Log-convexity of Aigner–Catalan–Riordan numbers

Linear Algebra and its Applications 463 (2014) 45–55

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Log-convexity of Aigner–Catalan–Riordan numbers Yi Wang ∗ , Zhi-Hai Zhang School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 24 February 2014 Accepted 4 September 2014 Available online 16 September 2014 Submitted by R. Brualdi MSC: 15A45 05A20 05A10 15B36 05A15 Keywords: Log-convexity TP2 matrix Catalan-like number Riordan array Generating function

a b s t r a c t Let T = [tn,k ]n,k≥0 be an infinite lower triangular matrix defined by

t0,0 = 1,

tn+1,0 =

n  j=0

zj tn,j ,

tn+1,k+1 =

n 

aj,k tn,j

j=k

for n, k ≥ 0, where all zj , aj,k are nonnegative and aj,k = 0 unless j ≥ k ≥ 0. We show that the sequence (tn,0 )n≥0 is log-convex if the coefficient matrix [ζ, A] is TP2 , where ζ = [z0 , z1 , z2 , . . .] and A = [ai,j ]i,j≥0 . This gives a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schröder numbers, the Bell numbers, and so on. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let (an )n≥0 be a sequence of nonnegative numbers. We say that the sequence is log-convex (log-concave, resp.) if am an+1 ≥ am+1 an (am an+1 ≤ am+1 an , resp.) for 0 ≤ m < n. Log-convex and log-concave sequences arise often in combinatorics. An effect * Corresponding author. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.laa.2014.09.007 0024-3795/© 2014 Elsevier Inc. All rights reserved.

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approach to attack the log-concavity and log-convexity problems comes from the theory of total positivity. We say that an infinite matrix of nonnegative numbers is TP 2 if its minors of order 2 are all nonnegative. Let (an )n≥0 be an infinite sequence of nonnegative numbers and with no internal zeros. Then it is log-concave if and only if its Toeplitz matrix [ai−j ]i,j≥0 is TP2 , and it is log-convex if and only if its Hankel matrix [ai+j ]i,j≥0 is TP2 . We refer the reader to [5–8,13,19,22,23] for total positivity and log-concavity problems. In the present paper we use the concept of total positivity to establish a criterion for the log-convexity of the 0th column (tn,0 )n≥0 of an infinite lower triangular matrix ⎡ ⎤ t0,0 ⎢ ⎥ ⎢ t1,0 t1,1 ⎥ ⎢ ⎥ T = [tn,k ]n,k≥0 = ⎢ t ⎥ t2,2 2,0 t2,1 ⎣ ⎦ .. . ··· defined by the recursive system t0,0 = 1,

tn+1,0 =

n 

zj tn,j ,

tn+1,k+1 =

j=0

n 

aj,k tn,j

(1.1)

j=k

for n, k ≥ 0, where all zj , aj,k are nonnegative and aj,k = 0 unless j ≥ k ≥ 0. The triangles defined by (1.1) are ubiquitous in combinatorics. A basic example is the famous Pascal triangle. We will consider two classes of particular interesting generalizations of the Pascal triangle. The first class of triangles [cn,k ]n,k≥0 , introduced by Aigner [2–4], is defined by c0,0 = 1,

c0,k = 0 (k > 0),

cn+1,k = cn,k−1 + sk cn,k + tk+1 cn,k+1

(n, k ≥ 0).

The elements cn,0 are called the Catalan-like numbers corresponding to (σ, τ ), where σ = (s0 , s1 , s2 , . . .),

τ = (t1 , t2 , t3 , . . .).

The Catalan-like numbers unify many well-known counting coefficients. For example, cn,0 are (1) the Catalan numbers Cn corresponding to σ = (1, 2, 2, . . .) and τ = (1, 1, 1, . . .); (2) the Motzkin numbers Mn corresponding to σ = τ = (1, 1, 1, . . .);

(3) the central binomial coefficients 2n corresponding to σ = (2, 2, 2, . . .) and τ = n (2, 1, 1, . . .); (4) the Schröder numbers Sn corresponding to σ = (2, 3, 3, . . .) and τ = (2, 2, 2, . . .); (5) the Bell numbers Bn corresponding to σ = τ = (1, 2, 3, 4, . . .).

Y. Wang, Z.-H. Zhang / Linear Algebra and its Applications 463 (2014) 45–55

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The second class of triangles concerned is the Riordan arrays introduced by Shapiro et al. in [16]. A (proper) Riordan array, denoted by (g(x), f (x)), is an infinite lower triangular matrix whose generating function of the kth column is xk f k (x)g(x) for k = 0, 1, 2, . . . , where g(0) = 1 and f (0) = 0. For example, the Pascal triangle  P =

1 1 , , 1−x 1−x

the Catalan triangle

C = C(x), C(x) ,

C(x) =

1−

√ 1 − 4x , 2x

and the Motzkin triangle

M = M (x), M (x) ,

M (x) =

1−x−

√

1 − 2x − 3x2 . 2x2

The Riordan arrays arise in the enumeration of lattice paths, e.g., the Dyck paths, the Motzkin paths, and the Schröder paths and so on [9,14,16]. The 0th column of such a Riordan array counts the corresponding lattice paths, including the Catalan numbers, the Motzkin numbers, and the Schröder numbers. It is known that a Riordan array R = (g(x), f (x)) = [rn,k ]n,k≥0 can be characterized by two sequences A = (an )n≥0 and Z = (zn )n≥0 such that r0,0 = 1,

rn+1,0 =



zj rn,j ,

rn+1,k+1 =

j≥0



aj rn,k+j

(1.2)

j≥0

for n, k ≥ 0 (see [1,9–11,14,15,17] for instance). Let A(x) and Z(x) be the generating functions of A- and Z-sequences respectively. Then it follows from (1.2) that g(x) =

1 , 1 − xZ(xf (x))



f (x) = A xf (x) .

(1.3)

For example, consider the ballot table [bn,k ]n,k≥0 defined by the recursive system b0,0 = 1,

b0,k = 0

(k > 0),

bn+1,k = bn,k−1 + bn,k + · · · + bn,n

(n ≥ 0).

(1.4)

Then A(x) = Z(x) = 1/(1 − x) since A = Z = (1, 1, 1, . . .). It follows from (1.3) that f (x) = g(x) = C(x). So the ballot table coincides with the Catalan triangle, and in

n particular, Cn+1 = k=0 bn,k , a classical result [4].

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We come back to the general triangle T = [tn,k ]n,k≥0 defined by (1.1). Clearly, such a triangle is determined completely by the coefficient matrix ⎡

z0 ⎢z ⎢ 1 ⎢ z [ζ, A] = ⎢ ⎢ 2 ⎢ z3 ⎣ .. .

⎤

a0,0 a1,0 a2,0 a3,0 .. .

a1,1 a2,1 a3,1 .. .

a2,2 a3,2 .. .

a3,3 .. .

..

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(1.5)

.

For convenience, we call tn,0 the Aigner–Catalan–Riordan numbers corresponding to the matrix [ζ, A]. It is well known that the Catalan numbers, the Motzkin numbers, the center binomial coefficients, the Bell numbers, the Schröder numbers and the Fine numbers are log-convex respectively (see Liu and Wang [12] for instance). Recently, Zhu [24, Theorem 3.1] gave a criterion for the log-convexity of the (generalized) Catalan-like numbers (see Corollary 2.2). The main objective of the present paper is to extend Zhu’s result to the Aigner–Catalan–Riordan numbers. Theorem 1.1. Let T = [ti,j ]i,j≥0 be the infinite lower triangular matrix defined by (1.1). If the coefficient matrix [ζ, A] is TP2 , then the sequence (tn,0 )n≥0 is log-convex. In the next section we first prove the theorem and then give its applications to the (generalized) Catalan-like numbers and the Riordan arrays. In Section 3 we present the q-version of Theorem 1.1. 2. Proof and applications of Theorem 1.1 We first give the proof of the theorem. The key observation behind the proof is that two recursive relations in (1.1) can be written as: ⎡

⎤ ⎡ t1,0 t0,0 ⎢ ⎥ ⎢ t t ⎢ 2,0 ⎥ ⎢ 1,0 ⎢ ⎥ ⎢ ⎢ t3,0 ⎥ = ⎢ t2,0 ⎣ ⎦ ⎣ .. .

⎤ z0 ⎥⎢ ⎥ ⎥ ⎢ z1 ⎥ ⎥⎢ ⎥ ⎥ ⎢ z2 ⎥ ⎦⎣ ⎦ .. .. . . ⎤⎡

t1,1 t2,1

t2,2

···

(2.1)

and ⎡

t1,1 ⎢ ⎢ t2,1 ⎢ ⎢ t3,1 ⎣

⎤ t2,2 t3,2 ···

t0,0 ⎥ ⎢ ⎥ ⎢ t1,0 ⎥=⎢ ⎥ ⎢ t2,0 ⎦ ⎣

t3,3 ..

⎡

.

⎤⎡ t1,1 t2,1

a0,0 ⎥⎢ ⎥ ⎢ a1,0 ⎥⎢ ⎥ ⎢ a2,0 ⎦⎣

t2,2

···

So the recursive system (1.1) is equivalent to

..

.

⎤ a1,1 a2,1 ···

⎥ ⎥ ⎥. ⎥ ⎦

a2,2 ..

.

(2.2)

Y. Wang, Z.-H. Zhang / Linear Algebra and its Applications 463 (2014) 45–55

⎤

⎡

t0,0 ⎢t ⎢ 1,0 ⎢ ⎢ t2,0 ⎢ ⎢ t3,0 ⎣

49

t1,1 t2,1 t3,1

t2,2 t3,2

t3,3 ..

···

⎡

1 ⎢0 t 0,0 ⎢ ⎢ 0 t1,0 =⎢ ⎢ ⎢ 0 t2,0 ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ⎤⎡

t1,1 t2,1

1 ⎥⎢z ⎥⎢ 0 ⎥⎢ ⎥ ⎢ z1 ⎥⎢ ⎥ ⎢ z2 ⎦⎣ .. .. . .

t2,2

···

⎤ a0,0 a1,0 a2,0

a1,1 a2,1 ···

⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

a2,2 ..

.

or briefly,  T =

 1 O

O T

1 ζ

 O . A

(2.3)

Proof of Theorem 1.1. Clearly, to show the log-convexity of the sequence (tn,0 )n≥0 , it suffices to show that the matrix ⎡ ⎤ t0,0 t1,0 ⎢ ⎥ ⎢ t1,0 t2,0 ⎥ ⎢ ⎥ ⎢ t2,0 t3,0 ⎥ ⎣ ⎦ .. .. . . is TP2 . By (2.1), we have ⎡

t0,0 ⎢ ⎢ t1,0 ⎢ ⎢ t2,0 ⎣ .. .

⎤ ⎡ t0,0 t1,0 ⎥ ⎢ t2,0 ⎥ ⎢ t1,0 ⎥ ⎢ t3,0 ⎥ = ⎢ t2,0 ⎦ ⎣ .. .

⎤⎡

t1,1 t2,1

1 ⎥⎢ ⎥⎢0 ⎥⎢ ⎥⎢0 ⎦⎣ .. .. . .

t2,2 ···

⎤ z0 ⎥ z1 ⎥ ⎥ z2 ⎥ . ⎦ .. .

(2.4)

Note that the product of TP2 matrices is still TP2 by the famous Cauchy–Binet formula. Now the second matrix in the right hand side of (2.4) is TP2 since all zj are nonnegative. Hence it suffices to show that the first matrix T = [tn,k ]n,k≥0 in the right hand side of (2.4) is TP2 . It is clear that an infinite matrix is TP2 if and only if its leading principal submatrices are all TP2 . Let ⎡

t0,0 ⎢t ⎢ 1,0 ⎢ t Tn = ⎢ ⎢ 2,0 ⎢ ⎣ tn,0

⎤ t1,1 t2,1 ··· tn,1

t2,2 ..

tn,2

. · · · tn,n

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

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Then T is TP2 if and only if all Tn are TP2 for n ≥ 1. So it suffices to show that all Tn are TP2 . We proceed by induction on n. Obviously, T1 is TP2 . Assume that Tn is TP2 . By (2.3), we have  Tn+1 =

 1 O

O Tn

 1 ζn

O An

,

(2.5)

where ζn = [z0 , z1 , . . . , zn ] and An = [ai,j ]0≤i,j≤n . By the assumption that Tn is TP2 , the first matrix in the right hand side of (2.5) is TP2 . By the assumption that the matrix [ζ, A] is TP2 , the submatrix [ζn , An ] is TP2 , and so is the second matrix in the right hand side of (2.5). Thus the product Tn+1 is TP2 , and so is the matrix T , as desired. This completes the proof of the theorem. 2 Remark 2.1. From the proof of the theorem it is easy to see that if the matrix T = [tn,k ]n,k≥0 defined by (1.1) is TP2 and all zj are nonnegative, then the sequence (tn,0 )n≥0 is log-convex. In what follows we present some applications of Theorem 1.1. We first apply Theorem 1.1 to the (generalized) Catalan numbers. Corollary 2.2. (See [24, Theorem 3.1].) Let (rk )k≥0 , (sk )k≥0 , (tk )k≥1 be three sequences of nonnegative numbers and let [cn,k ]n,k≥0 be the triangle defined by the recursion c0,0 = 1,

cn+1,k = rk−1 cn,k−1 + sk cn,k + tk+1 cn,k+1

for n, k ≥ 0. Assume that sk sk+1 ≥ rk tk+1 for all k ≥ 0. Then the sequence (cn,0 )n≥0 is log-convex. Corollary 2.2 yields the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schröder numbers, the Bell numbers, and so on. See [24] for more examples. We also refer the reader to [21] for a combinatorial interpretation of this result. Then we apply Theorem 1.1 to the Riordan arrays. Corollary 2.3. Let R = [rn,k ]n,k≥0 be a Riordan array. Assume that the A-sequence (an )n≥0 and Z-sequence (zn )n≥0 of R satisfy the conditions (A) a0 , a1 , a2 , . . . is a log-concave sequence with no internal zeros, and (B) aj zi ≥ ai zj for j ≥ i ≥ 0. Then the sequence (rn,0 )n≥0 is log-convex.

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Proof. The coefficient matrix of R is ⎡

z0 ⎢z ⎢ 1 ⎢ z J(R) = ⎢ ⎢ 2 ⎢ z3 ⎣ .. .

a0 a1 a2 a3 .. .

⎤ a0 a1 a2 .. .

a0 a1 .. .

a0 .. .

..

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ .

By the condition (A), the Toeplitz matrix [ai−j ]i,j≥0 of the sequence (an )n≥0 is TP2 , which implies that each minor of order 2 taken from the (i + 1)th and (j + 1)th columns of J(R) is nonnegative for j ≥ i ≥ 0. By the condition (B), each minor of order 2 taken from the 0th and 1th columns of J(R) is nonnegative. So, to show that the matrix J(R) is TP2 , it suffices to show that each minor of order 2 taken from the 0th and (j + 1)th columns of J(R) is nonnegative for j ≥ 0. In other words, we need to show that ai+k zj+k ≥ ak zi+j+k

(2.6)

for i, j, k ≥ 0. We may assume, without loss of generality, that ak = 0 and zi+j+k = 0 otherwise the inequality (2.6) is trivial. It implies that ai+j+k = 0 since ai+j+k zj+k ≥ aj+k zi+j+k by the condition (B). Thus all an = 0 for k ≤ n ≤ i + j + k by the condition (A). The inequality (2.6) follows since zi+j+k ai+j+k zj+k aj+k zj+k zi+j+k = · ≤ · = ai+k ai+j+k ai+k aj+k ak ak by the conditions (A) and (B) again. This completes the proof. 2 Remark 2.4. Applying Corollary 2.3 to the ballot table [bn,k ]n,k≥0 defined by (1.4), the log-convexity of the Catalan numbers immediately follows. Next we investigate the generating function of a log-convex sequence by means of Corollary 2.3. For the sake of brevity, we only consider a class of Riordan arrays D = [dn,k ]n,k≥0 defined by d0,0 = 1,

d0,k = 0 (k > 0);

dn+1,0 = pdn,0 + qdn,1

(n ≥ 0);

dn+1,k+1 = rdn,k + sdn,k+1 + tdn,k+2

(n, k ≥ 0).

(2.7)

Such Riordan arrays give one of the most interesting class of Catalan-like numbers dn = dn,0 , including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Fine numbers, the Schröder numbers, and so on. See Aigner [2–4] for details. Let D = (d(x), h(x)). Then by (1.3), we have

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1 , 1 − x(p + qxh(x))

d(x) =

h(x) = r + sxh(x) + tx2 h2 (x).

It follows that h(x) =

1 − sx −

 1 − 2sx + (s2 − 4rt)x2 2tx2

and d(x) =

2t  . 2t − q + (qs − 2pt)x + q 1 − 2sx + (s2 − 4rt)x2

An immediate consequence of Corollary 2.3 is the following corollary. Corollary 2.5. Let p, q, r, s, t be all nonnegative and 

dn x n =

n≥0

2t  . 2t − q + (qs − 2pt)x + q 1 − 2sx + (s2 − 4rt)x2

Assume that s2 ≥ rt and ps ≥ qr. Then the sequence (dn )n≥0 is log-convex. Corollary 2.5 unifies and strengthens a series of results of Zhu [24]. For example, taking q = t in Corollary 2.5, we have the following result. Corollary 2.6. (See [24, Proposition 3.2].) Let p, r, s, t be all nonnegative and 

dn x n =

n≥0

1 + (s − 2p)x +



2 1 − 2sx + (s2 − 4rt)x2

.

Assume that s2 ≥ rt and ps ≥ rt. Then the sequence (dn )n≥0 is log-convex. On the other hand, taking s = p and r = t = q/2 in Corollary 2.5, we obtain the following result. This result strengthens Zhu’s corresponding result [24, Proposition 3.4] √ in which the condition assumed is s ≥ 1+2 5 t. Corollary 2.7. Let s, t be nonnegative and 

1 dn x n =  . 1 − 2sx + (s2 − 4t2 )x2 n≥0 Assume that s ≥

√

2t. Then the sequence (dn )n≥0 is log-convex.

n Recall that the Catalan numbers Cn+1 = k=0 bn,k , where [bn,k ]n,k≥0 is the ballot table defined by (1.4). Many famous combinatorial numbers arise by considering the row

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53

sums of the Riordan arrays (see Aigner [2–4] for instance). In the final part of this section we consider the log-convexity of such numbers. It is well known that [17,18] the concept of Riordan arrays is a powerful tool to find generating functions of combinatorial sums (see Sprugnoli [17,18] for instance). Let R = [rn,k ]n,k≥0 = (g(x), f (x)) be a Riordan

array. Define its row sum matrix R = [rn,k ]n,k≥0 by rn,k = i≥k rn,i . Then 

rn,k xn =

n≥k

 n≥k i≥k

rn,i xn =



rn,i xn =

i≥k n≥i



xi f i (x)g(x) = xk f k (x)

i≥k

g(x) . 1 − xf (x)

Thus R is also a Riordan array and  R=

g(x) , f (x) . 1 − xf (x)

Furthermore, if A = (an ) and Z = (zn ) be A- and Z-sequences of R, then A- and Z-sequences of R are A = A and Z = (a0 + z0 , a1 + z1 − z0 , . . . , aj + zj − zj−1 , . . .). Thus Corollaries 2.3 and 2.5 can be applied to deal with the log-convexity of the sequence

n rn,0 = i=0 rn,i . On the other hand, we have R = RE by definition, where E is the infinite lower triangular matrix whose nonzero elements are all equal to 1. It is obvious that E is TP2 . Assume that R is TP2 . Then R is also TP2 , and the sequence (rn,0 )n≥0 is therefore log-convex if the sequence Z is nonnegative by Remark 2.1. For example, let D = [dn,k ] be the Riordan array defined by (2.7). Then the row sum matrix D = [dn,k ] of D satisfies d0,0 = 1,

d0,k = 0 (k > 0);

dn+1,0 = (r + p)dn,0 + (s + q − p)dn,1 + (t − q)dn,2 dn+1,k+1 = rdn,k + sdn,k+1 + tdn,k+2

(n ≥ 0);

(n, k ≥ 0).

It follows that (dn,0 )n≥0 is log-convex if s +q−p ≥ 0, t −q ≥ 0, s2 ≥ rt and p(r+s) ≥ rt by Corollary 2.5. This result generalizes and strengthens Zhu’s result [24, Proposition 3.3] in which the special case t = q is considered. 3. Remarks The Pascal triangle P has many fascinating properties of total positivity. For example, P is a totally positive matrix and each row of P is a Pólya frequency sequence. We refer the reader to Su and Wang [20] for various unimodality in the Pascal triangle. The Catalan triangle C and the Motzkin triangle M have similar properties. A further and interesting problem is to consider the total positivity of the triangle defined by (1.1). On the other hand, we may give a q-version of Theorem 1.1. For two polynomials f (q) and g(q), denote f (q) ≤q g(q) if the coefficients of the difference g(q) − f (q) are all nonnegative. Let (fn (q))n≥0 be a sequence of polynomials with nonnegative coefficients.

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It is called q-log-convex if fn2 (q) ≤q fn+1 (q)fn−1 (q) for all n ≥ 1. It is called strongly q-log-convex if fm (q)fn (q) ≤q fn+1 (q)fm−1 (q) for any n ≥ m ≥ 1. It is well known that the strong q-log-convexity implies the q-log-convexity but the converse is not true. The concept of the q-TP2 matrix may be defined similarly. Consider a q-array ⎡

  T = tn,k (q) n,k≥0

t0,0 (q) ⎢ ⎢ t1,0 (q) =⎢ ⎢ t2,0 (q) ⎣

⎤ t1,1 (q) t2,1 (q) ···

⎥ ⎥ ⎥ ⎥ ⎦

t2,2 (q) ..

.

defined by

t0,0 (q) = 1,

tn+1,0 (q) =

n 

zj (q)tn,j (q),

j=0

tn+1,k+1 (q) =

n 

aj,k (q)tn,j (q)

j=k

(3.1) for n, k ≥ 0, where all zj (q), aj,k (q) ≥q 0 and aj,k (q) = 0 unless j ≥ k ≥ 0. Denote ⎡

z0 (q) ⎢ z (q) ⎢ 1   ⎢ z (q) ζ(q), A(q) = ⎢ ⎢ 2 ⎢ z3 (q) ⎣ .. .

a0,0 (q) a1,0 (q) a2,0 (q) a3,0 (q) .. .

⎤ a1,1 (q) a2,1 (q) a3,1 (q) .. .

a2,2 (q) a3,2 (q) a3,3 (q) .. .. . .

..

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(3.2)

.

The method of proof used in Theorem 1.1 can be carried over verbatim to its q-analogue. Theorem 3.1. The q-Aigner–Catalan–Riordan sequence (tn,0 (q))n≥0 defined by (3.1) is strongly q-log-convex if the matrix [ζ(q), A(q)] defined by (3.2) is q-TP2 . Theorem 3.1 unifies and strengthens a series of Zhu’s results in [24], including the q-log-convexity of Bell polynomials, Eulerian polynomials, and Narayana polynomials of two types. We refer the reader to [24] for details. Acknowledgements The authors would like to thank the referee for his careful reading and valuable comments. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11071030, 11371078) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110041110039).

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