The concavity and convexity of the Boros Moll sequences

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1 The concavity and convexity of the Boros Moll sequences Ernest X.W. Xia Departent of Matheatics Jiangsu University Zhenjiang, Jiangsu 1013, P.R. China Subitted: Oct 1, 013; Accepted: Dec 17, 014; Published: Jan 9, 015 Matheatics Subject Classifications: 05A0, 05A10, 11B83 Abstract In their study of a quartic integral, Boros and Moll discovered a special class of sequences, which is called the Boros Moll sequences. In this paper, we consider the concavity and convexity of the Boros Moll sequences {d i ()} i=0. We show that for any integer 6, there exist two positive integers t 0 () and t 1 () such that d i ()+d i+ () > d i+1 () for i [0, t 0 ()] [t 1 (), ] and d i ()+d i+ () < d i+1 () for i [t 0 ()+1, t 1 () 1]. When is a square, we find t 0 () = 4 and t 1 () = +. As a corollary of our results, we show that card{i d i () + d i+ () < d i+1 (), 0 i } li = 1. + Keywords: Boros Moll sequences; concavity; convexity; log-concavity; log-convexity 1 Introduction and Main Results The object of this paper is to study the concavity and convexity of the Boros Moll sequences. Boros and Moll [4, 5, 6, 7, 8] explored a special class of Jacobi polynoials in their study of a quartic integral. They have shown that for any a > 1 and any nonnegative integer, 1 (x 4 + ax + 1) dx = π +1 +3/ (a + 1) P (a), (1) +1/ 0 This work was supported by the National Natural Science Foundation of China ( ). the electronic journal of cobinatorics (1) (015), #P1.8 1

2 where P (a) = j, k ( )( )( ) + 1 j k + j (a + 1) j (a 1) k. () j k k + j 3(k+j) Using Raanujan s Master Theore, Boros and Moll [7, 18] derived the following forula P (a) = ( )( ) k + k k (a + 1) k, (3) k k k which indicates that the coefficients of a i in P (a) are positive for 0 i. Chen, Pang and Qu [1] gave a cobinatorial proof to show that () is equal to (3). Let d i () be defined by P (a) = d i ()a i. (4) i=0 The polynoials P (a) will be called the Boros Moll polynoials, and the sequences {d i ()} i=0 of the coefficients will be called the Boros Moll sequences. It follows fro (3) and (4) that d i () = k=i k ( k k )( + k k )( ) k. (5) i The readers can find in [3] any proofs of this forula. Recall that P (a) can be expressed as a hypergeoetric function ( ( P (a) = )F 1, + 1; 1 ; a + 1 ), (6) fro which one sees that P (a) can be viewed as the Jacobi polynoial P (α,β) (a) with α = + 1 and β = ( + 1 (α,β) ), where P (a) is given by ( )( ) ( ) k + β + k + α + β 1 + a P (α,β) (a) = ( 1) k. (7) k k k=0 Soe cobinatorial properties of the Boros Moll sequences have been established. Boros and Moll [5] proved that the sequence {d i ()} i=0 is uniodal and the axiu eleent appears in the iddle, naely, d 0 () < d 1 () < < d [ ] () > d [ ]+1 () > > d (). (8) They also established the uniodality of the sequence {d i ()} i=0 by taking a different approach [6]. Adeberhan, Dixit, Guan, Jiu and Moll [1] presented another proof of (8). Adeberhan, Manna and Moll [] analyzed properties of the -adic valuation of an the electronic journal of cobinatorics (1) (015), #P1.8

3 integer sequence and gave a cobinatorial interpretation of the valuations of the integer sequence which is related to the Boros Moll sequences. Moll [18] conjectured that the sequence {d i ()} i=0 is log-concave. Kauers and Paule [16] proved this conjecture based on the following four recurrence relations found by using the WZ-ethod [0]: d i ( + 1) = + i + 1 d i 1() + d i ( + 1) = (4 i + 3)( + i + 1) d i () ( + 1)( + 1 i) (4 + i + 3) d i (), 0 i + 1, (9) ( + 1) i(i + 1) ( + 1)( + 1 i) d i+1(), 0 i, (10) d i ( + ) = 4i d i ( + 1) ( + i)( + ) and for 0 i + 1, ( + i)( + i 1)d i () ( + i + 1)(4 + 3)(4 + 5) 4( + i)( + 1)( + ) d i(), 0 i + 1, (11) (i 1)( + 1)d i 1 () + i(i 1)d i () = 0. (1) In fact, the recurrences (11) and (1) are also derived independently by Moll [19] by using the WZ-ethod [0]. Chen and Gu [11] showed that the Boros Moll sequences satisfy the reverse ultra log-concavity. Chen and Xia [13] proved that the Boros Moll sequences satisfy the ratio onotone property which iplies the log-concavity and the spiral property. They [14] also confired a conjecture given by Moll in [19]. By constructing an interediate function, Chen and Xia [15] proved the -log-concavity of the Boros Moll sequences. Chen, Dou and Yang [10] proved two conjectures of Brändén [9] on the real-rootedness of the polynoials Q n (x) and R n (x) which are related to the Boros Moll polynoials P n (x). The first conjecture iplies the -log-concavity of the Boros Moll sequences, and the second conjecture iplies the 3-log-concavity of the Boros Moll sequences. In this paper, we consider the concavity and convexity of the Boros Moll sequences. Let {a i } n i=0 be a sequence of real nubers. Recall that the sequence {a i } n i=0 is said to be convex (resp. concave) if a i + a i+ a i+1 (resp. a i + a i+ a i+1 ) (13) for 0 i n. It is easy to see that for positive sequences, the log-convexity iplies the convexity and the concavity iplies the log-concavity. The ain results of this paper can be stated as follows. Theore 1. Let, i be integers and 6. We have d i () + d i+ () > d i+1 () for i [0, t 0 ()] [t 1 (), ] and d i ()+d i+ () < d i+1 () for i [t 0 ()+1, t 1 () 1], the electronic journal of cobinatorics (1) (015), #P1.8 3

4 where t 0 () = and t 1() = + 1 when is a square; t 0() = [ ] or [ 1] and t 1 () = [ + 1] or [ + ] when is not a square. In order to prove Theore 1, we establish the following two Theores: Theore. Let, i be integers and 6. We have d i () + d i+ () > d i+1 () for i [0, ] [ + 1, ]. Theore 3. Let, i be integers and 6. We have d i () + d i+ () < d i+1 () for i [ 1, + ]. Note that is an integer if and only if is a square. Therefore, fro Theores and 3, we iediately prove Theore 1. By Theores and 3, we can obtain the following corollary: Corollary 4. We have card{i d i () + d i+ () < d i+1 (), 0 i } li = 1. (14) + To conclude this section, we propose an open proble. Deterine the signs of the differences d [ 1]()+d [ +1]() d [ ]() and d [ + 1]()+d [ + +1]() d [ + ]() when is not a square. Proofs of the Main Results In this section, we present proofs of the ain results. We first represent d i ()+d i+ () d i+1 () in ters of d i () and d i ( + 1). Lea 5. For 1 i, we have where A(, i) = d i () + d i+ () d i+1 () =A(, i)d i ( + 1) + B(, i)d i () (15) ( + 1)( + 1 i)( i 3), (16) i(i + 1)(i + ) B(, i) = ( i 8i 6 i 0i + + 1i + 8i 3 9). (17) i(i + 1)(i + ) Proof. It follows fro (10) and (1) that for 1 i 1, d i+1 () = (4 i + 3)( + i + 1) d i () i(i + 1) ( + 1 i)( + 1) d i ( + 1) (18) i(i + 1) the electronic journal of cobinatorics (1) (015), #P1.8 4

5 and d i+ () = + 1 i + d i+1() ( i)( + i + 1) d i (). (19) (i + 1)(i + ) Lea 5 follows fro (18) and (19). This copletes the proof. Now, we are ready to prove Theore. Proof of Theore. It is a routine to verify that Theore holds for 6 9. So, we can assue that 10. It is easy to check that for 1 i, B(, i) ( + 1)( + 1 i) A(, i) ( i ) = i(4i 4i + + 6i 6 + 1) ( i 3) > 0. (0) It is easy to verify that where ( i(4i 4i + + 6i 6 + 1) ( i 3) ) i (4 + 4i + 1) = 4i G(, i) ( i 3), (1) G(, i) =( )i ( )i () In Section 3, we will prove that for i [0, ] [ + 1, ] and 10, Cobining (0), (1) and (3), we deduce that G(, i) 0. (3) i(4i 4i + + 6i 6 + 1) ( i 3) > i 4 + 4i + 1. (4) It follows fro (0) and (4) that for i [1, ] [ + 1, ] and 10, B(, i) A(, i) > i + i 4 + 4i + 1. (5) ( + 1)( + 1 i) In order to establish the reverse ultra log-concavity of {d i ()} i=0, Chen and Gu [11] gave an upper bound of the ratio d i (+1)/d i (). They proved that for and 0 i, d i ( + 1) d i () i + i 4 + 4i + 1. (6) ( + 1)( + 1 i) the electronic journal of cobinatorics (1) (015), #P1.8 5

6 By (5) and (6), we see that for i [1, ] [ + 1, ] and 10, B(, i) A(, i) > d i( + 1). (7) d i () It should be noted that A(, i) < 0 for 1 i 1. Thus, the above inequality can be rewritten as A(, i)d i ( + 1) + B(, i)d i () > 0. (8) By Lea 5 and (8), we see that d i () + d i+ () d i+1 () > 0 for 10 and i [1, ] [ + 1, ]. It reains to verify that d 0 () + d () > d 1 () for 10. In (1), let i =, we have Therefore, d 0 () = + 1 ( + 1) d 1() ( + 1) d (). (9) d 0 () + d () d 1 () = + ( + 1) d () 1 ( + 1) d 1(). (30) Eploying (5), we deduce that and d () = ( 1)( ) + ( 1) ( ) d 3 () = ( )( + 1)(4 + 3) 4( 1) (31) ( ). (3) By (31), (3) and the ratio onotone property of the Boros Moll sequences established by Chen and Xia in [13], we have d () d 1 () > d 3() d () = ( )( + 1)(4 + 3) 6( 1)( ) > 1 +, (33) which iplies that the left hand side of (30) is positive for 10. This copletes the proof. Now we turn to prove Theore 3. Proof of Theore 3. It is easy to check that Theore 3 is true for by Maple. In the following, we assue that 93. It is easy to verify that i + 4 i+ B(, i) + ( + 1)( + 1 i) A(, i) H(, i) = ( + 1)( + 1 i)(i + )( i 3), (34) the electronic journal of cobinatorics (1) (015), #P1.8 6

7 where H(, i) =8i 4 8i 3 + i + 8i 3 0i + i + 7i 11i + 9i (35) We can prove that for 93 and i [ 1, + ], H(, i) < 0. (36) The proof of (36) is analogous to the proof of (3), and hence is oitted. In Section 4, we will prove that for 55 and [ ] + 1 i 1, 5 d i ( + 1) d i () i+ i + 4 ( + 1)( + 1 i). (37) In view of (34), (36) and (37), we find that for i [ 1, + ] and 93, which iplies d i ( + 1) d i () i+ i + 4 ( + 1)( + 1 i) > B(, i) A(, i), (38) A(, i)d i ( + 1) + B(, i)d i () < 0. (39) This is because A(, i) < 0 for 1 i 1. In view of Lea 5 and (39), we deduce that d i () + d i+ () d i+1 () < 0 for i [ 1, + ] and 93. This copletes the proof. Proof of Corollary 4. It follows fro Theore that Theore 3 iplies that card{i d i () + d i+ () < d i+1 (), 0 i } li 1. (40) + card{i d i () + d i+ () < d i+1 (), 0 i } li 1. (41) + Corollary 4 follows fro (40) and (41). The proof is coplete. 3 Proof of (3) In this section, we present a proof of (3). It is a routine to verify that for 10, ( ) ( ) ( ) = 4 5/ / / 4 < 0. (4) the electronic journal of cobinatorics (1) (015), #P1.8 7

8 Also, it is easy to check that for 6, ( ) ( ) + 1 ( ) =4 5/ / / > 0. (43) It follows fro (4) and (43) that for 10, < ( ) < + 1. (44) It should be noted that > 0 for 6. Therefore, for 10 and i [0, ] [ + 1, ], we obtain G(, i) in {G(, ), G(, } + 1). (45) It is easy to verify that for 10, and G(, ) = 35/ / / + > 0 (46) G(, + 1) = 5/ / / 1 > 0. (47) Inequality (3) follows fro (45), (46) and (47). This copletes the proof. 4 Proof of (37) In this section, we provide a proof of (37). We are ready to prove (37) by induction on. It is easy to check that (37) holds for = 55. We assue that (37) is true for n 55, i.e., d i (n + 1) d i (n) 4n + 7n + n i+ i + 4 (n + 1 i)(n + 1), We ai to prove that (37) holds for n + 1, that is, d i (n + ) d i (n + 1) 4(n + 1) + 7(n + 1) + n+1i + 4 i+, (n + i)(n + ) It is easy to check that [ n 5 ] + 1 i n 1. (48) [ ] (n + 1) + 1 i n. (49) 5 4n + 7n + n i+ i + 4 (n + 1 i)(n + 1) ( + i)(4n + 5)(4n + 3)(n + i + 1) (8i + 4i 3 8n 4n i 18n 8ni 8 3i)(n + 1) F (n, i) = (8n + 4n i + 18n + 8ni i 8i 4i 3 )( + i)(n + 1)(n + 1 i), (50) the electronic journal of cobinatorics (1) (015), #P1.8 8

9 where F (n, i) is given by F (n, i) = 4 4i + 8n 7i 4n i 3n i 6ni 4i 3 i ni + 8ni 3. (51) Let f(i) and g(i) be defined by f(i) = 8ni 3n i, g(i) = 10ni 4n 4i. (5) For [ ] n i n 1, we find that ( ) n f(i) f = n, g(i) g 5 5 ( n 5 ) = 16n 5. (53) In view of (51), (5) and (53), we deduce that for [ ] n i n 1 and n 55, F (n, i) =f(i)i + g(i)i + (8n + 4 4i) (7i + 6ni) > 1 5 n i 16 5 n i 13ni ni ( ) n 16n 35 > 0. (54) 5 5 It follows fro (50) and (54) that for [ ] n i n 1 and n 55, 4n + 7n + n i+ i + 4 (n + 1 i)(n + 1) ( + i)(4n + 5)(4n + 3)(n + i + 1) (8i + 4i 3 8n 4n i 18n 8ni 8 3i)(n + 1). (55) By (48) and (55), we deduce that for [ n+ 5 ] + 1 i n 1, d i (n + 1) d i (n) ( + i)(4n + 5)(4n + 3)(n + i + 1) (8i + 4i 3 8n 4n i 18n 8ni 8 3i)(n + 1). (56) It is a routine to verify that 4i + 8n + 4n + 19 (n + i)(n + ) which iplies that (n + i + 1)(4n + 3)(4n + 5) 4(n + i)(n + 1)(n + ) = 4i + 8n + 4n + 19 (n + i)(n + ) 4(n + 1) + 7(n + 1) + n+1 i+ i + 4 (n + )(n + i) ( + i)(4n + 5)(4n + 3)(n + i + 1) (8n + 4n i + 18n + 8ni i 8i 4i 3 )(n + 1), (57) 4(n + 1) + 7(n + 1) + n+1 i+ i + 4 (n + )(n + i) > 0. (58) the electronic journal of cobinatorics (1) (015), #P1.8 9

10 By (57), we can rewrite (56) as follows d i (n + 1) 4i + 8n + 4n + 19 (n + i)(n + ) (n + i + 1)(4n + 3)(4n + 5) 4(n + i)(n + 1)(n + ) 4(n + 1) + 7(n + 1) + n+1i + 4 d i (n). (59) i+ (n + )(n + i) It follows fro (58) and (59) that for [ ] n i n 1, 4i + 8n + 4n + 19 (n + i + 1)(4n + 3)(4n + 5) d i (n + 1) (n + i)(n + ) 4(n + i)(n + 1)(n + ) d i(n) 4(n + 1) + 7(n + 1) + n+1i + 4 i+ d i (n + 1). (60) (n + i)(n + ) By (11), we find that the left hand side of (60) equals d i (n + ). Thus we have verified the inequality (49) for [ ] n i n 1. It is still necessary to show that (49) is true for i = n, that is, d n (n + ) d n (n + 1) 4(n + 1) + 7(n + 1) + n+1n + 4 n+. (61) 4(n + ) By (5), we get ( ) n + d n (n + 1) = n (n + 3). (6) n + 1 It follows fro (31) and (6) that d n (n + ) d n (n + 1) = (n + 1)(4n + 18n + 1) (n + )(n + 3) which yields (61). Hence the proof is coplete by induction. Acknowledgeents 4(n + 1) + 7(n + 1) + n+1n + 4 n+, (63) 4(n + ) The author would like to thank the anonyous referee for valuable suggestions and coents. References [1] T. Adeberhan, A. Dixit, X. Guan, L. Jiu, and V.H. Moll. The uniodality of a polynoial coing fro a rational integral. Back to the original proof. J. Math. Anal. Appl., 40: , 014. [] T. Adeberhan, D. Manna, and V.H. Moll. The -adic valuation of a sequence arising fro a rational integral. J. Cobin. Theory Ser. A, 115: , 008. the electronic journal of cobinatorics (1) (015), #P1.8 10

11 [3] T. Adeberhan and V.H. Moll. A forula for a quartic integral: a survey of old proofs and soe new ones. Raanujan J., 18:91 10, 009. [4] G. Boros and V.H. Moll. An integral hidden in Gradshteyn and Ryzhik. J. Coput. Appl. Math., 106: , [5] G. Boros and V.H. Moll. A sequence of uniodal polynoials. J. Math. Anal. Appl. 37:7 85, [6] G. Boros and V.H. Moll. A criterion for uniodality. Electron. J. Cobin. 6: R3, [7] G. Boros and V.H. Moll. The double square root, Jacobi polynoials and Raanujan s Master Theore. J. Coput. Appl. Math., 130: , 001. [8] G. Boros and V.H. Moll. Irresistible Integrals. Cabridge University Press, Cabridge, 004. [9] P. Brändén. Iterated sequences and the geoetry of zeros. J. Reine Angew. Math. 658: , 011 [10] W.Y.C. Chen, D.Q.J. Dou, and A.L.B. Yang. Brändén s conjectures on the Boros Moll polynoials. Inter. Math. Res. Notices, 013: , 013. [11] W.Y.C. Chen and C.C.Y. Gu. The reverse ultra log-concavity of the Boros Moll polynoials. Proc. Aer. Math. Soc., 137: , 009. [1] W.Y.C. Chen, S.X.M. Pang, and E.X.Y. Qu. On the cobinatorics of the Boros Moll polynoials. Raanujan J., 1:41 51, 010. [13] W.Y.C. Chen and E.X.W. Xia. The ratio onotonicity of Boros Moll polynoials. Math. Coput., 78:69 8, 009. [14] W.Y.C. Chen and E.X.W. Xia. Proof of Moll s iniu conjecture. European J. Cobin., 34: , 013. [15] W.Y.C. Chen and E.X.W. Xia. -log-concavity of the Boros Moll polynoials. Proc. Edinburgh Math. Soc., 56:701 7, 013. [16] M. Kauers and P. Paule. A coputer proof of Moll s log-concavity conjecture. Proc. Aer. Math. Soc., 135: , 007. [17] D.V. Manna and V.H. Moll. A rearkable sequence of integers. Expo. Math., 7:89 31, 009. [18] V.H. Moll. The evaluation of integrals: A personal story. Notices Aer. Math. Soc., 49: , 00. [19] V.H. Moll. Cobinatorial sequences arising fro a rational integral. Online J. Anal. Cobin., :Article 4, 007. [0] H.S. Wilf and D. Zeilberger. An algorithic proof theory for hypergeoetric (ordinary and q ) ultisu/integral identities. Invent. Math., 108: , 199. the electronic journal of cobinatorics (1) (015), #P1.8 11