Log-Concavity and LC-Positivity for Generalized Triangles

Moussa Ahmia LMAM Laboratory Department of Mathematics UMSB Jijel 18000 Algeria

                 ahmiamoussa@gmail.com

Hacène Belbachir USTHB Faculty of Mathematics RECITS Laboratory BP 32 El Alia 16111 Bab Ezzouar Algiers Algeria

               hacenebelbachir@gmail.com

Abstract

In this paper, we propose the generalized triangles called s-triangles for s given positive integer, as a bi-indexed sequence of nonnegative numbers {a_{s}(n, k)}_{0≤k≤ns} satisfying a_{s}(n, k) = 0 for k < 0. We extend some results of Wang and Yeh, and show that if the s-triangle is LC-positive (resp., doubly LC-positive) then it preserves (resp., it doubly preserves) the log-concavity of the sequences. Applications related to bi^{s}nomial coefficients are given.

1 Introduction

A sequence of nonnegative numbers (x_{k})_{k} is log-concave (LC for short) if x_{i−1}x_{i+1} ≤ x_{i}^{2} for all i > 0, which is equivalent to x_{i−1}x_{j+1} ≤ x_{i}x_{j} for all j ≥ i ≥ 1; see [11]. Log-concave sequences arise often in combinatorics, algebra, geometry, analysis, probability and statistics and have been extensively investigated; see Stanley [26] and Brenti [11] for details.

For two polynomials with real coefficients A(q) and B(q), we write A(q) ≥_{q} B(q) if the difference A(q) − B(q) has only nonnegative coefficients. A polynomial sequence (A_{n}(q))_{n≥0} is called q-log-concave (as introduced by Sagan [23]) if

A_{n−1}(q)A_{n+1}(q) ≤_{q} A_{n}(q)^{2}

for n ≥ 1.

It is easy to see that if the sequence (A_{n}(q))_{n≥0} is q-log-concave, then for each fixed nonnegative number q, the sequence (f_{n}(q))_{n≥0} is log-concave. The q-log-concavity of polynomials have been extensively studied; see Butler [14], Krattenthaler [18], Leroux [19] and Sagan [23, 24], for instance.

Let {a_{s}(n, k)}_{0≤k≤ns} be a s-triangle of nonnegative numbers with s ≥ 1. We illustrate a 4-triangle as follows:

n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0 ⋆
1 ⋆ ⋆ ⋆ ⋆ ⋆
2 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
4 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆

Table 1. The 4-triangle.

A nice example of such s-triangles is the triangle given by the ordinary multinomials or bi^{s}nomial coefficients [9]: let s ≥ 1 and n ≥ 0 be two integers, and k = 0, 1, …, sn, the bi^{s}nomial number \binom{n}{k}_{s} is defined as the k-th coefficient in the expansion

(1 + x + x^{2} + ⋯ + x^{s})^{n} = ∑_{k≥0} \binom{n}{k}_{s}x^{k}.  (1)

Below we list some related identities for the bi^{s}nomial coefficients. For more details see [9] and references therein.

• Expression of bi^{s}nomial coefficients in terms of binomial coefficients,

\binom{n}{k}_{s} = ∑_{j_{1}+j_{2}+···+j_{s}=k} \binom{n}{j_{1}}\binom{j_{1}}{j_{2}} ⋯ \binom{j_{s−1}}{j_{s}}.  (2)

• The symmetry relation

\binom{n}{k}_{s} = \binom{n}{sn − k}_{s}.  (3)

• The longitudinal recurrence relation

\binom{n}{k}_{s} = ∑_{j=0}^{s} \binom{n − 1}{k − j}_{s}.  (4)

These coefficients, as for usual binomial coefficients, are defined as in the Pascal triangle known as the “s-Pascal triangle”. One can find the first values of the s-Pascal triangle in the On-Line Encyclopedia of Integer Sequences (OEIS) [25] as \underline{A027907} for s = 2, as \underline{A008287} for s = 3, and as \underline{A035343} for s = 4.

n\k 0 1 2 3 4 5 6  7  8  9 10 11 12
0 1
1 1 1 1
2 1 2 3 2 1
3 1 3 6 7 6 3 1
4 1 4 10 16 19 16 10 4  1
5 1 5 15 30 45 51 45 30 15 5 1
6 1 6 21 50 90 126 141 126 90 50 21 6 1
7 1 7 28 77 161 266 357 393 357 266 161 77 …
8 1 8 36 112 266 504 784 1016 1107 1016 784 504 …

Table 2. Triangle of trinomial coefficients: s = 2.

Brondarenko [12] gives a combinatorial interpretation of the bi^{s}nomial coefficient \binom{n}{k}_{s} as the number of different ways of distributing “k” balls among “n” cells where each cell contains at most “s” balls. Using this combinatorial argument, one can easily establish the following relation

\binom{n}{k}_{s} = ∑_{n_{1}+2n_{2}+···+sn_{s}=k} \binom{n}{n_{0}, n_{1}, …, n_{s−1}}.

These coefficients are also naturally linked to generalized Fibonacci sequence: the “multibonacci” sequence, given for s ≥ 1, by

\begin{cases}Φ_{0} = Φ_{1} = ⋯ = Φ_{s−1} = 0, Φ_{s} = 1, \\ Φ_{n} = Φ_{n−1} + Φ_{n−2} + ⋯ + Φ_{n−s−1} (n ≥ 1) . \end{cases}

We have the following identity [9]

Φ_{n+1} = ∑_{k} \binom{n − k}{k}_{s}.

The case s = 1 provides a nice identity for Fibonacci numbers (sequence \underline{A000045}):

F_{n+1} = ∑_{k} \binom{n − k}{k}.

One of the extensions of binomial coefficients are q-binomial coefficients. Several works and applications were done in this area. For Fibonacci sequences, see Carlitz [15] and Cigler [16]. For Lucas sequences, see Belbachir and Benmezai [7]. For a variant of q-bi^{s}nomials, see Belbachir and Benmezai [6] or our paper [5], and for a recent application to the determinant, see Arikan and Kiliç [4].

Let us consider the following two linear transformations of sequences:

t_{n} = ∑_{k=0}^{ns} a_{s}(n, k)x_{k}, (n ≥ 0),  (5)

z_{n} = ∑_{k=0}^{ns} a_{s}(n, k)x_{k}y_{sn−k}, (n ≥ 0).  (6)

We say that the linear transformation (5) (resp., (6)) has the PLC (resp., double PLC) property if it preserves log-concavity of sequences, i.e., the log-concavity of (x_{n}) (resp., (x_{n}) and (y_{n})) implies that of (t_{n}) (resp., (z_{n})). The corresponding s-triangle {a_{s}(n, k)} is also called PLC (resp., double PLC).

This is a good way to obtain log-concavity by linear transformations or some operators. For instance, Menon [21] demonstrated that log-concavity is preserved under the ordinary convolution. Walkup in [27], and later, Wang and Yeh [28] also proved that log-concavity is preserved under the binomial convolution. It is also established that the q-binomial convolution preserves log-concavity; see [30]. In [1, 2, 3], we established the preserving log-convexity and log-concavity properties, respectively, for the bi^{s}nomial coefficients and the p, q-binomial coefficients.

In this paper, we generalize the aforementioned results for the generalized triangles like the s-Pascal triangle. In § 2, we give the necessary conditions to establish the PLC (resp., double PLC) property of the generalized triangles {a_{s}(n, k)}. In § 3, some examples of the both properties are given include the s-Pascal triangle.

2 LC-positivity and preservation of log-concavity

In this section, we give a relation between LC-positivity (resp., double LC-positivity) and the PLC property (resp., the double PLC property) for generalized triangles. We start with the concept of LC-positivity introduced by Wang and Yeh [28].

Definition 1. Let s ≥ 1 and n ≥ 0 be two integers. For 0 ≤ r ≤ sn, define the polynomial

A_{s,r}(n; q) := ∑_{k=r}^{ns} a_{s}(n, k)q^{k}.

We say that the s-triangle {a_{s}(n, k)} has the LC-positive property if for each r ≥ 0, the sequence of polynomials (A_{s,r}(n; q))_{n≥r} is q-log-concave in n.

Definition 2. Let s ≥ 1 and n ≥ 0 be two integers. For 0 ≤ k ≤ sn, define the reciprocal triangle {a_{s}^{∗}(n, k)} of {a_{s}(n, k)} by

a_{s}^{∗}(n, k) = a_{s}(n, sn − k)

and for 0 ≤ r ≤ sn, the polynomial

A_{s,r}^{∗}(n; q) := ∑_{k=r}^{ns} a_{s}^{∗}(n, k)q^{k}.

We say that the s-triangle {a_{s}(n, k)} has the double LC-positive property if for each r ≥ 0, the sequence of polynomials (A_{s,r}(n; q))_{n≥r} and (A_{s,r}^{∗}(n; q))_{n≥r} are q-log-concave in n.

We shall need the following lemma due to Wang and Yeh [28].

Lemma 3. Let h ∈ ℕ. Suppose that two sequences a_{0}, …, a_{h} and X_{0}, …, X_{h} of real numbers satisfy the following two conditions:

1 ∑_{k=r}^{h} a_{k} ≥ 0 (0 ≤ r ≤ h);

2 0 ≤ X_{0} ≤ ⋯ ≤ X_{h}.

Then

∑_{k=0}^{h} a_{k}X_{k} ≥ X_{0} ∑_{k=0}^{h} a_{k} ≥ 0.

Let {a_{s}(n, k)}_{0≤k≤ns} be a s-triangle of nonnegative numbers and (x_{k})_{k≥0} be a log-concave sequence. Let (z_{n})_{n≥0} be the sequence defined by (5) and let us consider the difference

△_{n} := (∑_{k=0}^{ns} a_{s}(n, k)x_{k})^{2} − (∑_{k=0}^{ns−s} a_{s}(n − 1, k)x_{k}) (∑_{k=0}^{ns+s} a_{s}(n + 1, k)x_{k}) .  (7)

Then △_{n} is a quadratic form in ns + s + 1 variables x_{0}, x_{1}, …, x_{ns+s}.

Let S_{t} be the sum of terms x_{k}x_{t−k} in △_{n}. For 0 ≤ k ≤ ⌊t/2⌋ with 0 ≤ t ≤ 2ns, let a_{s,k}(n, t) be the coefficient of the term x_{k}x_{t−k} in △_{n}. Then

△_{n} = ∑_{t=0}^{2ns} S_{t} with S_{t} = ∑_{k=0}^{⌊t/2⌋} a_{s,k}(n, t)x_{k}x_{t−k}.  (8)

Thus, it suffices to show that S_{t} ≥ 0 (0 ≤ t ≤ 2ns). We have the following inequalities x_{0}x_{t} ≤ x_{1}x_{t−1} ≤ x_{2}x_{t−2} ≤ · · ·. Hence by Lemma 3, it suffices to establish that

A_{s,r}(n, t) := ∑_{k=r}^{⌊t/2⌋} a_{s,k}(n, t) ≥ 0, (0 ≤ r ≤ ⌊t/2⌋) .  (9)

Using relation (7), for k < t/2, we obtain

a_{s,k}(n, t) = 2a_{s}(n, k)a_{s}(n, t − k) − a_{s}(n − 1, k)a_{s}(n + 1, t − k)
− a_{s}(n + 1, k)a_{s}(n − 1, t − k),  (10)

and for t even and k = t/2, we have

a_{s,k}(n, t) = a_{s}(n, k)^{2} − a_{s}(n − 1, k)a_{s}(n + 1, k).  (11)

Let us remark that A_{s,r}(n, t) is precisely the coefficient of q^{t} in the polynomial A_{s,r}^{2}(n; q)− A_{s,r}(n − 1; q)A_{s,r}(n + 1; q), i.e.,

A_{s,r}^{2}(n; q) − A_{s,r}(n − 1; q)A_{s,r}(n + 1; q) = ∑_{t=2r}^{2ns} A_{s,r}(n, t)q^{t}.  (12)

Hence, the following characterization of positivity holds:

Lemma 4. The s-triangle {a_{s}(n, k)}_{0≤k≤ns} is LC-positive if and only if A_{s,r}(n, t) ≥ 0 for all 2r ≤ t ≤ 2ns.

Now, from the discussion above, we obtain the following:

Theorem 5. The LC-positive s-triangles are PLC.

The relation between double LC-positivity and the double PLC property is given by the following proposition.

Proposition 6. Given a s-triangle {a_{s}(n, k)}_{0≤k≤ns} of nonnegative numbers and two logconcave sequences (x_{k})_{k≥0} and (y_{k})_{k≥0}.

Define three s-triangles {b_{s}(n, k)}, {c_{s}(n, k)} and {d_{s}(n, k)} by

b_{s}(n, k) = a_{s}(n, k)x_{k}, c_{s}(n, k) = a_{s}(n, k)y_{ns−k}, d_{s}(n, k) = a_{s}(n, k)x_{k}y_{ns−k}.

For 2r ≤ t ≤ 2ns, define B_{s,r}(n, t), C_{s,r}(n, t) and D_{s,r}(n, t) similar to A_{s,r}(n, t) in (12).

1. If the s-triangle {a_{s}(n, k)} is LC-positive, then the s-triangle {b_{s}(n, k)} is LC-positive and B_{s,r}(n, t) ≥ A_{s,r}(n, t)x_{r}x_{t−r}.

2. If the s-triangle {a_{s}(n, k)} is double LC-positive, then the s-triangle {c_{s}(n, k)} is LCpositive and C_{s,r}(n, t) ≥ A_{s,r}(n, t)y_{ns−t+r}y_{ns−r} for t ≤ ns + r.

3. If the s-triangle {a_{s}(n, k)} is double LC-positive, then the s-triangle {d_{s}(n, k)} is LCpositive and D_{s,r}(n, t) ≥ A_{s,r}(n, t)x_{r}x_{t−r}y_{ns−t+r}y_{ns−r} for t ≤ ns + r.

Proof.

1. Let 0 ≤ t ≤ 2ns. It is easy to see by definition that b_{s,k}(n, t) = a_{s,k}(n, t)x_{k}x_{t−k} for 0 ≤ k ≤ ⌊t/2⌋. Hence for 0 ≤ r ≤ ⌊t/2⌋

B_{s,r}(n, t) := ∑_{k=r}^{⌊t/2⌋} b_{s,k}(n, t) = ∑_{k=r}^{⌊t/2⌋} a_{s,k}(n, t)x_{k}x_{t−k},

Now {a_{s}(n, k)} is LC-positive and x_{0}x_{t} ≤ x_{1}x_{t−1} ≤ ⋯ by the log-concavity of (x_{k}). From Lemma 3 it follows that

B_{s,r}(n, t) ≥ x_{r}x_{t−r} ∑_{k=r}^{⌊t/2⌋} a_{s,k}(n, t) = A_{s,r}(n, t)x_{r}x_{t−r} ≥ 0,

So the s-triangle {b_{s}(n, k)} is LC-positive.

2. Let 2r ≤ t ≤ 2ns. We need to prove C_{s,r}(n, t) ≥ 0. For brevity, we do this only for the case t odd since the same technique is still valid for the case where t is even.

Let t = 2l + 1 for 0 ≤ k ≤ l. Then we define

α_{k} = a_{s}(n, k)a_{s}(n, t − k),
β_{k} = a_{s}(n − 1, k)a_{s}(n + 1, t − k),
γ_{k} = a_{s}(n + 1, k)a_{s}(n − 1, t − k),
Y_{k} = c_{ns−t+k}y_{ns−k}.

Then

a_{s,k}(n, t) = 2α_{k} − β_{k} − γ_{k},

and

c_{s,k}(n, t) = 2α_{k}Y_{k} − β_{k}Y_{k+s} − γ_{k}Y_{k−s}

by definition. It follows that

C_{s,r}(n, t) = ∑_{k=r}^{l}(2α_{k}Y_{k} − β_{k}Y_{k+s} − γ_{k}Y_{k−s})

= ∑_{k=r}^{l}(2α_{k} − β_{k−s} − γ_{k+s})Y_{k} + ∑_{j=1}^{s} β_{r−j}Y_{r+s−j}

− ∑_{j=1}^{s} γ_{r+j−1}Y_{r−s+j−1} − ∑_{j=1}^{s} β_{l−j+1}Y_{l+s−j+1} + ∑_{j=1}^{s} γ_{l+j}Y_{l−s+j},

where we use the fact that Y_{l+s−j+1} = Y_{l−s+j} and β_{l−j+1} = γ_{l+j} for j = \overline{1, s}. Note that (Y_{k}) is nondecreasing by the log-concavity of (y_{k}) and

2α_{k} − β_{k−s} − γ_{k+s} = 2a_{s}^{⋆}(n, ns − k)a_{s}^{⋆}(n, np − t + k)

− a_{s}^{⋆}(n − 1, ns − k)a^{⋆}(n + 1, ns − t + k) − a_{s}^{⋆}(n + 1, np − k)a_{s}^{⋆}(n − 1, ns − t + k)

= a_{s,ns−t+k}^{⋆}(n, 2ns − t).

Hence by the LC-positivity of {a_{s}^{⋆}(n, k)}, we have

C_{s,r}(n, t) = ∑_{j=ns−t+r}^{⌊(2ns−t)/2⌋} a_{s,j}^{⋆}(n, 2ns − t)Y_{j−ns+t} + ∑_{j=1}^{s} β_{r−j}Y_{r+s−j}

− ∑_{j=1}^{s} γ_{r+j−1}Y_{r−s+j−1}

≥ Y_{r} ∑_{j=ns−t+r}^{⌊(2ns−t)/2⌋} a_{s,j}^{⋆}(n, 2ns − t) + Y_{r} ∑_{j=1}^{s} β_{r−j} − Y_{r−s} ∑_{j=1}^{s} γ_{r+j−1}

= Y_{r} ∑_{k=r}^{s}(2α_{k} − β_{k−s} − γ_{k+s}) + Y_{r} ∑_{j=1}^{s} β_{r−j} − Y_{r−s} ∑_{j=1}^{s} γ_{r+j−1}

= Y_{r} ∑_{k=r}^{s}(2α_{k} − β_{k} − γ_{k}) + (Y_{r} − Y_{r−s}) ∑_{j=1}^{s} γ_{r+j−1}

= A_{s,r}(n, t)Y_{r} + (Y_{r} − Y_{r−s}) ∑_{j=1}^{s} γ_{r+j−1}.

Thus C_{s,r}(n, t) ≥ A_{s,r}(n, t)y_{ns−t+r}y_{ns−r}.

3. We have d_{s}(n, k) = a_{s}(n, k)x_{k}y_{ns−k} = c_{s}(n, k)x_{k} and

D_{s,r}(n, t) = ∑_{k=r}^{⌊t/2⌋} d_{s,k}(n, t) = ∑_{k=r}^{⌊t/2⌋} c_{s,k}(n, t)x_{k}x_{t−k},

by 1 and 2, so

D_{s,r}(n, t) ≥ C_{s,r}(n, t)x_{r}x_{t−r} ≥ A_{s,r}(n, t)x_{r}x_{t−r}y_{ns−t+r}y_{ns−r}.

Now we establish the second result.

Theorem 7. The double LC-positive s-triangles are double PLC.

Proof. Let the s-triangle {a_{s}(n, k)} be doubly LC-positive. Suppose that both (x_{k}) and (y_{k}) are log-concave. Then the s-triangle {a_{s}(n, k)x_{k}y_{ns−k}} is LC-positive by Proposition 6 (3) and is therefore PLC by Theorem 5. Thus the row-sum sequence

z_{n} = ∑_{k=0}^{ns} a_{s}(n, k)x_{k}y_{ns−k}, n = 0, 1, 2, …

is log-concave. In other words, the s-triangle {a_{s}(n, k)} is double PLC.

By Lemma 4, {a_{s}(n, k)} is LC-positive if and only if the inequality ∑_{k=r}^{⌊t/2⌋} a_{s,k}(n, t) ≥ 0 for all 2r ≤ t ≤ 2ns, so the following corollary is immediate.

Corollary 8. Suppose that the following two conditions hold:

A There exists an index m = m(n, t) such that a_{k}(n, t) < 0 for k < m and a_{s,k}(n, t) ≥ 0 for k ≥ m;

B The sequence (A_{s,0}(n; q))_{n≥0} is q-log-concave.

Then the s-triangle {a_{s}(n, k)} is LC-positive and therefore PLC.

Corollary 9. Suppose that s-triangle {a_{s}(n, k)} satisfies Conditions (A) and (B) in Corollary 8 and {a_{s}^{∗}(n, k)} satisfies Condition (A). Then {a_{s}(n, k)} is doubly LC-positive and therefore double PLC.

Proof. It suffices to show that (A_{s,0}^{⋆}(n; q)) is q-log-concave. We have

A_{s,0}^{⋆}(n; q) = ∑_{k=0}^{ns} a_{s}(n, ns − k)q^{k} = ∑_{k=0}^{ns} a_{s}(n, k)q^{ns−k} = q^{ns}A_{s,0}(n; q^{−1})

It follows that

A_{s,0}^{⋆2}(n; q)−A_{s,0}^{⋆}(n−1; q)A_{s,0}^{⋆}(n+1; q) = q^{2ns} (A_{s,0}^{2}(n; q^{−1}) − A_{s,0}(n − 1; q^{−1})A_{s,0}(n + 1; q^{−1}))

which has nonnegative coefficients by the q-log-concavity of (A_{s,0}(n; q)).

3 Application to linear operators of finite order

In this section, for selected examples of s-triangles we show their LC-positivity leading to the PLC property.

Let 𝔖 denote the set of sequences (u_{k})_{k∈ℤ} of nonnegative numbers. Given (s + 1) nonnegative numbers λ_{0}, λ_{1}, …, λ_{s}, define the linear operator L = L[λ_{0}, λ_{1}, …, λ_{s}], on 𝔖 by

L(u_{k}) = ∑_{j=0}^{s} λ_{j}u_{k−j} (k ∈ ℤ).

For n ≥ 2, define L^{n} := L(L^{n−1}) by induction. It is convenient to view L^{0} as the identity operator. Let (u_{k})_{k∈ℤ} be a log-concave sequence.

Lemma 10. If the sequence (λ_{0}, λ_{1}, …, λ_{s}) is log-concave, then so is the sequence (L^{n}(u_{k}))_{k∈ℤ}.

Proof. In fact

(L(u_{k}))^{2} − L(u_{k−1})L(u_{k+1}) = (∑_{j=0}^{s} λ_{j}u_{k−j})^{2} − ∑_{j=0}^{s} λ_{j}u_{k−j−1} ∑_{j=0}^{s} λ_{j}u_{k−j+1}

= ∑_{j=0}^{s} λ_{j}^{2}(u_{k−j}^{2} − u_{k−j−1}u_{k−j+1}) + ∑_{0≤l<j≤s} λ_{j}λ_{l}(u_{k−j}u_{k−l} − u_{k−j−1}u_{k−l+1})

+ ∑_{0≤l<j≤s} λ_{j}λ_{l}u_{k−j}u_{k−l} − ∑_{0≤l<j≤0} λ_{j}λ_{l}u_{k−j+1}u_{k−l−1}

= T_{1} + T_{2} + T_{3},

with

T_{1} = ∑_{j=0}^{s} λ_{j}^{2}(u_{k−j}^{2} − u_{k−j−1}u_{k−j+1}),
T_{2} = ∑_{0≤l<j≤s} λ_{j}λ_{l}(u_{k−j}u_{k−l} − u_{k−j−1}u_{k−l+1})

and

T_{3} = − ∑_{1≤l+1<j≤s} λ_{j}λ_{l}(u_{k−j+1}u_{k−l−1} − u_{k−j}u_{k−l}).

It follows that

(L(u_{k}))^{2} − L(u_{k−1})L(u_{k+1}) = ∑_{j=1}^{s−1} (λ_{j}^{2} − λ_{j−1}λ_{j+1}) × (u_{k−j}^{2} − u_{k−j−1}u_{k−j+1})

+ λ_{0}^{2} (u_{k}^{2} − u_{k−1}u_{k+1}) + λ_{s}^{2} (u_{k−s}^{2} − u_{k−s−1}u_{k−s+1})

+ ∑_{2≤l+2<j≤s−1} (λ_{l}λ_{j} − λ_{l−1}λ_{j+1}) × (u_{k−j}u_{k−l} − u_{k−j−1}u_{k−l+1})

+ ∑_{l=0}^{s−1} λ_{l}λ_{l+1} (u_{k−l−1}u_{k−l} − u_{k−l−2}u_{k−l+1})

+ ∑_{l=0}^{s−2} λ_{l}λ_{l+2} (u_{k−l−2}u_{k−l} − u_{k−l−3}u_{k−l+1})

≥ 0.

By induction, the polynomial sequence (L^{n}(u_{k}))_{k∈ℤ} is also log-concave for n ≥ 0.

This brings us to the following theorem.

Theorem 11. Given (s + 1) nonnegative numbers λ_{0}, λ_{1}, …, λ_{s} and a log-concave sequence (u_{k})_{k∈ℤ}, define

a_{s}(n, k) = L^{n}(u_{k}), (0 ≤ k ≤ ns) .

If (λ_{0}, λ_{1}, …, λ_{s}) is log-concave. Then the s-triangle {a_{s}(n, k)} is doubly LC-positive and therefore double PLC.

Proof. Denote a_{k} = L^{n−1}(u_{k}) for k ∈ ℤ and A_{s,r}(n − 1; q) = ∑_{k=r}^{ns−s} a_{k}q^{k}. If (λ_{0}, λ_{1}, …, λ_{s}) is log-concave, then by Lemma 10 so is the sequence (a_{k})_{k∈ℤ}. We have

A_{s,r}(n; q) = λ_{0} ∑_{k=r}^{ns} a_{k}q^{k} + λ_{1} ∑_{k=r}^{ns} a_{k−1}q^{k} + ⋯ + λ_{s} ∑_{k=r}^{ns} a_{k−s}q^{k}

= A_{s,r}(n − 1; q) ∑_{j=0}^{s} λ_{j}q^{j} + ∑_{j=1}^{s} λ_{j} ∑_{l=1}^{j} a_{r−l}q^{r+j−l}

+ ∑_{j=0}^{s−1} λ_{j} ∑_{l=1}^{s−j} a_{ns−s+l}q^{ns−s+l+j},

thus

A_{s,r}(n; q)^{2} − A_{s,r}(n − 1; q)A_{s,r}(n + 1; q) =

∑_{j=1}^{s} ∑_{l=1}^{j} ∑_{f=0}^{s} λ_{j}λ_{f} (∑_{k=r}^{ns}[a_{r−l}a_{k−f} − a_{r−f−l}a_{k}]q^{k+r+j−l}

+ ∑_{k=ns−s+1}^{ns} a_{r−f−l}a_{k}q^{k+r+j−l})

+ ∑_{j=0}^{s−1} ∑_{l=1}^{s−j} ∑_{f=0}^{s} λ_{j}λ_{f} (∑_{k=r}^{ns} (a_{ns−s+l}a_{k−f} − a_{ns+l−f}a_{k−s}) q^{k+ns−s+l+j}

+ ∑_{k=r}^{r+s−1} a_{ns+l−f}a_{k}q^{k+ns−s+l+j}) ,

which has nonnegative coefficients by the log-concavity of the sequence (a_{k}). Hence the s-triangle {a_{s}(n, k)}_{0≤k≤ns} is LC-positive.

On the other hand, let u_{k}^{⋆} = u_{−k} for k ∈ ℤ. Then the sequence (u_{k}^{⋆})_{k∈ℤ} is log-concave and a_{s}^{⋆}(n, k) = L^{n}[λ](u_{k}^{⋆}). Thus the s-triangle {a_{s}^{⋆}(n, k)}_{0≤k≤ns} is also LC-positive, and the s-triangle {a_{s}(n, k)}_{0≤k≤ns} is therefore doubly LC-positive.

Corollary 12. Let a and b be two nonnegative integers with a ≥ b. If the sequences (x_{k}) and (y_{k}) are log-concave, then so is the sequence

z_{n} = ∑_{k=0}^{ns} \binom{a + n}{b + k}_{s}x_{k}y_{sn−k}, (n ≥ 0).

Proof. Using relation (4), we have \binom{a+n}{b+k}_{s} = ∑_{j=0}^{s} \binom{a+n−1}{b+k−j}_{s}, and taking u_{k} = \binom{a}{b+k}_{s} with λ_{j} = 1, (1 ≤ j ≤ s) in Theorem 11, we obtain the result.

When s = 1, we obtain the result of Y. Wang [29, Corollary 3.4]. Taking a = b = 0 in Corollary 12, we obtain the following nice result.

Corollary 13. If the sequences (x_{k}) and (y_{k}) are log-concave, then so is

z_{n} = ∑_{k=0}^{ns} \binom{n}{k}_{s}x_{k}y_{sn−k} (n ≥ 0).

The following theorem is in a sense dual to Theorem 11.

Theorem 14. Let λ_{0}, λ_{1}, …, λ_{s}, (s + 1) nonnegative numbers and {a_{s}(n, k)} an s-triangle of nonnegative numbers. Suppose that each row of {a_{s}(n, k)} is log-concave and satisfies the following recurrence relation

a_{s}(n, k) = ∑_{j=0}^{s} λ_{j}a_{s}(n + 1, k + j), (0 ≤ k ≤ ns).  (13)

Then the s-triangle {a_{s}(n, k)} is LC-positive and therefore double PLC. Proof. Denote a_{s}(n + 1, k) = v_{k} (0 ≤ k ≤ ns + s). Then the sequence (v_{k}) is log-concave and A_{s,r}(n + 1; q) = ∑_{k=r}^{ns+s} v_{k}q^{k}. By the recurrence relation (13) we have

A_{s,r}(n; q) = A_{s,r}(n + 1; q) ∑_{j=0}^{s} λ_{j}q^{−j} − ∑_{j=0}^{s−1} ∑_{l=1}^{s−j} λ_{j}v_{ns+j+l}q^{ns+l} − ∑_{j=1}^{s} ∑_{l=0}^{j−1} λ_{j}v_{r+l}q^{r+l−j}.

It follows that A_{s,r}^{2}(n; q) − A_{s,r}(n − 1; q)A_{s,r}(n + 1; q)

= A_{s,r}(n; q) (A_{s,r}(n + 1; q) ∑_{j=0}^{s} λ_{j}q^{−j} − ∑_{j=0}^{s−1} ∑_{l=1}^{s−j} λ_{j}v_{ns+j+l}q^{ns+l}

− ∑_{j=1}^{s} ∑_{l=0}^{j−1} λ_{j}v_{r+l}q^{r+l−j})

− A_{s,r}(n + 1; q) (A_{s,r}(n; q) ∑_{j=0}^{s} λ_{j}q^{−j} − ∑_{j=0}^{s−1} ∑_{l=1}^{s−j} ∑_{f=0}^{s} λ_{j}λ_{f}v_{ns−s+j+l+f}q^{ns−s+l}

− ∑_{j=1}^{s} ∑_{l=0}^{j−1} ∑_{f=0}^{s} λ_{j}λ_{f}v_{r+l+f}q^{r+l−j})

= S_{1} + S_{2} + S_{3},

with

S_{3} = ∑_{j=0}^{s−1} ∑_{f=0}^{s} λ_{j}λ_{f} (∑_{l=1}^{s−j} ∑_{k=r}^{r+s−1} v_{ns−s+j+l+f}v_{k}q^{k+ns−s+l} + ∑_{l=0}^{j−1} ∑_{k=ns+1}^{ns+s} v_{r+l+f}v_{k}q^{k+r+l−j}) ,

and

S_{1} = ∑_{j=0}^{s−1} ∑_{f=0}^{s} ∑_{l=1}^{s−j} λ_{j}λ_{f} (( ∑_{k=r+s}^{ns} + ∑_{k=ns+1}^{ns+j} + ∑_{k=ns+j+1}^{ns+s}) (v_{ns−s+j+l+f}v_{k}

− v_{ns+j+l}v_{k+f−s}) q^{k+ns−s+l})

= ∑_{j=0}^{s−1} ∑_{f=0}^{s} ∑_{l=1}^{s−j} q^{ns−s+l}λ_{j}λ_{f} ( ∑_{k=r+s}^{ns}(v_{ns−s+j+l+f}v_{k} − v_{ns+j+l}v_{k+f−s})q^{k}

+ ∑_{k=ns+j}^{ns+s}(v_{ns−s+j+l+f}v_{k} − v_{k+(ns+j+l−k)}v_{ns−s+j+l+f−(ns+j+l−k)})q^{k})  (14)

since

∑_{j=0}^{s−1} ∑_{f=0}^{s} ∑_{l=1}^{s−j} ∑_{k=ns+j+1}^{ns+s} λ_{j}λ_{f}(v_{ns−s+j+l+f}v_{k} − v_{ns+j+l}v_{k+f−s})q^{k+ns−s+l}

= ∑_{j=0}^{s−1} ∑_{f=0}^{s} λ_{j}λ_{f} ∑_{l=1}^{s−j} (( ∑_{k=ns+j+1}^{ns+j+l} + ∑_{k=ns+j+l}^{ns+s})

(v_{ns−s+j+l+f}v_{k} − v_{ns+j+l}v_{k+f−s})q^{k+ns−s+l}) = 0,

by setting, l′ = k−ns−j and k′ = l+j+ns in the second term. The sum (14) has nonnegative coefficients by log-concavity of (v_{k})_{k}, and the first term of (14) gives the following: if ns − s + j + l + f ≤ k, then

v_{ns−s+j+l+f}v_{k} − v_{ns+j+l}v_{k+f−s} = v_{ns−s+j+l+f}v_{k} − v_{k+(ns+j+l−k)}v_{ns−s+j+l+f−(ns+j+l−k)} ≥ 0;

and otherwise,

v_{ns−s+j+l+f}v_{k} − v_{ns+j+l}v_{k+f−s} = v_{ns−s+j+l+f}v_{k} − v_{ns−s+j+f+(s−f)}v_{k−(s−f)} ≥ 0.

S_{2} = ∑_{f=1}^{s} λ_{f} ( ∑_{k=r+1}^{ns} λ_{j}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j}

+ ∑_{j=2}^{s} λ_{j} ( ∑_{k=r+1}^{r+j−1}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j}

+ ∑_{k=r+j}^{ns}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j} + ∑_{l=1}^{j−1} ((v_{r+l+f}v_{r} − v_{r+l}v_{r+f})q^{2r+l−j}

+ ∑_{k=r+1}^{r+l}(v_{r+l+f}v_{k} − v_{r+l}v_{k+f})q^{k+r+l−j} + ∑_{k=r+l}^{r+j−1}(v_{r+l+f}v_{k} − v_{r+l}v_{k+f})q^{k+r+l−j}

+ ∑_{k=r+j}^{ns}(v_{r+l+f}v_{k} − v_{r+l}v_{k+f})q^{k+r+l−j})))

= ∑_{f=1}^{s} λ_{f} ( ∑_{k=r+1}^{ns} λ_{j}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j} + ∑_{j=2}^{s} ∑_{k=r+j}^{ns} λ_{j}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j}

+ ∑_{j=2}^{s} ∑_{l=1}^{j−1} ∑_{k=r+j}^{ns} λ_{j}(v_{r+l+f}v_{k} − v_{r+l}v_{k+f})q^{k+r+l−j})  (15)

since, by setting k′ = l + r in the second term

∑_{j=2}^{s} ∑_{f=1}^{s} λ_{j}λ_{f} ( ∑_{k=r+1}^{r+j−1}(v_{r+f}v_{k} − v_{r}v_{k+f})q^{k+r−j} + ∑_{l=1}^{j−1}(v_{r+l+f}v_{r} − v_{r+l}v_{r+f})q^{2r+l−j}) = 0,

also, by setting k′ = l + r and l′ = k − r in second term

∑_{j=2}^{s} ∑_{f=1}^{s} ∑_{l=1}^{j−1} λ_{j}λ_{f} (( ∑_{k=r+1}^{r+l} + ∑_{k=r+l}^{r+j−1}) (v_{r+l+f}v_{k} − v_{r+l}v_{k+f})q^{k+r+l−j}) = 0.

The sum (15) has nonnegative coefficients by the log-concavity of (v_{k})_{k}. Hence the polynomial A_{s,r}^{2}(n; q)−A_{s,r}(n−1; q)A_{s,r}(n+1; q) has nonnegative coefficients. So the triangle {a_{s}(n, k)} is LC-positive.

Clearly, the reciprocal s-triangle {a_{s}^{∗}(n, k)} possesses the same property as {a_{s}(n, k)} does. Hence {a_{s}^{∗}(n, k)} is also LC-positive. Thus the s-triangle {a_{s}(n, k)} is doubly LCpositive and therefore double PLC.

In Theorem 14, the choice λ_{j} = 1 (1 ≤ j ≤ s) and a_{s}(n, k) = \binom{a−n}{b−k}_{s} (0 ≤ k ≤ ns), leads to the following:

Corollary 15. Let a, b ∈ ℕ with a ≥ b. If the sequences (x_{k}) and (y_{k}) are log-concave, then so is the sequence

z_{n} = ∑_{k=0}^{ns} \binom{a − n}{b − k}_{s}x_{k}y_{sn−k}, (n ≥ 0).

By setting s = 1 in the above result, we obtain the result of Wang [29, Corollary 3.9]. We conclude this paper with the following.

Conjecture 16. The s-triangle (\binom{n}{k}_{s}\binom{a−n}{b−k}_{s})_{k} is double PLC.

4 Acknowledgments

The authors would like to thank the anonymous referees for their careful reading and helpful comments which have hopefully led to a clearer paper.

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2000 Mathematics Subject Classification: Primary 05A20; Secondary 05A10, 15A04, 11B65. Keywords: log-concavity, LC-positivity, q-log-concavity, ordinary multinomial, linear transformation, convolution.

(Concerned with sequences \underline{A000045}, \underline{A008287}, \underline{A027907}, and \underline{A035343}.)

Received May 30 2019; revised versions received January 2 2020; January 3 2020. Published in Journal of Integer Sequences, May 11 2020.

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