A New Criterion for the Total Positivity of Riordan Arrays

Lili Mu

School of Mathematics and Statistics

Jiangsu Normal University

Xuzhou, 221116

PR China

lilimu@jsnu.edu.cn

Yuanyuan Xing School of Mathematics Liaoning Normal University Dalian, 116029 PR China

                          xyy-yy@outlook.com

Sai-Nan Zheng^{1}

School of Data Science and Artificial Intelligence Dongbei University of Finance and Economics

Dalian, 116025 PR China

     zhengsainan@dufe.edu.cn

Abstract

This paper establishes a new criterion for the total positivity of Riordan arrays with parameterized A- and Z-sequence generating functions. We demonstrate the effectiveness of this criterion by proving the total positivity of several Riordan array classes.

1 Introduction

Riordan arrays play an important unifying role in enumerative combinatorics [19, 22, 20]. Many combinatorial properties of Riordan arrays have been studied in the literature; see, for instance, [4, 6, 7, 10, 11, 12, 13, 16]. A (proper) Riordan array, denoted by (d(x), h(x)), is an infinite lower triangular matrix whose generating function of the kth column is h^{k}(x)d(x) for k = 0, 1, 2, …, where d(0) = 1, h(0) = 0 and h′(0) ≠ 0.

1Corresponding author.

A Riordan array R = [r_{n,k}]_{n,k≥0} can also be characterized by two sequences (a_{n})_{n≥0} and (z_{n})_{n≥0} such that

r_{0,0} = 1, r_{n+1,0} = ∑_{j≥0} z_{j}r_{n,j}, r_{n+1,k+1} = ∑_{j≥0} a_{j}r_{n,k+j}  (1)

for n, k ≥ 0. We call (a_{n})_{n≥0} and (z_{n})_{n≥0} the A- and Z-sequences of R, respectively. Let A(x) and Z(x) be the generating functions of A- and Z-sequences, respectively. Then it is known that [15]

A(x) = \frac{x}{h̄(x)}, Z(x) = \frac{1}{h̄(x)} (1 − \frac{1}{d(h̄(x))})  (2)

where h̄(x) is the compositional inverse of h(x), i.e., h(h̄(x)) = h̄(h(x)) = x. We call the matrix

J(Z(x), A(x)) = [\begin{matrix}z_{0} & a_{0} \\ z_{1} & a_{1} & a_{0} \\ z_{2} & a_{2} & a_{1} & a_{0} \\ z_{3} & a_{3} & a_{2} & a_{1} & a_{0} \\ ⋮ & & & & ... \end{matrix}]

the coefficient matrix of the Riordan array R.

Let M = [m_{n,k}]_{n,k≥0} be a finite or infinite matrix of real numbers. We say that M is totally positive (TP for short) if its minors of all orders are nonnegative. Total positivity has a wide variety of applications throughout the realm of applied mathematics. We refer the reader to [3, 9, 14, 17] for details.

An infinite nonnegative sequence (a_{n})_{n≥0} is a Pólya frequency sequence (PF for short) if its Toeplitz matrix

[a_{i−j}]_{i,j≥0} = [\begin{matrix}a_{0} \\ a_{1} & a_{0} \\ a_{2} & a_{1} & a_{0} \\ ⋮ & & & ... \end{matrix}]

is TP. A fundamental characterization for PF sequences is due to Schoenberg and Edrei, which states that a sequence (a_{n})_{n≥0} is PF if and only if its generating function

∑_{n≥0} a_{n}t^{n} = Ct^{k}e^{γt} \frac{∏_{j≥0}(1 + α_{j}t)}{∏_{j≥0}(1 − β_{j}t)},

where C > 0, k ∈ ℕ, α_{j}, β_{j}, γ ≥ 0 and ∑_{j≥0}(α_{j} + β_{j}) < +∞ (see [14, p. 412] for instance). This generating function is called a Pólya frequency formal power series. If we identify a finite sequence a_{0}, a_{1}, …, a_{n} with the infinite sequence a_{0}, a_{1}, …, a_{n}, 0, …, then a finite sequence of nonnegative numbers is PF if and only if its generating function has only real zeros (see [23] for instance).

In recent years, there have been some papers concerning the total positivity of Riordan arrays. In [6], by means of A- and Z-sequences, Chen et al. gave the following criterion for the total positivity of Riordan arrays.

Proposition 1 ([6, Theorem 2.1]). Let R be a Riordan array defined by (1). Then R is TP if its coefficient matrix J(Z(x), A(x)) is TP.

Recently, Chen et al. [7] provided a new criterion for the total positivity of Riordan arrays in terms of formal power series d(x) and h(x): if both d(x) and h(x) are Pólya frequency formal power series, then R is totally positive. However, many Riordan arrays do not meet these conditions but are still totally positive. For example, the Riordan array (see [21, \underline{A113955}] for details)

[r_{n,k}]_{n,k≥0} = (\frac{1}{(1 − 4x)c(x)}, \frac{xc(x)}{\sqrt{1 − 4x}}) = [\begin{matrix}1 \\ 3 & 1 \\ 11 & 6 & 1 \\ 42 & 30 & 9 & 1 \\ ⋮ & & & & ... \end{matrix}] ,

where c(x) = \frac{1−\sqrt{1−4x}}{2x} is the generating function of Catalan numbers, and its entries

r_{n,k} = ∑_{j=0}^{n} \binom{2n}{j}\binom{n − j}{k}.

The sequence consisting of sums of rows of R is ∑_{k=0}^{n} \binom{2n}{n+k}2^{k} [21, \underline{A100192}].

In this paper, we present a new criterion for the total positivity of Riordan arrays in terms of its decompositions, and give some applications.

2 Main results and applications

In this section we investigate the total positivity of a certain class of Riordan arrays R(P, Q; a, b, r, s, t), whose A- and Z-sequences have the generating functions

A(x) = \frac{P^{2}(x) + sxP (x) + tx^{2}}{P (x) − rx}  (3)

and

Z(x) = \frac{(P^{2}(x) + sxP (x) + tx^{2})Q(x) + (aP (x) + bx)(P (x) − xQ(x))}{P^{2}(x) − rxP (x)},  (4)

respectively, where P (x) and Q(x) are formal power series, a, b, r, s and t are nonnegative numbers.

Theorem 2. If J(Q(x), P (x)) is TP, s^{2} ≥ 4t, and as+a\sqrt{s^{2} − 4t} ≥ 2b, then R(P, Q; a, b, r, s, t) is TP.

Proof. Recall that if the generating functions of A- and Z-sequences of a Riordan array R satisfy

A(x) = A_{2}(x)A_{1} (\frac{x}{A_{2}(x)})  (5)

and

Z(x) = A_{1} (\frac{x}{A_{2}(x)}) Z_{2}(x) + Z_{1} (\frac{x}{A_{2}(x)}) (1 − \frac{x}{A_{2}(x)}Z_{2}(x)) ,  (6)

respectively, where A_{i}(x) and Z_{i}(x), i = 1, 2, are formal power series, then R can be characterized by the product of two Riordan arrays, i.e., R = R_{1}R_{2}, where the two Riordan arrays R_{i} are defined by A_{i}(x) and Z_{i}(x) exactly (this fact follows easily from [13, Theorems 3.3 and 3.4]).

Note that (3) can be expressed as

A(x) = P (x)\frac{1 + s\frac{x}{P (x)} + t\frac{x^{2}}{P^{2}(x)}}{1 − r\frac{x}{P (x)}}

which is equivalent to (5) with

A_{1}(x) = \frac{1 + sx + tx^{2}}{1 − rx} and A_{2}(x) = P (x).

Similarly, (4) can be expressed as

Z(x) = \frac{1 + s\frac{x}{P (x)} + t\frac{x^{2}}{P^{2}(x)}}{1 − r\frac{x}{P (x)}} Q(x) + \frac{a + b\frac{x}{P (x)}}{1 − r\frac{x}{P (x)}} (1 − \frac{x}{P (x)}Q(x))

which is equivalent to (6) with

Z_{1}(x) = \frac{a + bx}{1 − rx} and Z_{2}(x) = Q(x).

Hence R(P, Q; a, b, r, s, t) = R_{1}R_{2}, where R_{i} are defined by A_{i}(x) and Z_{i}(x) exactly. To prove the total positivity of R(P, Q; a, b, r, s, t), it suffices to show that R_{i} are TP by a fact which says that the product of two TP matrices is TP.

From Proposition 1, it follows immediately that R_{2} is TP, since A_{2}(x) = P (x), Z_{2}(x) = Q(x) and J(Q(x), P (x)) is TP.

We now prove that R_{1} is TP. From

A_{1}(x) = \frac{1 + sx + tx^{2}}{1 − rx} and Z_{1}(x) = \frac{a + bx}{1 − rx},

it follows that

J(Z_{1}(x), A_{1}(x)) = [\begin{matrix}a & 1 \\ ra + b & r + s & 1 \\ r^{2}a + rb & r^{2} + rs + t & r + s & 1 \\ r^{3}a + r^{2}b & r^{3} + r^{2}s + rt & r^{2} + rs + t & r + s & 1 \\ ⋮ & & & & ... \end{matrix}]

= [\begin{matrix}1 \\ r & 1 \\ r^{2} & r & 1 \\ r^{3} & r^{2} & r & 1 \\ ⋮ & & & & ... \end{matrix}] [\begin{matrix}a & 1 \\ b & s & 1 \\ 0 & t & s & 1 \\ 0 & 0 & t & s & 1 \\ ⋮ & & & & ... \end{matrix}] .

It is easy to check that the first matrix in the last equation is TP since r ≥ 0, and so is the second matrix since s^{2} ≥ 4t and as + a\sqrt{s^{2} − 4t} ≥ 2b (see [6, Prop. 2.6] for details). Then J(Z_{1}(x), A_{1}(x)) is TP, and so is R_{1} by Proposition 1, as required.

Example 3. Consider the Riordan array

R_{1} = (\frac{1}{(1 − 4x)c(x)}, \frac{xc(x)}{\sqrt{1 − 4x}})

which was mentioned in the introduction. Recall that c(x) = \frac{1−\sqrt{1−4x}}{2x} is the generating function of Catalan numbers. By (2), we have

A(x) = \frac{4x^{2} + 4x + 1}{1 + x} and Z(x) = \frac{3 + 8x + 4x^{2}}{1 + 2x + x^{2}} .

Then it is easy to check that

R_{1} = R(1 + x, 1; 2, 2, 0, 2, 1).

Clearly, R_{1} meets the conditions of Theorem 2, since

J(1, 1 + x) = [\begin{matrix}1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ ⋮ & & & & ... \end{matrix}]

is TP. Then R_{1} is TP.

Example 4. Consider the Riordan array

R_{2} = (\frac{1 + \sqrt{5 − 4c(x)}}{1 − 3xc(x) + (1 − xc(x))\sqrt{5 − 4c(x)}}, \frac{1 − \sqrt{5 − 4c(x)}}{2}) ,

where c(x) = \frac{1−\sqrt{1−4x}}{2x} is the generating function of Catalan numbers. Its first few entries are as follows.

R_{2} = [\begin{matrix}1 \\ 2 & 1 \\ 7 & 5 & 1 \\ 29 & 24 & 8 & 1 \\ ⋮ & & & & ... \end{matrix}] .

The first column of R is the sequence of antichains in rooted plane trees on n nodes (see [21, \underline{A007852}] for details). By (2), we have

A(x) = \frac{x^{4} − 2x^{3} − x^{2} + 2x + 1}{1 − x} and Z(x) = \frac{x^{3} − 3x^{2} + x + 2}{1 − x}.

Then it is easy to check that

R_{2} = R (\frac{1}{1 − x}, \frac{1}{1 − x}; 1, 1, 0, 2, 1) .

Clearly, R_{2} meets the conditions of Theorem 2, since

J (\frac{1}{1 − x}, \frac{1}{1 − x}) = [\begin{matrix}1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ ⋮ & & & & ... \end{matrix}]

is TP. Hence R_{2} is TP.

In the sequel we consider the application of Theorem 2 to a class of special interesting Riordan arrays, the Bell-type Riordan arrays. Following [11], we say that a Riordan array R is a Bell-type Riordan array if z_{i} = a_{i+1} for all i ∈ ℕ, i.e., the generating functions of its A- and Z-sequences satisfy

A(x) − a_{0} = xZ(x).

Many well-known triangles are actually Bell-type Riordan arrays. Examples include the Pascal triangle, the Catalan triangle [18], the Motzkin triangle [1], the ballot table [2], and the large Schröder triangle [8]. Chen and Wang gave the following criterion for the total positivity of Bell-type Riordan arrays.

Proposition 5 ([6, Theorem 2.7]). Let R be a Bell-type Riordan array. If the A-sequence of R is PF, then R is TP.

In the following, we consider the total positivity of Bell-type Riordan arrays of the form (3), and use R(P; r, s, t) to denote such matrices for convenience.

Corollary 6. If P (x) is a Pólya frequency formal power series and s^{2} ≥ 4t, then R(P; r, s, t) is TP.

Proof. As in the proof of Theorem 2, it is straightforward to show that R(P; r, s, t) can be expressed as the product of two Bell-type Riordan arrays, i.e., R(P; r, s, t) = R_{1}R_{2}, where R_{1} and R_{2} are defined by A_{1}(x) = \frac{1+sx+tx^{2}}{1−rx} and A_{2}(x) = P (x), respectively. Note that A_{1}(x) is a Pólya frequency formal power series since s^{2} ≥ 4t. From Proposition 5, it follows immediately that both R_{1} and R_{2} are TP, since A_{2}(x) is also a Pólya frequency formal power series. Thus R(P; r, s, t) is TP by the Cauchy-Binet formula.

As applications, we next give some Bell-type Riordan arrays whose A-sequences are not PF sequences, i.e., for their total positivity, Proposition 5 does not work, but Corollary 6 does.

Example 7. Consider the Bell-type Riordan array

R_{3} = (\frac{1 − \sqrt{1 − 4x} − 4x}{2x(4x − 1)}, \frac{1 − \sqrt{1 − 4x} − 4x}{2(4x − 1)}) .

Its first few entries are as follows.

R_{3} = [\begin{matrix}1 \\ 3 & 1 \\ 10 & 6 & 1 \\ 35 & 29 & 9 & 1 \\ ⋮ & & & & ... \end{matrix}] .

The first column is the sequence \binom{2n+1}{n+1}. By (2), we have

A(x) = \frac{(1 + 2x)^{2}}{1 + x} = \frac{(1 + x)^{2} + 2x(1 + x) + x^{2}}{1 + x} .

Clearly,

R_{3} = R(1 + x; 0, 2, 1),

and R_{3} is TP.

Example 8. Consider the Bell-type Riordan array

R_{4} = (\frac{−1 + x + \sqrt{1 − 6x + 5x^{2}}}{2x(x − 1)} , \frac{−1 + x + \sqrt{1 − 6x + 5x^{2}}}{2(x − 1)} )

(see [21, \underline{A104259}] for details). Its first entries are as follows.

R_{4} = [\begin{matrix}1 \\ 2 & 1 \\ 5 & 4 & 1 \\ 15 & 14 & 6 & 1 \\ ⋮ & & & & ... \end{matrix}] .

The first column is the binomial transform of Catalan numbers [21, \underline{A007317}]. It is easy to check that

R_{4} = R (\frac{1}{1 − x}; 0, 1, 0) ,

since A(x) = \frac{1+x−x^{2}}{1−x}. Clearly, R_{4} is TP.

Example 9. Consider the Bell-type Riordan array

R_{5} = (\frac{1 − \sqrt[4]{1 − 8x}}{2x}, \frac{1 − \sqrt[4]{1 − 8x}}{2})

(see [21, \underline{A202039}] for details). Its first few entries are as follows.

R_{5} = [\begin{matrix}1 \\ 3 & 1 \\ 14 & 6 & 1 \\ 77 & 37 & 9 & 1 \\ ⋮ & & & & ... \end{matrix}] .

The first column is the sequence \underline{A101478} in [21]. It follows from

R_{5} = R (\frac{1}{1 − x}; 2, 0, 0)

that R_{5} is TP, since A(x) = \frac{1}{(1−x)(1−2x+2x^{2})}.

3 Concluding remarks

In this paper we provide a criterion for the total positivity of Riordan arrays in terms of the decomposition of Riordan arrays. In particular, we derive a criterion for the Belltype Riordan arrays in the view of the generating function of A-sequence. Riordan arrays arise often in combinatorics and many well-known combinatorial triangles are such matrices, including the Pascal triangle, the Catalan triangle, the Motzkin triangle, the large and little Schröder triangles, the ballot table and so on. Such triangles arise often in enumeration of lattice paths, for example, the Dyck paths, the Motzkin paths, and the Schröder paths and so on [8, 15, 19]. Brenti [5] gave combinatorial proofs for the total positivity of many well-known matrices by means of lattice path techniques. It is natural to consider the combinatorial proofs for the total positivity of Riordan arrays.

4 Acknowledgments

The authors thank the anonymous referees for their careful reading and valuable suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 12271222, 12101110), the Scientific Research Fund of Liaoning Provincial Education Department (Grant No. LJ212410173071).

References

[1] M. Aigner, Catalan-like numbers and determinants, J. Combin. Theory Ser. A 87 (1999), 33–51.

[2] M. Aigner, Catalan and other numbers: a recurrent theme, in H. Crapo, D. Senato, eds., Algebraic Combinatorics and Computer Science, Springer, 2001, pp. 347–390.

[3] T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165–219.

[4] P. Barry, Riordan arrays, orthogonal polynomials as moments, and Hankel transforms, J. Integer Sequences 14 (2011), Article 11.2.2.

[5] F. Brenti, Combinatorics and total positivity, J. Combin. Theory Ser. A 71 (1995), 175–218.

[6] X. Chen, H. Liang, and Y. Wang, Total positivity of Riordan arrays, European J. Combin. 46 (2015), 68–74.

[7] X. Chen and Y. Wang, Notes on the total positivity of Riordan arrays, Linear Algebra Appl. 569 (2019), 156–161.

[8] G.-S. Cheon, H. Kim, and L. W. Shapiro, Combinatorics of Riordan arrays with identical A and Z sequences, Discrete Math. 312 (2012), 2040–2049.

[9] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, 2011.

[10] T.-X. He, Matrix characterizations of Riordan arrays, Linear Algebra Appl. 465 (2015), 15–42.

[11] T.-X. He, Shift operators defined in the Riordan group and their applications, Linear Algebra Appl. 496 (2016), 331–350.

[12] T.-X. He and L. W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra Appl. 507 (2016), 77–95.

[13] T.-X. He and R. Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), 3962–3974.

[14] S. Karlin, Total Positivity, Vol. 1, Stanford University Press, 1968.

[15] D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math. 49 (1997), 301–320.

[16] L. Mu, J. Mao, and Y. Wang, Row polynomial matrices of Riordan arrays, Linear Algebra Appl. 522 (2017), 1–14.

[17] A. Pinkus, Totally Positive Matrices, Cambridge University Press, 2010.

[18] L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), 83–90.

[19] L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Appl. Math. 34 (1991), 229–239.

[20] L. W. Shapiro, R. Sprugnoli, P. Barry, G.-S. Cheon, T.-X. He, D. Merlini, and W. Wang, The Riordan Group and Applications, Springer Monographs in Mathematics, 2022.

[21] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2024. Available

    at https://oeis.org.

[22] R. Sprugnoli, Riordan arrays and combinatorial sums, Discrete Math. 132 (1994), 267– 290.

[23] Y. Wang and Y.-N. Yeh, Polynomials with real zeros and Pólya frequency sequences, J. Combin. Theory Ser. A 109 (2005), 63–74.

2020 Mathematics Subject Classification: Primary 05A20; Secondary 15A45, 15B36. Keywords: Riordan array, total positivity.

(Concerned with sequences \underline{A007317}, \underline{A007852}, \underline{A100192}, \underline{A101478}, \underline{A104259}, \underline{A113955}, and \underline{A202039}.)

Received July 17 2025; revised versions received July 19 2025; December 5 2025; December 6 2025. Published in Journal of Integer Sequences, December 6 2025.

Return to Journal of Integer Sequences home page.