Linear transformations preserving the strong q-log-convexity of polynomials

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1 Linear transformations preserving the strong q-log-convexity of polynomials Bao-Xuan Zhu School of Mathematics and Statistics Jiangsu Normal University Xuzhou, PR China Hua Sun College of Sciences Dalian Ocean University Dalian, PR China Submitted: Apr 5, 2015; Accepted: Jul 25, 2015; Published: Aug 28, 2015 Mathematics Subject Classifications: 05A20; 15A15 Abstract In this paper, we give a sufficient condition for the linear transformation preserving the strong q-log-convexity As applications, we get some linear transformations (for instance, Morgan-Voyce transformation, binomial transformation, Narayana transformations of two inds preserving the strong q-log-convexity In addition, our results not only extend some nown results, but also imply the strong q-logconvexity of some sequences of polynomials Keywords: Log-concavity; Log-convexity; q-log-convexity; Strong q-log-convexity 1 Introduction Let a 0, a 1, a 2, be a sequence of nonnegative real numbers The sequence is called log-concave (resp log-convex if for all 1, a 1 a +1 a 2 (resp a 1a +1 a 2 The log-concave sequences arise often in combinatorics, algebra, geometry, analysis, probability and statistics and have been extensively investigated We refer the reader to [21, 3, 25] for log-concavity and [15, 26] for log-convexity For a polynomial f(q with real coefficients, denote f(q q 0 if it has only nonnegative coefficients For a sequence of polynomials with nonnegative coefficients {f n (q} n 0, it is called q-log-convex, introduced by Liu and Wang [15], if T (f n (q f n+1 (qf n1 (q f n (q 2 q 0 (11 Supported partially by the National Natural Science Foundation of China (Grant No the electronic journal of combinatorics 22(3 (2015, #P326 1

2 for n 1 If the opposite inequality in (11 holds, then it is called q-log-concave, first suggested by Stanley It is called strongly q-log-convex if f n+1 (qf m1 (q f n (qf m (q q 0 for any n m 1, see Chen et al [8] Clearly, their strong q-log-convexity of polynomial sequences implies the q-log-convexity However, the converse does not hold, see Chen et al [8] It is easy to see that if the sequence {f n (q} n 0 is q-log-convex, then for each fixed nonnegative number q, the sequence {f n (q} n 0 is log-convex The q-log-concavity and q-log-convexity of polynomials have been extensively studied, see [4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 19, 20, 22, 26, 27] for instance It is a good way to obtain the log-concavity or log-convexity by some operators For instance, Davenport and Pólya [9] demonstrated that the log-convexity is preserved under the binomial convolution Wang and Yeh [25] also proved that the log-concavity is preserved under the binomial convolution Brändén [2] studied some linear transformations preserving the Pólya frequency property of sequences Liu and Wang [15] also studied the linear transformation preserving the log-convexity However, there is no result about the linear transformation preserving the strong q-log-convexity This is our motivation of this paper It has been found that many polynomials have the strong q-log-convexity Let polynomials A n (q n a(n, q for n 0 Note that {q } 0 is a strongly q-log-convex sequence Thus it is natural to consider the strong q-log-convexity of B n (q a(n, f (q, for n 0 if {f n (q} n 0 is a strongly q-log-convex sequence The objective of this paper is to study the strong q-log-convexity of {B n (q} n 0 In Section 2, we first give a sufficient condition implying the strong q-log-convexity of B n (q, see Theorem 21 Then we apply it to some special linear transformations As consequences, on the one hand, we extend some nown results On the other hand, we also get some new results on the strong q-log-convexity of some sequences of polynomials 2 Strong q-log-convexity and linear transformations Given a triangular array {a(n, } 0 n of nonnegative real numbers and a strongly q-log-convex sequence {f n (q} n 0, define the polynomials A n (q a(n, q and B n (q a(n, f (q, for n 0 For convenience, let a(n, 0 unless 0 n Suppose m n For 0 t m + n, define a (m, n, t a(n 1, a(m + 1, t + a(m + 1, a(n 1, t a(m, a(n, t a(n, a(m, t the electronic journal of combinatorics 22(3 (2015, #P326 2

3 if 0 < t/2, and a (m, n, t a(n 1, a(m + 1, a(n, a(m, if t is even and t/2 Our main result of this paper is as follows Theorem 21 Suppose that the triangle {a(n, } of nonnegative real numbers satisfies the following two conditions: (C1 The sequence of polynomials {A n (q} n 0 is strongly q-log-convex (C2 There exists an index r r(m, n, t such that a (m, n, t 0 for r and a (m, n, t < 0 for > r If the sequence {f (q} 0 is strongly q-log-convex, then the polynomials B n (q form a strongly q-log-convex sequence Proof Let m n 0 By computation, we have A n1 (qa m+1 (q A n (qa m (q m+n t0 t/2 a (m, n, t q t, and B n1 (qb m+1 (q B n (qb m (q m+n t0 t/2 a (m, n, tf (qf t (q Let A(m, n, t t/2 a (m, n, t Then the condition (C1 is equivalent to A(m, n, t 0 for 0 t m + n Since {f (q} 0 is strongly q-log-convex, we have f 0 (qf t (q q f 1 (qf t1 (q q f 2 (qf t2 (q q By (C2 we find that t/2 t/2 a (m, n, tf (qf t (q q Thus {B n (q} n 0 is strongly q-log-convex a (m, n, tf r (qf tr (q A(m, n, tf r (qf tr (q In what follows we will give some applications of Theorem 21 Proposition 22 If {f (q} 0 is strongly q-log-convex, then the polynomials g n (q n + n f (q form a strongly q-log-convex sequence Proof Let a(n, + n for 0 n Then by Theorem 21, it suffices to show that the triangle {a(n, } satisfies the conditions (C1 and (C2 Let A n (q n + n q, which is the n-th Morgan-Voyce polynomial ([24] By the recurrence relation of the binomial coefficients, we can obtain n + 1 ( n n + 1 n n 1 the electronic journal of combinatorics 22(3 (2015, #P326 3

4 From this it follows that A n+1 (q (2 + qa n (q A n1 (q with the initial conditions A 0 (q 1, A 1 (q 1 + q and A 2 (q 1 + 3q + q 2 Thus, by induction on, it is easy to show A +1 (q q A (q for 0 This implies that Let m n + n Then we have A +2 (q (1 + qa +1 (q A +1 (q A (q q 0 A n1 (qa m+1 (q A n (qa m (q A n1 (qa n++1 (q A n (qa n+ (q A n1 (q[(2 + qa n+ (q A n+1 (q] [(2 + qa n1 (q A n2 (q]a n+ (q A n2 (qa n+ (q A n1 (qa n1+ (q A 0 (qa 2+ (q A 1 (qa 1+ (q A +1 (q A (q q 0 Thus the sequence {A n (q} n 0 is strongly q-log-convex, and so the condition (C1 is satisfied In what follows we consider (C2 condition Note that a(n, ( ( n+ n n+ 2 Let m n, 0 t m + n and 0 t/2 If t is even and t/2, then ( ( ( ( n 1 + m n + m + a (m, n, t < If 0 < t/2, then a (m, n, t + + {( ( ( ( } n 1 + m t n + m + t 2 2(t 2 2(t {( ( ( ( } m n 1 + t m + n + t 2 2(t 2 2(t ( ( 2[nt (m + n + 1] n + m + t (n + [m + 1 (t ] 2 2(t A ( ( 2[(m + n + 1 (m + 1t] m + n + t [n + (t ](m (t B It can be seen that if 0 < t(m+1, then A > 0 and B < 0; if < < t/2 < then A < 0 and B < 0 Thus a (m, n, t < 0 if < < t/2 And ( ( ( ( m t m + t n 1 + t n + t a 0 (m, n, t + 2t 2t 2t 2t ( 2t m + t 2t ( n + t > 0 m + 1 t 2t n + t 2t, the electronic journal of combinatorics 22(3 (2015, #P326 4

5 Note that if 0 <, then A > 0 and B < 0 In order to show that A + B changes sign at most once (from nonnegative to nonpositive for [0,, we consider the monotonicity of A/(B Let A/(B Then we have + ( m+t [nt (m + n + 1][n + (t ](m + 1 [(m + 1t (m + n + 1](n + [m + 1 (t ] 2 2(t ( m+ +t [nt (m + n + 1][n + (t ](m + 1 [(m + 1t (m + n + 1](n + [m + 1 (t ] Claim 23 is decreasing when is increasing 2 2(t ( m+(t ( m mn mn ( m+ ( m(t mn mn Proof If we assume that then it is not hard to prove that Y ( m+(t ( m mn mn ( m+ ( m(t mn mn, Y +1 Y (n + t (n (n + 1 t + (n (m + t (m (m + 1 t + (m , which implies that Y is decreasing in On the other hand, it is easy to see that both are decreasing in In addition, m+1 m+1(t and n+(t n+ nt (m + n + 1 (m + 1t (m + n (m n + 1t (m + 1t (m + n + 1 is also decreasing in Thus is decreasing when increasing By Claim 23, it follows that A 1 changes sign at most once (from nonnegative to B nonpositive for [0, t/2] So does A + B It follows that the triangle a(n, ( n+ n satisfies the condition (C2 The proof is complete In [1], Bonin, Shapiro and Simion introduced a q-analog of the large Schröder number r n, called the q-schröder number r n (q It is defined as: r n (q P q diag(p, where P taes over all Schröder paths from (0, 0 to (n, n and diag(p denotes the number of diagonal steps in the path P Obviously, the large Schröder numbers r n r n (1 In addition ( n + r n (q C q n, n where C 1 +1( 2 In [26], Zhu proved the strong q-log-convexity of q-schröder numbers, which also immediately follows from Proposition 22 the electronic journal of combinatorics 22(3 (2015, #P326 5

6 Corollary 24 [26] The q-schröder numbers r n (q form a strongly q-log-convex sequence The q-central Delannoy numbers D n (q + n ( 2 q n, see Sagan [18] Liu and Wang [15] proved that numbers D n (q form a q-log-convex sequence Zhu [26] demonstrated the strong q-log-convexity of q-central Delannoy numbers, which can also been obtained from Proposition 22 Corollary 25 [26] The q-central Delannoy numbers D n (q form a strongly q-log-convex sequence The Bessel polynomials are defined by B n (q (n +! ( q, (n!! 2 and they have been extensively studied Chen, Wang and Yang [8] obtained the strong q-log-convexity of B n (q Note that B n (q + n ( 2! ( q 2 and it is not hard to prove that the sequence { ( 2! ( q 2 } 0 is strongly q-log-convex Thus we have the following corollary by Proposition 22 Corollary 26 [8] The Bessel polynomials B n (q form a strongly q-log-convex sequence Proposition 27 If {f (q} 0 is strongly q-log-convex, then the polynomials b n (q f (q for n 0 form a strongly q-log-convex sequence n Proof Let a(n, for 0 n Then by Theorem 21, it suffices to show that the triangle {a(n, } satisfies the conditions (C1 and (C2 Let g n (q n q (1 + q n for n 0 It follows that g n1 (qg m+1 (q g n (qg m (q 0 for any m n Thus the sequence {g n (q} n 0 is strongly q-log-convex, and so the condition (C1 is satisfied We proceed to demonstrating the condition (C2 as follows Note that a(n, Let m n, 0 t m + n and 0 t/2 If t is even and t/2, then ( ( n 1 m + 1 a (m, n, t ( m < 0 the electronic journal of combinatorics 22(3 (2015, #P326 6

7 If 0 < t/2, then a (m, n, t + + {( ( ( ( } n 1 m + 1 n m t t {( ( ( ( } m + 1 n 1 m n t t ( ( nt (m + n + 1 n m n[m + 1 (t ] t A 1 ( ( (m + n + 1 (m + 1t m n n(m + 1 t B 1 We find that if 0 <, then A t(m+1 1 > 0 and B 1 < 0; if < t/2 < then A 1 0 and B 1 < 0 Thus a (m, n, t < 0 if < t/2 And a 0 (m, n, t ( ( ( m + 1 m n 1 + t t t ( t m t ( n > 0 m + 1 t t n t ( n t, In order to show that A 1 +B 1 changes sign at most once ( from nonnegative to nonpositive for [0,, we consider the monotonicity of A 1/(B 1 Let 1 A 1 /(B 1 Then we have m [nt (m + n + 1](m ( [(m + 1t (m + n + 1][m + 1 (t ] t ( m In the following we will prove that 1 is decreasing in If we assume that m y ( t ( m, then it is not hard to prove that t t y +1 y (n t + + 1(n (m t + + 1(m 1 It follows that y is decreasing in On the other hand, we have nown that nt( m+1 and m+1(t are decreasing in Thus (m+1t( 1 is decreasing in It follows that a (m, n, t changes sign at most once (from nonnegative to nonpositive As a consequence, we now that a(n, satisfies the condition (C2 of Theorem 21 This completes the proof the electronic journal of combinatorics 22(3 (2015, #P326 7

8 Let the Bell polynomial B(n, q n 0 S(n, q, where S(n, is the Stirling number of the second ind It was proved that the polynomials B(n, q form a strongly q-log-convex sequence, see Chen et al [8] and Zhu [26, 27] Note that B(n + 1, q B(n, q Thus, by induction and Proposition 27, we can give a new proof for n the strong q-log-convexity of B(n, q Let the polynomial W n (q 0 ( 2 n q, which is called the Narayana polynomials of type B, see [6] Liu and Wang [15] conjectured that {W n (q} n 0 is q-log-convex, which was proved by Chen et al [6] using the theory of symmetric functions In addition, Zhu [26] proved the strong q-log-convexity of W n (q Now we can extend it to the following by Theorem 21 Proposition 28 If {f (q} 0 is strongly q-log-convex, then the polynomials s n (q 2f (q form a strongly q-log-convex sequence n Proof Note that the strong q-log-convexity of W n (q has been proved, see Zhu [26] So the condition (C1 of Theorem 21 is satisfied Note that a(n, 2, m n, 0 t m + n and 0 t/2 If t is even and t/2, then ( 2 ( 2 n 1 m + 1 a (m, n, t 2 ( m 2 < 0 If 0 < t/2, then { (n 2 ( 2 ( 2 ( } 2 1 m + 1 n m a (m, n, t t t { (m 2 ( 2 ( 2 ( } n 1 m n + t t ( 2 ( 2 nt (m + n + 1 n(2m + 2 t (m n + 1 n m n[m + 1 (t ] n[m + 1 (t ] t A 2 ( 2 ( 2 (m + n + 1 (m + 1t (m + 1(2n t + (m n + 1 m n + n(m + 1 n(m + 1 t B 2 We find that if 0 <, then A t(m+1 2 > 0 and B 2 < 0; if < t/2 <, then A 2 0 and B 2 < 0 Thus a (m, n, t < 0 if < t/2 And ( 2 ( 2 ( 2 ( 2 m + 1 m n 1 n a 0 (m, n, t + t t t t ( {( ( } t m m + 1 m + t ( {( ( } n n 1 n + > 0 m + 1 t t t t n t t t the electronic journal of combinatorics 22(3 (2015, #P326 8

9 Note that if 0 < have 2 y 1, then A 2 > 0 and B 2 < 0 Let 2 A 2 /(B 2 Then we n(2m + 2 t (m n + 1 (m + 1(2n t + (m n + 1 m + 1 m + 1 (t Noticing that when is increasing, we have nown that y, 1 and decreasing, respectively Moreover, it is easy to see that n(2m+2t(mn+1 (m+1(2nt+(mn+1 m+1 m+1(t are is decreasing when increasing So is 2 As a result, we obtain that a(n, 2 satisfies the condition (C2 of Theorem 21 This completes the proof In [12], Drivera, et al proved that a conjecture of C Greene and H Wilf that all zeros of the hypergeometric polynomial P n (q 2 ( 2 are real It is nown that P n (1 is related to the Franel numbers, and Sun [23] studied the congruence properties of P n (q In [11], it was proved that polynomials P n (q form a q-log-convex sequence By Proposition 28, we have the following stronger result Corollary 29 The hypergeometric polynomials P n (q form a strongly q-log-convex sequence Note that the roo polynomial of a square of side n, denoted by S n (q, is given by q S n (q ( 2 n!q, see [17, Chapter 3 Problems 18] for instance, which also has only real zeros in term of the roo theory The following result is immediate from Proposition 28 Corollary 210 The roo polynomials S n (q form a strongly q-log-convex sequence The Narayana number N(n, is defined as the number of Dyc paths of length 2n with exactly peas (a pea of a path is a place at which the step (1, 1 is directly followed by the step (1, 1 The Narayana numbers have an explicit expression N(n, n( 1 n 1 Liu and Wang [15] conjectured that the Narayana polynomials N n (q n N(n, q are q-log-convex Using the technique of the symmetric functions, Chen, et al [6] proved the strong q-log-convexity of N n (q Recently, Zhu [26] gave a simple proof based on certain recurrence relation Now, we can extend it to the next result, whose proof is omitted for brevity Proposition 211 If {f (q} 0 is strongly q-log-convex, then the polynomials N n (q n N(n, f (q form a strongly q-log-convex sequence the electronic journal of combinatorics 22(3 (2015, #P326 9

10 References [1] J Bonin, L Shapiro, R Simion, Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths, J Statist Plann Inference 34 ( [2] P Brändén, On linear transformations preserving the Pólya frequency property, Trans Amer Math Soc 358 ( [3] F Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp Math 178 ( [4] LM Butler, The q-log concavity of q-binomial coeffcients, J Combin Theory Ser A 54 ( [5] LM Butler, WP Flanigan, A note on log-convexity of q-catalan numbers Ann Comb 11 ( [6] WYC Chen, RL Tang, LXW Wang, ALB Yang, The q-log-convexity of the Narayana polynomials of type B, Adv in Appl Math 44(2 ( [7] WYC Chen, LXW Wang, ALB Yang, Schur positivity and the q-log-convexity of the Narayana polynomials, J Algebraic Combin 32(3 ( [8] WYC Chen, LXW Wang, ALB Yang, Recurrence relations for strongly q-logconvex polynomials, Canad Math Bull 54 ( [9] H Davenport and G Pólya, On the product of two power series, Canadian J Math 1 ( [10] DQJ Dou, AXY Ren, On the q-log-convexity conjecture of Sun, arxiv: [11] DQJ Dou, AXY Ren, The q-log-convexity of Domb s polynomials, arxiv: [12] K Drivera, K Jordaanb, A Martínez-Finelshtein, Pólya frequency sequences and real zeros of some 3 F 2 polynomials, J Math Anal Appl 332 ( [13] P Leroux, Reduced matrices and q-log-concavity properties of q-stirling numbers, J Combin Theory Ser A 54 ( [14] Z Lin, J Zeng, Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials, Adv in Appl Math 53 ( [15] LL Liu, Y Wang, On the log-convexity of combinatorial sequences, Adv in Appl Math 39 ( [16] SM Ma, Y Wang, q-eulerian polynomials and polynomials with only real zeros Electron J Combin 15 (2008, no 1, Research Paper 17 [17] J Riordan, Combinatorial Identities, New Yor, 1979 [18] BE Sagan, Unimodality and the reflection principle, Ars Combin 48 ( [19] BE Sagan, Inductive proofs of q-log concavity, Discrete Math 99 ( the electronic journal of combinatorics 22(3 (2015, #P326 10

11 [20] BE Sagan, Log concave sequences of symmetric functions and analogs of the Jacobi- Trudi determinants, Trans Amer Math Soc 329 ( [21] RP Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann New Yor Acad Sci 576 ( [22] XT Su, Y Wang, YN Yeh, Unimodality Problems of Multinomial Coefficients and Symmetric Functions, Electron J Combin 18(1 (2011, Research Paper 73 [23] Z-W Sun, Congruences for Franel numbers, Adv in Appl Math 51(4 ( [24] MNS Swamy, Further properties of Morgan-Voyce polynomials, Fibonacci Quart 6 ( [25] Y Wang and Y-N Yeh, Log-concavity and LC-positivity, J Combin Theory Ser A 114 (2007, [26] B-X Zhu, Log-convexity and strong q-log-convexity for some triangular arrays, Adv in Appl Math 50(4 ( [27] B-X Zhu, Some positivities in certain triangular array, Proc Amer Math Soc 142(9 ( the electronic journal of combinatorics 22(3 (2015, #P326 11

On the log-convexity of combinatorial sequences arxiv:math/ v3 [math.co] 27 Nov 2006

On the log-convexity of combinatorial sequences arxiv:math/0602672v3 [math.co] 27 Nov 2006 Lily L. Liu, Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China