10)

where $a_{n,k}=0$ unless $2n\ge k\ge 0$. We call $a_{n}:=a_{n,0}$ the two-Motzkin-like numbers corresponding to $({{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau })$.

Example 1

The two-Motzkin-like numbers unify many well-known counting coefficients, such as

• 1. the Motzkin numbers of even index $M_{2n}$ if ${{\mathcal {F}}}=(1,1,1,\ldots )$, ${{\mathcal {P}}}=(2,2,2,\ldots )$, ${\gamma }=(2,3,3$, $\ldots )$, ${\sigma }=(2,2,2,\ldots )$ and ${\tau }=(1,1,1,\ldots )$;

• 2. the sequence of the numbers A291090 in OEIS if ${{\mathcal {F}}}=(1,1,1,\ldots )$, ${{\mathcal {P}}}=(1,1,1,\ldots )$, ${\gamma }=(1,2,2,\ldots )$, ${\sigma }=(1,1,1,\ldots )$ and ${\tau }=(1,1,1,\ldots )$;

• 3. the sequence of the numbers A047098 in OEIS if ${{\mathcal {F}}}=(1,1,1,\ldots )$, ${{\mathcal {P}}}=(1,1,1,\ldots )$, ${\gamma }=(2,2,2,\ldots )$, ${\sigma }=(2,2,1,\ldots )$ and ${\tau }=(2,1,1,\ldots )$;

• 4. the central trinomial coefficients of even index $\left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}$ if ${{\mathcal {F}}}=(1,1,1,\ldots )$, ${{\mathcal {P}}}=(2,2,2,\ldots )$, ${\gamma }=(3,4,3$, $\ldots )$, ${\sigma }=(4,2,2,\ldots )$ and ${\tau }=(2,1,1,\ldots )$;

• 5. the central pentanomial coefficients $\left( {\begin{array}{c}n\\ 2n\end{array}}\right) _{4}$ if ${{\mathcal {F}}}=(1,1,\ldots )$, ${{\mathcal {P}}}=(1,1,1,\ldots )$, ${\gamma }=(1,2,1$, $\ldots )$, ${\sigma }=(2,1,1,\ldots )$ and ${\tau }=(2,1,1,\ldots )$.

Taking ${{\mathcal {F}} }={\tau }=(0,0,0,\ldots )$ and ${{\mathcal {P}}}=(1,1,1,\ldots )$ in (10), we obtain the recursive relation (9) associated with the Catalan-like numbers corresponding to $({\gamma ,\sigma })$.

Remark 1

The Catalan-like numbers unify many well-known counting coefficients, such as the Motzkin numbers $M_n$, the Catalan numbers $C_n$, the central binomial coefficients $\left( {\begin{array}{c}2n\\ n\end{array}}\right)$, the central trinomial coefficients $\left( {\begin{array}{c}n\\ n\end{array}}\right) _{2}$, the central Delannoy numbers $D_n$, the Fine numbers $F_n$, the (restricted) hexagonal numbers $h_n$, the Riordan numbers $R_n$, the factorial numbers n!, the Schröder numbers and the Bell numbers $B_n$.

2.2 Infinite Log-Convexity of Two-Motzkin-Like Numbers

Let ${{\mathcal {F}}}=(f_{k})_{k\ge 0}$, ${{\mathcal {P}}}=(p_{k})_{k\ge 0}$, ${\gamma }=(r_{k})_{k\ge 0}$, ${\sigma }=(s_{k})_{k\ge 1}$ and ${\tau }=(t_{k})_{k\ge 2}$ be five sequences of nonnegative numbers, and define an infinite generalized triangular matrix $A:=A^{{{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau }}=[a_{n,k}]_{n,k\ge 0}$ by the recurrence

$$\begin{aligned} a_{0,0}=1, a_{n+1,k}=f_{k-2}a_{n,k-2}+p_{k-1}a_{n,k-1}+r_{k}a_{n,k}+s_{k+1}a_{n,k+1}+t_{k+2}a_{n,k+2},\nonumber \\ \end{aligned}$$

(11)

where $a_{n,k}=0$ unless $2n\ge k\ge 0$.

The coefficient matrix of (11) is

$$\begin{aligned} J:=J^{{{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau }}=\left[ \begin{array}{cccccc} r_{0}\ &{} p_{0}\ &{} f_{0}\ &{} &{} &{} \\ s_{1}\ &{} r_{1}\ &{} p_{1}\ &{} f_{1}\ &{} \\ t_{2}\ &{} s_{2}\ &{} r_{2}\ &{} p_{2}\ &{} \ddots \\ &{} t_{3}\ &{} s_{3}\ &{} r_{3}\ &{} \ddots \\ &{} &{} \ddots &{} \ddots &{} \ddots \end{array} \right] . \end{aligned}$$

The nth leading principal submatrices of A and J are $A_{n}$ and $J_{n}$, respectively.

Definition 1

We define the recursive matrix associated with the generalized triangular matrix A by the recurrence

$$\begin{aligned} a_{0,0}=1, \ a_{n+1,0}=r_{0}a_{n,0}+s_{1}a_{n,1}+t_{2}a_{n,2}. \end{aligned}$$

(12)

Remark 2

All sequences of the two-Motzkin-like numbers in Example 1 satisfy the relation (12).

Now, we define the recursive matrix $A(\xi ,\beta ,\delta ,f,p,r,s,t)=[a_{n,k}]_{n,k\ge 0}$ by the recurrence

$$\begin{aligned} \left\{ \begin{array}{l} a_{0,0}=1, \ a_{n+1,0}=\xi a_{n,0}+\beta a_{n,1}+\delta a_{n,2} \\ a_{n+1,k}=f a_{n,k-2}+pa_{n,k-1}+ra_{n,k}+sa_{n,k+1}+ta_{n,k+2}, \end{array} \right. \end{aligned}$$

(13)

where $a_{n,k}=0$ unless $2n\ge k\ge 0$. Then we have the following result.

Proposition 1

The double generating function of $[a_{n,k}]_{n,k\ge 0}$ defined by the recurrence (13) is

$$\begin{aligned} A(x,y)=\sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}=\frac{y^{2}}{y^{2}-x(fy^{4}+py^{3}+ry^{2}+sy+t)}. \end{aligned}$$

(14)

Proof

Using (13), we get

$$\begin{aligned} A(x,y)&=1+\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}\\&=1+f\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n-1,k-2}x^{n}y^{k}+p\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n-1,k-1}x^{n}y^{k}\\&\quad +r\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n-1,k}x^{n}y^{k}\\&\quad +s\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n-1,k+1}x^{n}y^{k} +t\sum _{n\ge 1}\sum _{k\in {\mathbb {Z}}}a_{n-1,k+2}x^{n}y^{k}\\&=1+fxy^{2} \sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}+pxy\sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}\\&\quad +rx\sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}\\&\quad +\frac{sx}{y}\sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k} +\frac{tx}{y^{2}}\sum _{n\ge 0}\sum _{k\in {\mathbb {Z}}}a_{n,k}x^{n}y^{k}. \end{aligned}$$

It follows that

$$\begin{aligned} A(x,y)=\frac{y^{2}}{y^{2}-x(fy^{4}+py^{3}+ry^{2}+sy+t)}, \end{aligned}$$

as required. $\square$

Rewrite the recursive relation (11) as

$$\begin{aligned} \left[ \begin{array}{cccccccc} a_{1,0}\ &{} a_{1,1}\ &{} a_{1,2}\ &{} &{} &{} &{} &{} \\ a_{2,0}\ &{} a_{2,1}\ &{} a_{2,2}\ &{} a_{2,3}\ &{} a_{2,4}\ &{} &{} &{} \\ a_{3,0}\ &{} a_{3,1}\ &{} a_{3,2}\ &{} a_{3,3}\ &{} a_{3,4}\ &{} a_{3,5}\ &{} a_{3,6}\ &{} \\ &{} &{} &{} \cdots &{} &{} &{} &{} \ddots \end{array} \right]&= \left[ \begin{array}{cccccc} a_{0,0}\ &{} &{} &{} &{} &{} \\ a_{1,0}\ &{} a_{1,1}\ &{} a_{1,2}\ &{} &{} &{} \\ a_{2,0}\ &{} a_{2,1}\ &{} a_{2,2}\ &{} a_{2,3}\ &{} a_{2,4}\ &{} \\ &{} &{} \cdots &{} &{} &{} \ddots \end{array} \right] \\&\quad \times \left[ \begin{array}{ccccc} r_{0}\ &{} p_{0}\ &{} f_{0}\ &{} &{} \\ s_{1}\ &{} r_{1}\ &{} p_{1}\ &{} f_{1}\ &{} \\ t_{2}\ &{} s_{2}\ &{} r_{2}\ &{} p_{2}\ &{} \ddots \\ &{} t_{3}\ &{} s_{3}\ &{} r_{3}\ &{} \ddots \\ &{} &{} \ddots &{} \ddots &{} \ddots \end{array} \right] , \end{aligned}$$

or briefly,

$$\begin{aligned} {\overline{A}}=AJ, \end{aligned}$$

where ${\overline{A}}$ is obtained from A by deleting the 0th row and $J=J^{{{\mathcal {F}},{\mathcal {P}},\gamma ,\sigma ,\tau }}$ is a pentadiagonal matrix. Clearly, the recursive relation (11) is decided completely by the coefficient matrix J.

Lemma 1

If $p_{k-1}p_{k}=\frac{f_{k-1} s_{k}s_{k+1}}{t_{k+1}}$ for $k\ge 1$, then the Hankel Matrix $H_{n}({\alpha })=[a_{i+j}]_{0\le i,j\le n}$ of the two-Motzkin-like numbers $a_{n}$ satisfies that

$$\begin{aligned} H_{n}({\alpha })=A_{n} U_{n} A_{n}^{t}, \end{aligned}$$

where $U_{n}=\mathrm{{diag}}(u_{0},u_{1},\ldots , u_{2n})$ and

$$\begin{aligned} u_{i}=\left\{ \begin{array}{l} 1, \ \ \ \text { if }i=0, \\ \frac{s_{1}}{p_{0}}, \ \ \ \text { if }i=1, \\ \prod _{j=1}^{\frac{i}{2}}\frac{t_{2j}}{f_{0}},\ \ \ \text { if }i\text { even,} \\ \frac{p_{1}t_{2}}{f_{0}s_{2}}\prod _{j=1}^{\frac{i-1}{2}}\frac{t_{2j+1}}{f_{2j-1}},\ \ \text { else.} \end{array} \right. \end{aligned}$$

Proof

The case $n=0$ is trivial.

When $n=1$, through computation, we get

$$\begin{aligned} A_{1} U_{1}A_{1}^{t}=\left[ \begin{array}{c@{\quad }c} a_{0,0}\ &{} a_{1,0} \\ a_{1,0}\ &{} a_{1,0}^{2}+u_{1}a_{1,1}^{2}+u_{2}a_{1,2}^{2} \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} A_{1}=\left[ \begin{array}{ccc} a_{0,0}\ &{} &{} \\ a_{1,0}\ &{} a_{1,1}\ &{}a_{1,2} \end{array} \right] \ \ \ \text { and }\ \ U_{1}=\left[ \begin{array}{ccc} u_{0}\ &{} &{} \\ &{} u_{1}\ &{}\\ &{} &{}u_{2} \end{array} \right] . \end{aligned}$$

One can verify that

$$\begin{aligned} H_{1}({\alpha })=A_{1} U_{1}A_{1}^{t}, \end{aligned}$$

where $u_{1}=\frac{s_{1}}{p_{0}}$ and $u_{2}=\frac{t_{2}}{f_{0}}$.

Similarly, when $n=2$ we have

$$\begin{aligned}&A_{2} U_{2}A_{2}^{t}=\left[ \begin{array}{ccccc} a_{0,0}\ &{} a_{1,0}\ &{} a_{2,0} \\ a_{1,0}\ &{} a_{2,0}\ &{} a_{3,0}\\ a_{2,0}\ &{} a_{3,0}\ \ &{} a_{2,0}^{2}+\frac{s_{1}}{p_{0}}a_{2,1}^{2} +t_{2}a_{2,2}^{2}+u_{3}a_{2,3}^{2}+u_{4}a_{2,4}^{2} \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} A_{2}=\left[ \begin{array}{ccccc} a_{0,0}\ &{} &{} &{} &{} \\ a_{1,0}\ &{} a_{1,1}\ &{} a_{1,2}\ &{} &{} \\ a_{2,0}\ &{} a_{2,1}\ &{} a_{2,2}\ &{} a_{2,3}\ &{} a_{2,4} \end{array} \right] \ \ \ \text { and }\ \ U_{2}=\left[ \begin{array}{ccccc} u_{0}\ &{} &{} &{} &{} \\ &{} u_{1}\ &{} &{} &{} \\ &{} &{} u_{2}\ &{} &{} \\ &{} &{} &{} u_{3}\ &{} \\ &{} &{} &{} &{} u_{4} \end{array} \right] . \end{aligned}$$

If $p_{0}p_{1}=\frac{f_{0} s_{1}s_{2}}{t_{2}}$, we get

$$\begin{aligned} H_{2}({\alpha })=A_{2} U_{2}A_{2}^{t}, \end{aligned}$$

where $u_{3}=\frac{p_{1}t_{2}t_{3}}{f_{0}f_{1}s_{2}}$ and $u_{4}=\frac{t_{2}t_{4}}{f^{2}_{0}}$.

Hence, it suffices to proceed by induction over $n\ge 3$. That completes the proof. $\square$

Remark 3

The sequences of Example 1 verify Lemma 1.

Now, we are able to give the following interesting result.

Theorem 1

Let ${{\mathcal {F}}}=(f_{k})_{k\ge 0}$, ${{\mathcal {P}}}=(p_{k})_{k\ge 0}$, ${\gamma }=(r_{k})_{k\ge 0},$ ${\sigma }=(s_{k})_{k\ge 1}$ and ${\tau }=(t_{k})_{k\ge 2}$ be five sequences of nonnegative numbers. Assume that $p_{k-1}p_{k}=\frac{f_{k-1}s_{k}s_{k+1}}{t_{k+1}}$ for $k\ge 1$. Then the two-Motzkin-like numbers corresponding to $({{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau })$ form a Hamburger moment sequence.

Proof

Let $H({\alpha })= \left[ a_{i+j}\right] _{i,j\ge 0}$ be the Hankel matrix of the two-Motzkin-like numbers ${\alpha }=(a_{n})_{n\ge 0}$ corresponding to $({{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau })$. It suffices to show that all Hankel determinants $h_{n}({\alpha })=\det \left[ a_{i+j}\right] _{0\le i,j\le n}$ are nonnegative.

By the assumption, $p_{k-1}p_{k}=\frac{f_{k-1}s_{k}s_{k+1}}{t_{k+1}}$ for $k\ge 1$. Hence, by Lemma 1, we have

$$\begin{aligned} H_{n}({\alpha })=A_{n}U_{n}A_{n}^{t}, \end{aligned}$$

where $U_{n}=\mathrm{{diag}}(u_{0},u_{1},\ldots ,u_{2n})$ and

$$\begin{aligned} u_{i}=\left\{ \begin{array}{l} 1,\ \ \ \text { if }i=0, \\ \frac{s_{1}}{p_{0}},\ \ \ \text { if }i=1, \\ \prod _{j=1}^{\frac{i}{2}}\frac{t_{2j}}{f_{0}},\ \ \ \text { if }i\text { even,} \\ \frac{p_{1}t_{2}}{f_{0}s_{2}}\prod _{j=1}^{\frac{i-1}{2}}\frac{t_{2j+1}}{f_{2j-1}},\ \ \text {else}. \end{array} \right. \end{aligned}$$

This gives the evaluation of Hankel determinants of the two-Motzkin-like numbers $a_{n}$:

$$\begin{aligned} h_{n}({\alpha })=\det \left( A_{n}U_{n}A_{n}^{t}\right) . \end{aligned}$$

Obviously, $det\left( A_{n}\right) =\det \left( A_{n}^{t}\right)$ and $\det \left( U_{n}\right) \ge 0$ since all $u_{i}$ are nonnegative. Then applying Cauchy–Binet formula, we immediately get

$$\begin{aligned} h_{n}({\alpha })=\det \left( A_{n}U_{n}A_{n}^{t}\right) \ge 0. \end{aligned}$$

The proof of the theorem is complete. $\square$

Thus, the statement follows.

Corollary 1

All sequences of the two-Motzkin-like numbers in Example 1 are Hamburger moment sequences.

When ${{\mathcal {F}} }={\tau }=(0,0,0,\ldots )$ and ${{\mathcal {P}}}=(1,1,1,\ldots )$ in (11), we obtain the following result due to Liang et al. [22, Theorem 2.2].

Theorem 2

The Catalan-like numbers corresponding to $({\gamma ,\sigma })$ form a Hamburger moment sequence if and only if all $t_k$ are nonnegative.

Remark 4

All sequences of the Catalan-like numbers in Remark 1 are Hamburger moment sequences.

Theorem 3

Let ${{\mathcal {F}}}=(f_{k})_{k\ge 0}$, ${{\mathcal {P}}}=(p_{k})_{k\ge 0}$, ${\gamma }=(r_{k})_{k\ge 0},$ ${\sigma }=(s_{k})_{k\ge 1}$ and ${\tau }=(t_{k})_{k\ge 2}$ be five sequences of nonnegative numbers. Assume that $p_{k-1}p_{k}=\frac{f_{k-1}s_{k}s_{k+1}}{t_{k+1}}$ for $k\ge 1$. Then the two-Motzkin-like numbers corresponding to $({{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau })$ form a Stieltjes moment sequence if and only if all determinants of the coefficient matrix J are nonnegative.

Proof

By the definition, the two-Motzkin-like numbers ${\alpha }=\left[ a_{n}\right] _{n\ge 0}$ corresponding to $({{\mathcal {F}}, {\mathcal {P}},\gamma ,\sigma ,\tau })$ form a Stieltjes moment sequence if and only if both ${\alpha }=\left[ a_{n}\right] _{n\ge 0}$ and its shift $\overline{{\alpha }}$ $=\left[ a_{n+1}\right] _{n\ge 0}$ are Hamburger moment sequences.

By Theorem 1, the two-Motzkin-like numbers ${\alpha }=\left[ a_{n}\right] _{n\ge 0}$ form a Hamburger moment sequence. It remains to check that $\overline{{\alpha }}$ $=\left[ a_{n+1}\right] _{n\ge 0}$ forms a Hamburger moment sequence.

By Lemma 1, the Hankel matrix of $\overline{{\alpha }}$ $=\left[ a_{n+1}\right] _{n\ge 0}$ has the form

$$\begin{aligned} H(\overline{{\alpha }})=AU{\overline{A}}^{t}, \end{aligned}$$

where ${\overline{A}}=AJ$. Then to show that all Hankel determinants of $\overline{{\alpha }}$ are nonnegative, it suffices to do that for all determinants of the coefficient matrix J.

$\square$

Proposition 2

For $n\ge 1$, the central trinomial coefficients of even index $\left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}$ form a Stieltjes moment sequence, and are, therefore, infinitely log-convex.

Proof

By Theorem 3, it suffices to prove that all determinants $d_{n}$ of J are nonnegative.

$$\begin{aligned} d_{n}=\left| \begin{array}{cccccc} 3\ \ &{} 2\ \ &{} 1\ \ &{} &{} &{} \\ 4\ \ &{} 4\ \ &{} 2\ \ &{} 1\ \ &{} &{} \\ 2\ \ &{} 2\ \ &{} 3\ \ &{} 2\ \ &{} \ddots &{} \\ &{} 1\ \ &{} 2\ \ &{} 3\ &{} \ddots &{} \ \ 1 \\ &{} &{} \ddots &{} \ddots &{} \ddots &{} \ \ 2 \\ &{} &{} &{} \ \ 1\ \ &{} \ \ 2\ \ &{} \ \ 3 \end{array} \right| . \end{aligned}$$

By an inductive argument, we obtain

$$\begin{aligned} d_{n}=\left\{ \begin{array}{l} 6k+2,\ \ \ \text { if }n=3k, \\ 6k+3,\ \ \ \text { if }n=3k+1, \\ 6k+4,\ \ \ \text { if }n=3k+2. \end{array} \right. \end{aligned}$$

It is clear that all $d_{n}$ are nonnegative, as required. Thus, the central trinomial coefficients of even index $\left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}$ form a Stieltjes moment sequence, and are, therefore, infinitely log-convex. $\square$

Proposition 3

The Motzkin numbers of even index $M_{2n}$ form a Stieltjes moment sequence, and are, therefore, infinitely log-convex.

Proof

By Theorem 3, it suffices to prove that all determinants $d_{n}$ of J are nonnegative.

$$\begin{aligned} d_{n}=\left| \begin{array}{cccccc} 2\ \ &{} 2\ \ &{} 1\ \ &{} &{} &{} \\ 2\ \ &{} 3\ \ &{} 2\ \ &{} 1\ \ &{} &{} \\ 1\ \ &{} 2\ \ &{} 3\ \ &{} 2\ \ &{} \ddots &{} \\ &{} 1\ \ &{} 2\ \ &{} 3\ &{} \ddots &{} \ \ 1 \\ &{} &{} \ddots &{} \ddots &{} \ddots &{} \ \ 2 \\ &{} &{} &{} \ \ 1\ \ &{} \ \ 2\ \ &{} \ \ 3 \end{array} \right| . \end{aligned}$$

By an inductive argument, we obtain

$$\begin{aligned} d_{n}=\left\{ \begin{array}{l} 2k+1,\ \ \ \text { if }n=3k, \\ 2(k+1),\ \ \ \text { else}. \end{array} \right. \end{aligned}$$

It is clear that all $d_{n}$ are nonnegative, as required. Thus the Motzkin numbers of even index $M_{2n}$ form a Stieltjes moment sequence, and are, therefore, infinitely log-convex. $\square$

Lemma 2

[22, Proposition 3.2] If ${\alpha }$ is HM, then the subsequence $(a_{2n})$ is SM.

By the previous lemma and Corollary 1, we obtain the following result.

Corollary 2

The central pentanomial coefficients of even index $\left( {\begin{array}{c}2n\\ 4n\end{array}}\right) _{4}$ are Stieltjes moment sequence, and are therefore infinitely log-convex.

Remark 5

We know from [22] and Remark 4 that all sequences of the Catalan-like numbers given in Remark 1 are Hamburger moment sequences. Hence, by Lemma 2, the terms of even index of these sequences form a Stieltjes moment sequences, and are, therefore, infinitely log-convex.

3 Stieltjes Moment Sequences of Polynomials

Let f(q) and g(q) be two real polynomials in q. We say that f(q) is q-nonnegative if f(q) has nonnegative coefficients. Denote f(q) $\ge _{q}$ g(q) if $f(q)-g(q)$ is q-nonnegative. Let $A(q)=[a_{n,k}(q)]_{n,k\ge 0}$ be a matrix whose entries are all real polynomials in q. We say that A(q) is q-TP if all minors are q-nonnegative. Let $\alpha (q)=(a_{n}(q))_{n\ge 0}$ be a sequence of real polynomials in q. We say that the sequence is strongly q-log-convex (q-SLCX for short) if

$$\begin{aligned} a_{n+1}(q)a_{m-1}(q)\ge _{q}a_{n}(q)a_{m}(q) \end{aligned}$$

for $n\ge m\ge 0.$ If the Hankel matrix $H({\alpha }(q))=[a_{i+j}(q)]_{i,j\ge 0}$ is q-TP, then we say that ${\alpha }(q)$ is a q-Stieltjes moment (q-SM for short) sequence of polynomials. If ${\alpha }(q)$ is a Stieltjes moment sequence for any fixed $q\ge 0$, then we say that ${\alpha }(q)$ is a pointwise Stieltjes moment (PSM for short) sequence of polynomials. Clearly, a q-SM sequence is both q-SLCX and PSM.

The simplest non-trivial q-SM sequence in combinatorics should be $((q+1)^{n})_{n\ge 0}$, see [24]. Recently, Ahmia and Belbachir [1] proved that the sequence of polynomials of square trinomials $W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}$ is q-SLCX, Zhu [30] showed that the Bell polynomials, the Eulerian polynomials, the Narayana polynomials (of type B), the q-central Delannoy numbers and the q-Schröder numbers are q-SLCX.

If the Hankel matrix $H(\alpha (q))=[a_{i+j}(q)]_{i,j\ge 0}$ is q-positive semi-definite, i.e., their leading principal minors are all q-nonnegative, then we say that ${\alpha }(q)$ is a q-Hamburger moment (q-HM for short) sequence of polynomials. For example, the Fibonacci polynomials $F_{n}(q)$ defined by $F_{0}(q)=1,F_{1}(q)=q$ and $F_{n}(q)=qF_{n-1}(q)+F_{n-2}(q),$ is q-HM.

Let ${{\mathcal {F}}}(q)=(f_{k}(q))_{k\ge 0}$, ${{\mathcal {P}}}(q)=(p_{k}(q))_{k\ge 0}$, ${\gamma }(q)=(r_{k}(q))_{k\ge 0},$ ${\sigma }(q)=(s_{k}(q))_{k\ge 1}$ and ${\tau }(q)=(t_{k}(q))_{k\ge 2}$ be five sequences of polynomials. Define an infinite generalized triangular matrix $A(q):=[a_{n,k}(q)]_{n,k\ge 0}$ by the recurrence

$$\begin{aligned} a_{n+1,k}(q)= & {} f_{k-2}(q)a_{n,k-2}(q)+p_{k-1}(q)a_{n,k-1}(q)+r_{k}(q)a_{n,k}(q)\nonumber \\&+s_{k+1}(q)a_{n,k+1}(q)+t_{k+2}(q)a_{n,k+2}(q), \end{aligned}$$

(15)

where $a_{0,0}(q)=1$ and $a_{n,k}(q)=0$ unless $2n\ge k\ge 0.$ We say that $a_{n}(q):=a_{n,0}(q)$ by the q-analogue of two-Motzkin-like numbers corresponding to (${{\mathcal {F}}}(q)$, ${{\mathcal {P}}}(q)$, ${\gamma }(q)$, ${\sigma }(q)$, ${\tau }(q)$).

Example 2

The sequence of polynomials of square trinomials

$$\begin{aligned} W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k} \end{aligned}$$

is the q-analogue of two-Motzkin-like numbers corresponding to ${{\mathcal {F}}}(q)=(1$, $1,\ldots )$, ${{\mathcal {P}}}(q)=(1+q,1+q,\ldots )$, ${\gamma }(q)=(1+q+q^{2},1+2q+q^{2},1+q+q^{2},\ldots ),$ ${\sigma }(q)=(2q+2q^{2},q+q^{2},\ldots )$ and ${\tau }(q)=(2q^{2},q^{2},q^{2},\ldots )$.

Similar to Lemma 1, we give the following.

Lemma 3

If $p_{k-1}(q)p_{k}(q)=\frac{f_{k-1}(q)s_{k}(q)s_{k+1}(q)}{t_{k+1}(q)}$ for $k\ge 1,$ then the Hankel Matrix $H_{n}(q)=[a_{i+j}(q)]_{0\le i,j\le n}$ of the two-Motzkin-like numbers $a_{n}(q)$ satisfies that

$$\begin{aligned} H(q)=A(q)U(q)A^{t}(q), \end{aligned}$$

where $U_{n}(q)=diag(u_{0}(q),u_{1}(q),\ldots , u_{2n}(q))$ and

$$\begin{aligned} u_{i}(q)=\left\{ \begin{array}{l} 1,\ \ \ \text { if }i=0,\\ \frac{s_{1}(q)}{p_{0}(q)},\ \ \ \text { if }i=1,\\ \prod _{j=1}^{\frac{i}{2}}\frac{t_{2j}(q)}{f_{0}(q)},\ \ \ \text { if }i\text { even},\\ \frac{p_{1}(q)t_{2}(q)}{f_{0}(q)s_{2}(q)}\prod _{j=1}^{\frac{i-1}{2}} \frac{t_{2j+1}(q)}{f_{2j-1}(q)},\ \ \text {else}. \end{array} \right. \end{aligned}$$

Remark 6

The sequence of polynomials of square trinomials

$$\begin{aligned}W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}\end{aligned}$$

satisfies the previous lemma.

By Lemma 3, we can obtain the analogue of Theorem 1.

Theorem 4

Let ${{\mathcal {F}}}(q)=(f_{k}(q))_{k\ge 0}$, ${\gamma }(q)=(r_{k}(q))_{k\ge 0},$ ${\sigma }(q)=(s_{k}(q))_{k\ge 1}$ and ${\tau }(q)=(t_{k}(q))_{k\ge 2}$ be five sequences of polynomials of nonnegative coefficients. Assume that $p_{k-1}(q)p_{k}(q)=\frac{f_{k-1}(q)s_{k}(q)s_{k+1}(q)}{t_{k+1}(q)}$ for $k\ge 1$. Then the q-analogue of two-Motzkin-like numbers corresponding to $({{\mathcal {F}}}(q)$, ${{\mathcal {P}}}(q)$, ${\gamma }(q)$, ${\sigma }(q)$, ${\tau }(q))$ forms a q-Hamburger moment sequence of polynomials.

Corollary 3

The sequence of polynomials of square trinomials

$$\begin{aligned} W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k} \end{aligned}$$

is q-Hamburger moment sequence of polynomials.

When ${{\mathcal {F}}}(q)={\tau }(q)=(0,0,0,\ldots )$ and ${{\mathcal {P}}}(q)=(1,1,1,\ldots )$ in (15), we obtain the analogue of Theorem 2 due to Liang et al. [22, Theorem 2.5].

Theorem 5

The q-analogue of the Catalan-like numbers corresponding to $({\gamma }(q),{\sigma }(q))$ forms a q-Hamburger moment sequence of polynomials if and only if all $t_k (q)$ are q-nonnegative.

Remark 7

From [22, Corollary 2.6] the Bell polynomials, the Eulerian polynomials, the Narayana polynomials (of type A and type B), the q-central Delannoy numbers, the Morgan-Voyce polynomials and the q-Schröder numbers are q-Hamburger moment sequence of polynomials.

The method of the proof used in Theorem 3 can be carried over verbatim to its q-analogue. Here we omit the details for brevity.

Theorem 6

Let ${{\mathcal {F}}}(q)=(f_{k}(q))_{k\ge 0}$, ${{\mathcal {P}}}(q)=(p_{k}(q))_{k\ge 0}$, ${\gamma }(q)=(r_{k}(q))_{k\ge 0},$ ${\sigma }(q)=(s_{k}(q))_{k\ge 1}$ and ${\tau }(q)=(t_{k}(q))_{k\ge 2}$ be five sequences of polynomials of nonnegative coefficients. Assume that $p_{k-1}(q)p_{k}(q)=\frac{f_{k-1}(q) s_{k}(q)s_{k+1}(q)}{t_{k+1}(q)}$ for $k\ge 1$. Then the q-analogue of two-Motzkin-like numbers corresponding to $({{\mathcal {F}}}(q)$, ${{\mathcal {P}}}(q)$, ${\gamma }(q)$, ${\sigma }(q)$, ${\tau }(q))$ forms a q-Stieltjes moment sequence of polynomials if and only if all determinants of the coefficient matrix J(q) are q-nonnegative.

Proposition 4

The sequence of polynomials of square trinomials $W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}$ forms a q-Stieltjes moment sequence of polynomials.

Proof

By Theorem 6, it suffices to prove that all determinants $d_{n}(q)$ of J(q) are q-nonnegative.

$$\begin{aligned} d_{n}(q)=\left| \begin{array}{cccccc} 1+q+q^{2} &{} 1+q &{} 1 &{} &{} &{} \\ 2q+2q^{2} &{} 1+2q+q^{2} &{} 1+q &{} 1 &{} &{} \\ 2q^{2} &{} q+q^{2} &{} 1+q+q^{2} &{} 1+q &{} \ddots &{} \\ &{} q^{2} &{} q+q^{2} &{} 1+q+q^{2} &{} \ddots &{} 1 \\ &{} &{} \ddots &{} \ddots &{} \ddots &{} 1+q \\ &{} &{} &{} q^{2} &{} q+q^{2} &{} 1+q+q^{2} \end{array} \right| . \end{aligned}$$

By an inductive argument, we obtain

$$\begin{aligned} d_{n}(q)=\left\{ \begin{array}{l} \sum _{i=0}^{6k}q^{i}+q^{3k},\ \ \ \text { if }n=3k, \\ \sum _{i=0}^{6k+2}q^{i},\ \ \ \text { if }n=3k+1, \\ \sum _{i=0}^{6k+4}q^{i}-q^{3k+2}, \ \ \ \text { if }n=3k+2. \end{array} \right. \end{aligned}$$

It is clear that all $d_{n}(q)$ are q-nonnegative, as required. Thus the sequence of polynomials of square trinomials $W_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}$ forms a q-Stieltjes moment sequence of polynomials. $\square$

4 Linear Transformations and Convolutions for Generalized Triangles

Let $A=[a_{n,k}]$ be an infinite nonnegative lower triangular matrix. Define the A-linear transformation

$$\begin{aligned} t_{n}=\sum _{k=0}^{n}a_{n,k}x_{k},\ \ \ n=0,1,2,\ldots \end{aligned}$$

(16)

and the A-convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{n}a_{n,k}x_{k}y_{n-k},\ \ \ n=0,1,2,\ldots \end{aligned}$$

(17)

We start with the following result due to Wang and Zhu [28, Theorem 4.2].

Proposition 5

Let $A_{n}(q)=\sum _{k=0}^{n}a_{n,k}q^{k}$ be the nth row generating function of the triangle A. Assume that $(A_{n}(q))_{n\ge 0}$ is a pointwise Stieltjes moment (PSM for short) sequence of polynomials. Then both the A-linear transformation (16) and the A-convolution (17) preserve the SM property.

Now, we consider an infinite nonnegative lower triangular matrix denoted by $A=\left[ a_{s}(n,k)\right] .$ Define the A-linear transformation

$$\begin{aligned} t_{n}=\sum _{k=0}^{sn}a_{s}(n,k)x_{k}, \quad n=0,1,2,\ldots \end{aligned}$$

(18)

and the A-convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{sn}a_{s}(n,k)x_{k}y_{sn-k},\quad n=0,1,2,\ldots \end{aligned}$$

(19)

Hence, we can extend the previous proposition as follows.

Theorem 7

Let $A_{n}(q)=\sum _{k=0}^{sn}a_{s}(n,k)q^{k}$ be the nth row generating function of the triangle A. Assume that $(A_{n}(q))_{n\ge 0}$ is a pointwise Stieltjes moment (PSM for short) sequence of polynomials. Then both the A-linear transformation (18) and the A-convolution (19) preserve the SM property.

Corollary 4

If both $(x_{n})_{n\ge 0}$ and $(y_{n})_{n\ge 0}$are Stieltjes moment sequences, then so is the convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}x_{k}y_{2n-k}, \quad n=0,1,2,\ldots \end{aligned}$$

Proof

By Proposition 4, $\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}$ is q-Stieltjes moment sequence of polynomials, hence by Theorem 7 the following convolution:

$$\begin{aligned} z_{n}=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}x_{k}y_{2n-k}, \quad n=0,1,2,\ldots \end{aligned}$$

preserves the Stieltjes moment sequences. $\square$

Corollary 5

If both $(x_{n})_{n\ge 0}$ and $(y_{n})_{n\ge 0}$ are Stieltjes moment sequences, then so are their bi$^{s}$nomial convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}x_{k}y_{sn-k}, \quad n=0,1,2,\ldots \end{aligned}$$

and bi$^{s}$nomial transformation

$$\begin{aligned} t_{n}=\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}x_{k}, \quad n=0,1,2,\ldots \end{aligned}$$

Proof

We know that the nth row generating function of the s-Pascal triangle is $\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}q^{k}=\left( 1+q+\cdots +q^{s}\right) ^{n}$.

Taking $s=2$, we have

$$\begin{aligned} \sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}q^{k}=\left( 1+q+q^{2}\right) ^{n}=(1+q\left( 1+q\right) )^{n}=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) q^{k}(1+q)^{k}. \nonumber \\ \end{aligned}$$

(20)

Form Wang and Zhu [28] $(1+q)^{k}$ is a Stieltjes moment sequence for $q\ge 0,$ Hence $\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}q^{k}$ so is it by Proposition 5.

Suppose that this hypothesis is true for $s=l$. We show that it remains true for $s=l+1$. For $s=l$, we have

$$\begin{aligned} \sum _{k=0}^{ln}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{l}q^{k}=\left( 1+q+\cdots +q^{l}\right) ^{n} \end{aligned}$$

(21)

is a Stieltjes moment sequence for $q\ge 0$. So, for $s=l+1$, it follows that

$$\begin{aligned} \sum _{k=0}^{(l+1)n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{l}q^{k}= & {} \left( 1+q+\cdots +q^{l+1}\right) ^{n}=\left( 1+q(1+q+\cdots +q^{l})\right) ^{n}\\= & {} \sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) q^{k}(1+q+\cdots +q^{l})^{k} \end{aligned}$$

by (20) and (21), $\sum _{k=0}^{(l+1)n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{l}q^{k}$ is a Stieltjes moment sequence for $q\ge 0$. Hence by Theorem 7, the bi$^{s}$nomial convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}x_{k}y_{sn-k}, \quad n=0,1,2,\ldots \end{aligned}$$

and the bi$^{s}$nomial transformation

$$\begin{aligned} t_{n}=\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}x_{k}, \quad n=0,1,2,\ldots \end{aligned}$$

preserve the Stieltjes moment sequences. $\square$

5 Conclusion and Perspectives

In this paper, we established that the two-Motzkin-like numbers are Stieltjes moment sequences. They are defined as the 0th column of the infinite generalized triangular array $A=[a_{n,k}]_{n,k\ge 0}$ of the form

$$\begin{aligned} a_{n+1,k}=f_{k-2}a_{n,k-2}+p_{k-1}a_{n,k-1}+r_{k}a_{n,k}+s_{k+1}a_{n,k+1}+t_{k+2}a_{n,k+2},\nonumber \\ \end{aligned}$$

(22)

where $a_{0,0}=1$ and $a_{n,k}=0$ unless $2n\ge k\ge 0$. As applications, we showed that the central trinomial, Motzkin and central pentanomial numbers of even indices are Stieltjes moment sequences and therefore infinitely log-convex sequences in a unified approach. Next, we established that the sequence of polynomials of square trinomials forms a q-Stieltjes moment sequence of polynomials. Finally, we proved that the square trinomial, the bi$^s$nomial convolutions and transformations preserve the Stieltjes moment sequences property.

It would be very appealing to investigate the Stieltjes moment properties of the sequences defined by an infinite generalized triangulars array of recurrence order greater than that of the triangular array defined by the recurrence relation (22), and the study of the preservation of the Stieltjes moment sequences under the square bi$^{s}$nomial convolution remains a question to be answered. This let us to conclude our paper by the following conjectures.

Conjecture 1

The sequence of polynomials of square bi$^{s}$nomial coefficient $\sum _{k=0}^{sn} \left( {\begin{array}{c}n\\ k\end{array}}\right) ^{2}_{s}q^{k}$ forms a q-Stieltjes moment sequence of polynomials for $s\ge 3$.

Conjecture 2

If both $(x_{n})_{n\ge 0}$ and $(y_{n})_{n\ge 0}$ are Stieltjes moment sequences, then so is their square bi$^{s}$nomial convolution

$$\begin{aligned} z_{n}=\sum _{k=0}^{sn}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{s}^{2}x_{k}y_{sn-k}, \ \ \ n=0,1,2,\ldots \end{aligned}$$

for $s\ge 3$.

Appendix A: Other Stieltjes Moment Sequences in OEIS

$$\begin{aligned} \begin{array}{|c|c|c|c|c|} \hline a_{n} &{} \sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) M_{k} &{} \sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) C_{k} &{} \sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}\left( {\begin{array}{c}2k\\ k\end{array}}\right) &{} \sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}B_{k} \\ \hline \text {OEIS} &{} A270561 &{} A007317 &{} A082760 &{} A207864\\ \hline \end{array} \end{aligned}$$