1 Introduction↩︎

Following Karlin [1] and Pinkus [2], an infinite matrix is called totally positive (abbreviately, TP), if its minors of all orders are nonnegative. An infinite sequence of nonnegative real numbers \((a_n)_{n\geq 0}\) is called a Pólya frequency sequence (abbreviately, PF), if its Toeplitz matrix

\[\left[a_{i-j}\right]_{i,j\geq 0}=\left[ \begin{array} {lllll} a_0 & & & & \\ a_1& a_0 & & & \\ a_2 &a_1& a_0& & \\ a_3& a_2 & a_1& a_0 & \\ \vdots& \vdots &\vdots&\vdots & \ddots \end{array}\right]\]

is TP. We say that a finite sequence \((a_0, a_1,\ldots, a_n)\) is PF if the corresponding infinite sequence \((a_0, a_1, \ldots, a_n, 0, \ldots)\) is PF. Denote by \({{\mathbb{N}}}\) the set of all nonnegative integers. A fundamental characterization for PF sequences is given by Schoenberg et al.[3], [4] (see also Karlin [1]), which states that a sequence \((a_n)_{n\geq 0}\) is PF if and only if its generating function can be written as

\[\label{0} \sum_{n\geq 0} a_n t^n=C t^ke^{\gamma t}\frac{\Pi_{j\geq 0} (1+\alpha_j t)}{\Pi_{j\geq 0} (1-\beta_j t)},\tag{1}\] where \(C>0\), \(k\in {{\mathbb{N}}}\), \(\alpha_j\), \(\beta_j\), \(\gamma \geq 0\), and \(\sum_{j\geq 0}(\alpha_j+\beta_j)<\infty\). In this case, the above generating function is called a Pólya frequency formal power series. For more preliminary materials and more relevant results see, for example, Brenti [5], Mao, Mu, and Wang [6] and Pinkus [2].

Let \(A\) and \(B\) be \(m\times m\) and \(n\times n\) matrices, respectively. Then we define the direct sum of \(A\) and \(B\) by

\[\label{2465} A\oplus B =\left[ \begin{matrix} A &0 \\ 0 &B\end{matrix}\right]_{(m+n)\times(m+n)}.\tag{2}\]

Let us recall some basic definitions and introduce the notation.

First, the Riordan arrays (or Riordan matrices) are infinite, lower triangular matrices defined by the generating function of their columns. With the matrix multiplication, the set of all Riordan arrays form a group, called the Riordan group (see Shapiro, Getu, Woan and Woodson [7]).

More formally, let us consider the set of formal power series ring \({\mathcal{F}} = {{\mathbb{K}}}[\![\) \(t\) \(]\!]\), where \({{\mathbb{K}}}\) is the field \({{\mathbb{R}}}\) or \({{\mathbb{C}}}\). The order of \(f(t) \in {\mathcal{F}}\), \(f(t) =\sum_{k=0}^\infty f_kt^k\) (\(f_k\in {{\mathbb{K}}}\)), is the minimum number \(r\in{{\mathbb{N}}}\) such that \(f_r \neq 0\), where \({{\mathbb{N}}}=\{0,1,2\ldots\}\) is the set of all natural numbers. We denote by \({\mathcal{F}}_r\) the set of formal power series of order \(r\). Let \(g(t) \in {\mathcal{F}}_0\) and \(f(t) \in {\mathcal{F}}_1\); the pair \((g,\,f )\) defines the (proper) Riordan array \(R=(d_{n,k})_{n,k\in {{\mathbb{N}}}}=(g, f)\) having

\[\label{Radef} d_{n,k} = [t^n]g(t) f(t)^k\tag{3}\] or, in other words, having \(g f^k\) as the generating function of the \(k\)th column of \((g,f)\). Hence, \[\label{2466-0} (g,f):=(g,gf,gf^2,gf^3,\ldots),\tag{4}\] where \(g\), \(gf\), \(gf^2\), \(gf^3\cdots\) are the generating functions of the \(0\)th, \(1\)st, \(2\)nd, \(3\)rd, \(\cdots\), columns of the matrix \((g,f)\), respectively. The first fundamental theorem of Riordan arrays can be represented as

\[(g(t), f(t)) h(t)=g(t) (h\circ f)(t),\] which can also be simplified to \((g,f)h=gh(f)\). Thus we immediately see that the usual row-by-column product of two Riordan arrays is also a Riordan array: \[\label{Proddef} (g_1,\,f_1 ) (g_2,\,f_2 ) = (g_1 g_2(f_1),\,f_2(f_1)).\tag{5}\] The Riordan array \(I = (1,\,t)\) is the identity matrix because its entries are \(d_{n,k} = [t^n]t^k=\delta_{n,k}\).

Let \((g\left( t\right),\,f(t))\) be a Riordan array. Then its inverse is

\[(g\left( t\right),\,f(t))^{-1}=\left( \frac{1}{g(\overline{{f}}(t))}, \overline{f}(t),\right) \label{Invdef}\tag{6}\] where \(\overline{f}(t)\) is the compositional inverse of \(f(t)\), i.e., \((f\circ \overline{f})(t)=(\overline{f}\circ f)(t)=t\). In this way, the set \(\mathcal{ R}\) of all proper Riordan arrays forms a group (see [7]) called the Riordan group.

Here is a list of six important subgroups of the Riordan group (see [7]–[9]).

• the Appell subgroup \(\{ (g(t),\,t):g\in {\mathcal{F}}_0\}\).

• the Lagrange (associated) subgroup \(\{(1,\,f(t): f\in{\mathcal{F}}_1)\}\).

• the \(k\)-Bell subgroup \(\{(g(t),\, t(g(t))^k):g\in {\mathcal{F}}_0\}\), where \(k\) is a fixed positive integer.

• the hitting-time subgroup \(\{(tf'(t)/f(t),\, f(t)): f\in {\mathcal{F}}_1\}\).

• the derivative subgroup \(\{ (f'(t), \, f(t)):f\in {\mathcal{F}}_1\}\).

• the checkerboard subgroup \(\{ (g(t),\, f(t)):g\in {\mathcal{F}}_0, f\in {\mathcal{F}}_1\},\) where \(g\) is an even function and \(f\) is an odd function.

The \(1\)-Bell subgroup is referred to as the Bell subgroup for short, and the Appell subgroup can be considered as the \(0\)-Bell subgroup if we allow \(k=0\) to be included in the definition of the \(k\)-Bell subgroup.

An infinite lower triangular matrix \([d_{n,k}]_{n,k\in{{{\mathbb{N}}}}}\) is a Riordan array if and only if a unique sequence \(A=(a_0\not= 0, a_1, a_2,\ldots)\) exists such that for every \(n,k\in{{{\mathbb{N}}}}\) \[\label{eq:1461} d_{n+1,k+1} =a_0 d_{n,k}+a_1d_{n,k+1}+\cdots +a_nd_{n,n}.\tag{7}\] This is equivalent to \[\label{eq:1462} f(t)=tA(f(t))\quad \text{or}\quad t=\bar f(t) A(t).\tag{8}\] Here, \(A(t)\) is the generating function of the \(A\)-sequence. The first formula of 8 is also called the second fundamental theorem of Riordan arrays.

Moreover, there exists a unique sequence \(Z=(z_0, z_1,z_2,\ldots)\) such that every element in column \(0\) can be expressed as the linear combination \[\label{eq:1463} d_{n+1,0}=z_0 d_{n,0}+z_1d_{n,1}+\cdots +z_n d_{n,n},\tag{9}\] or equivalently, \[\label{eq:1464} g(t)=\frac{1}{1-tZ(f(t))},\tag{10}\] in which and thoroughly we always assume \(g(0)=g_0=1\), a usual hypothesis for proper Riordan arrays. From 10 , we may obtain

\[Z(t)=\frac{g(\bar f(t))-1}{\bar f(t)g(\bar f(t))}.\]

\(A\)- and \(Z\)-sequence characterizations of Riordan arrays were introduced, developed, and/or studied in Merlini, Rogers, Sprugnoli, and Verri [10], Roger [11], Sprugnoli and the author [12], Jean-Louis and Nkwanta [13], Luzón, Morón, Prieto-Martinez [14], [15], and [16] as well as their references.

Secondly, we recall the definition of quasi-Riordan arrays given in [17].

The relationship between the Riordan arrays and quasi-Riordan arrays can be presented in the following theorem.

For an integer \(r\geq 0\), denote the \(r\)-th truncation of a formal power series \(h=\sum_{n\geq 0} h_n t^n\) by \(h|_r:=\sum^r_{n=0} h_nt^n\). For an integer \(n\geq 0\), denote by \((g, f)_n\) the \(n\)-th order leading principle submatrix of the Riordan array \((g(t),f(t))_n\). This notation is introduced by Luzón et al in [18] and to the one used later by Luzón and her collaborators. Similarly, \([g,f]_n\) denotes the \(n\)-th order leading principle submatrix of the quasi-Riordan array \([g,f]\) defined by ?? , namely \[\label{24612} [g,f]_n=\left( \begin{matrix} g(0) & 0\\ ((g-g(0))/t)|_{n-1} & (f,t)_{n-1}\end{matrix}\right),\tag{11}\] where the \(n-1\)-st truncation of \((g-g(0)/t)|_{n-1}\) is \[(g-g(0)/t)|_{n-1}=\sum^{n-1}_{k=1}g_kt^{k-1},\] and \((f,t)_{n-1}=(f, tf, t^2f, \ldots, t^{n-1} f).\) We call \([g,f]_n\) the \(n\)-th truncation of the quasi-Riordan array \([g,f]\).

Let \(R=(g(t), f(t))\) be a Riordan array, where \(g(t)=\sum_{n\geq 0} g_nt^n\) and \(f(t)=\sum_{n\geq 1}f_nt^n\). If the lower triangular matrix

\[\label{eq:1} Q=\left [ \begin{array}{llllll} g_0& 0& 0& 0& 0& \cdots\\ g_1& f_1 & 0& 0& 0&\cdots\\ g_2 &f_2& f_1& 0& 0& \cdots\\ g_3& f_3& f_2& f_1&0&\cdots\\ \vdots &\vdots& \vdots& \vdots&\vdots&\ddots\end{array}\right]=\left( g(t), f(t), tf(t), t^{2}f(t),\ldots\right)\tag{12}\] is totally positive (TP), then so is \(R\) [6]. This can be proved easily by using Corollary 1 and mathematical induction.

The question is when \(Q\) is totally positive. It is obvious that the Pólya frequency property for the formal power series \(g\), \(f\) is not sufficient to ensure TP of \(Q\).

In this paper, we search for the conditions of TP quasi-Riordan arrays and the sequence characterization of TP quasi-Riordan arrays. From the last example, we know that the formal power series \(g(t)\) plays an important role in identifying whether a quasi-Riordan array is TP. Hence, in the next section, we discuss the conditions and their equivalents for TP quasi-Riordan arrays \([g,f]\) with some special power series \(g\). In Section 3, we study the sequence characterization of TP quasi-Riordan arrays. In particular, we present some generalizations of results given in [19]. More specifically, Theorems 5 and 6 in this section are generalizations of Theorem 2.1 and Theorem 2.3 of [19] for quasi-Riordan arrays. In Section 4 we will present some more examples with various \(g\)’s for TP and non-TP quasi-Riordan arrays \([g,f]\), respectively. From a list of examples showing in this section and the results in the previous sections, one may see a kind of independence of a TP quasi-Riordan array \([g,f]\) and the properties of that its formal power series \(g\) and \(f\) are the Pólya frequency formal power series.

2 TP Quasi-Riordan Arrays↩︎

Let us consider two special types of Riordan arrays and discuss their TP property and the TP property for the corresponding quasi-Riordan array.

If \(g(t)=1\), then from ?? we have

\[\label{2460} [g,f]=\left( \begin{matrix} 1 & 0\\ 0& (f/t,t)\end{matrix}\right).\tag{13}\] Hence,

Consider the case when \(g(t)=1+g_1t\) and \(f(t)=f_1t+f_2t^2\). Then

\[[g,f]=\left[\begin{array}{llll} 1& 0 & & \\ g_1 & f_1 &0 & \\ & f_2& f_1 & \\ & & f_2 &\\ & \ddots &\ddots &\ddots\end{array} \right].\] Hence, from the proof of Proposition 2.6 of [19] we have

Proposition 2 can be somewhat generalized to the following.

Clearly, the above necessary condition is absolutely not sufficient for total positivity of \([g,f]\). Primarily, if the terms of \(g\) are not nonnegative, then \([g,f]\) is definitely not TP, even if \(f\) is a Pólya frequency sequence. Moreover, it should also be emphasized again that, although the fact that both \(g\) and \(f\) are Pólya frequency sequences implies that the Riordan array \((g,f)\) is TP [20], the analogous result for quasi-Riordan arrays does not hold as we have seen in Example 1.

3 Sequence Characterization of the TP of Quasi-Riordan Arrays↩︎

Let us now bring the attention to the fact that the quasi-Riordan array \([g,f]\) can be treated as almost-Riordan array \(\left(g|\frac{f(t)}{t},t\right)\) [21]. From [22], we obtain the generating functions of the \(A\)-, \(Z\)-, and \(W\)-sequences of the latter one as follows:

\[\begin{align} &A(t)=1, \tag{14}\\ &Z(t)=\frac{f(t)-z_0tg(t)}{f(t)}+z_0, \tag{15}\\ &W(t)=\frac{(1-w_0t)g(t)-1}{f(t)}+w_0,\tag{16} \end{align}\] where \(z_0=f_1\) and \(w_0=g_1\). For instance, for quasi-Riordan array \([1/(1-t), t/(1-t)]\), \(A(t)=Z(t)=W(t)=1\).

We call the following matrix the production matrix of the quasi-Riordan matrix \([g,f]\):

\[\label{eq:2} J=\left[ \begin{array}{llllll} w_0& z_0& & & & \\ w_1& z_1& a_0& & & \\ w_2& z_2& a_1& a_0 & & \\ w_3& z_3& a_2& a_1& a_0 &\\ \vdots &\vdots &\vdots &\vdots& \vdots &\ddots \end{array}\right]\tag{17}\]

Clearly, by (14 )16 the matrix \(J\) shown above is of a special form. Namely

\[\label{eq:J_for_[g,f]} J=\left[ \begin{array}{lllllll} w_0&z_0&&&&&\\ w_1&z_1&1&&&&\\ w_2&z_2&0&1&&&\\ w_3&z_3&0&0&1&&\\ w_4&z_4&0&0&0&1&\\ \vdots&\vdots&&&&&\ddots\\ \end{array} \right].\tag{18}\]

Then, straight from the definition of total positivity one derives:

Joining the results of Theorems 5, 6 and \((\ref{eq:A95Z95W-1})\)-\((\ref{eq:A95Z95W-3})\) one can derive the following.

Note that since the discriminant of the denominator \(D(t)\) of the obtained \(g\) and \(f\) is equal to \[\Delta=w_0^2+z_1^2-2w_0z_1+4w_1z_0=(w_0-z_1)^2+4w_1z_0,\] all the coefficients are nonnegative, so \(D(t)\) has two real roots. Moreover, according to the assumption of \(w_0z_1-w_1z_0\geqslant 0\), their sum \(\frac{w_0+z_1}{w_0z_1-w_1z_0}\) and their product \(\frac{1}{w_0z_1-w_1z_0}\) are nonnegative. Then, they must be nonnegative as well. Thus, \(f\) is of form 1 , so by [1], [3], [4] this confirms that \(f\) must be Pólya frequency formal power series. If \(z_1=0\), then by \(w_0z_1-w_1z_0\geq 0\) we have \(w_1z_0=0\), which implies \(w_1=0\) due to \(z_0\not= 0\). Hence, if \(z_1=0\), then \(z_0> 0\), \(w_1=0\), and \(w_0> 0\), and all those make \([g,f]=\left[ \frac{1}{1-w_0t}, \frac{z_0t}{1-w_0t}\right]\), where both \(f\) and \(g\) are Pólya frequency. If \(z_1>0\), then \(g\) is not a Pólya frequency formal power series, although both \([g,f]\) and, hence, \((g,f)\) are TP. This gives another counterexample for Item 2 in [23]. As what we have seen, if \(z_1>0\), then \(g\) is not a Pólya frequency formal power series– its numerator has a positive root. However, as Proposition 5 holds, \([g(t),f(t)]\) is TP, so the coefficients of the series \(g(t)\) are nonnegative. Moreover, we have already seen in Example 1 that, in general, fact that both series \(g(t)\) and \(f(t)\) are Pólya frequency is not sufficient for \([g,f]\) to be TP, so one could mistrust the necessity of this condition as well.

4 Counterexamples↩︎

Note that in Example 1 we have already presented a non-TP quasi-Riordan array \([g,f]\), where \(g\) and \(f\) are Pólya frequency power series. Here, in the first subsection, we give some more examples like that. In the second subsection, we give some more examples for TP of quasi-Riordan arrays \([g,f]\) with non-Pólya frequency power series \(g\).

4.1 Quasi-Riordan arrays that are not TP↩︎

In general, Proposition 7 implies the following.

Series \(g\), that in from Proposition 7 is equal to \(\frac{1}{1-\alpha t}\), can be generalized to various other forms. A formula for this general form seems difficult to be found, yet let us present one more example.

4.2 Quasi-Riordan arrays that are TP↩︎

Example 6 can be extended to the following.

Note that replacing \(g(t)=1+t+t^2\) from Example 6 by, for instance, \(g(t)=t^2+1\) (that does not satisfy the assumption \(g_1>0\) of Corollary 6) yields the array of the form \[\left[\begin{array}{lllll} 1&&&&\\ 0&f_1&&&\\ 1&f_2&f_1&&\\ 0&f_3&f_2&f_1&\\ \vdots&&&&\ddots\\ \end{array}\right]\] whose minor \(M^{1,2}_{0,1}\) is equal to \(-f_1<0\), so the above array is not TP.

5 Closing comments↩︎

In the present work we have studied total positivity of quasi-Riordan arrays. As we have mentioned, these are, in fact, some particular cases of almost-Riordan arrays. Thus, it is natural question whether the presented results can be generalized for almost-Riordan arrays. Considering the capacity of the paper, the TP of almost-Riordan arrays will be discussed in a subsequent paper. For the same reason, some other related topics, such as analogues of \(TPr\) Riordan arrays in [19], \(TP_r\) quasi-Riordan arrays and \(TPr\) almost-Riordan arrays, will also be discussed in a subsequent paper.

6 Data availability statement↩︎

All data generated or analyzed during this study are included in this article.

References↩︎