The 2-log-convexity of the Apery Numbers
aa r X i v : . [ m a t h . C O ] S e p THE 2-LOG-CONVEXITY OF THE AP´ERY NUMBERS
WILLIAM Y. C. CHEN AND ERNEST X. W. XIA
Abstract.
We present an approach to proving the 2-log-convexity of se-quences satisfying three-term recurrence relations. We show that the Ap´erynumbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers,the Franel numbers of order 3 and 4 and the large Schr¨oder numbers are all2-log-convex. Numerical evidence suggests that all these sequences are k -log-convex for any k ≥ Introduction
In his proof of the irrationality of ζ (2) and ζ (3), Ap´ery [2] introduced the fol-lowing numbers A n and B n as given by A n = n X k =0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19) , (1.1) B n = n X k =0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19) . (1.2)The numbers A n and B n are often called the Ap´ery numbers. It has been shownby Ap´ery [2] that A n and B n satisfy the following three-term recurrence relationsfor n ≥ A n = 34 n − n + 27 n − n A n − − ( n − n A n − , (1.3) B n = 11 n − n + 3 n B n − + ( n − n B n − , (1.4)where A = 1 , A = 5 , B = 1 , B = 3; see also [10, 13]. Congruences of theAp´ery numbers have been investigated by Ahlgren, Ekhad, Ono, and Zeilberger[1], Beukers [3, 4], Chowla and Clowes[5] and Gessel [9]. Note that the recurrencerelations (1.3) and (1.4) can be derived by using Zeilberger’s algorithm [14].Cohen [6] and Rhin obtained the following recurrence relation of the numbers U n in connection with the rational approximation of ζ (4), see also [11], U n +1 = R ( n ) U n + G ( n ) U n − , n ≥ , (1.5) Date : September 4, 2009 and, in revised form, May 11, 2010.2000
Mathematics Subject Classification.
Primary 05A20; 11B37, 11B83.
Key words and phrases.
Ap´ery number, log-convexity, 2-log-convexity, infinite log-convexity.The authors wish to thank the referee, Tomislav Doˇsli´c and Tanguy Rivoal for helpful com-ments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry ofEducation, and the National Science Foundation of China. where U = 1, U = 12 and R ( n ) = 3(2 n + 1)(3 n + 3 n + 1)(15 n + 15 n + 4)( n + 1) , G ( n ) = 3 n (3 n − n + 1)( n + 1) . Expressions of U n as double sums of products of binomial coefficients have beenderived by Krattenthaler and Rivoal [11] and Zudilin [15, 16].In this paper, we shall establish the 2-log-convexity of the sequences of theAp´ery numbers A n , B n , the Cohen-Rhin numbers U n and some other combinatorialsequences based on the three-term recurrence relations. Recall that an infinitepositive sequence { a n } ∞ n =0 is said to be log-convex if for all n ≥ a n ≤ a n − a n +1 . We say that { a n } ∞ n =0 is 2-log-convex if { a n } ∞ n =0 is log-convex and for all n ≥ (cid:0) a n a n +2 − a n +1 (cid:1) ≤ (cid:0) a n − a n +1 − a n (cid:1) (cid:0) a n +1 a n +3 − a n +2 (cid:1) . (1.7)Meanwhile, the sequence { a n } ∞ n =0 is called strictly log-convex (2-log-convex) if theinequality in (1.6) ((1.7)) is strict for all n ≥
1. Doˇsli´c [7] proved the log-convexityof A n by induction. In fact, using similar arguments one can show that { B n } ∞ n =0 and { U n } ∞ n =0 are log-convex.This paper is organized as follows. In Section 2, we give a general framework toprove the 2-log-convexity of a sequence { S n } ∞ n =0 based on a lower bound f n andan upper bound g n for the ratio S n /S n − , where the numbers S n satisfy a three-term recurrence relation. Section 3 demonstrates how to find the bounds f n and g n . Section 4 is devoted to the computations of the upper bounds for the ratios A n /A n − , B n /B n − and U n /U n − . In Section 5, we show that the sequences of A n , B n , U n , the Motzkin numbers, the Fine numbers, the Franel numbers of order3 and 4 and the large Schr¨oder numbers are all 2-log-convex. We conclude thispaper with a conjecture on the infinite log-convexity in the spririt of the infinitelog-concavity introduced by Moll [12].2. A criterion
In this section, we present a criterion for the 2-log-convexity of a sequence { S n } ∞ n =0 satisfying a three-term recurrence relation. We need the assumption thatthe ratio S n /S n − has a lower bound f n and an upper bound g n . Theorem 2.1.
Suppose { S n } ∞ n =0 is a positive log-convex sequence that satisfies therecurrence relation S n = b ( n ) S n − + c ( n ) S n − (2.1) for n ≥ . Let a ( n ) =2 b ( n + 2) b ( n + 1) + 2 b ( n + 1) c ( n + 2) − b ( n + 1) − b ( n + 1) b ( n + 2) b ( n + 3) − b ( n + 3) c ( n + 2) − c ( n + 3) b ( n + 1) ,a ( n ) =4 b ( n + 1) b ( n + 2) c ( n + 1) + 2 c ( n + 1) c ( n + 2) + b ( n + 1) b ( n + 2) b ( n + 3)+ b ( n + 1) b ( n + 3) c ( n + 2) + b ( n + 1) c ( n + 3) − c ( n + 1) b ( n + 1) − b ( n + 3) b ( n + 2) c ( n + 1) − c ( n + 3) c ( n + 1) − b ( n + 2) b ( n + 1) HE 2-LOG-CONVEXITY OF THE AP´ERY NUMBERS 3 − b ( n + 2) b ( n + 1) c ( n + 2) − c ( n + 2) ,a ( n ) = − c ( n + 1) (cid:0) b ( n + 2) c ( n + 2) − b ( n + 2) c ( n + 1) − b ( n + 3) b ( n + 2) b ( n + 1) − b ( n + 3) c ( n + 2) − c ( n + 3) b ( n + 1)+ 3 c ( n + 1) b ( n + 1) + 2 b ( n + 2) b ( n + 1) (cid:1) ,a ( n ) = − c ( n + 1) (cid:0) c ( n + 1) − b ( n + 2) b ( n + 3) − c ( n + 3) + b ( n + 2) (cid:1) and ∆( n ) = 4 a ( n ) − a ( n ) a ( n ) . Assume that a ( n ) < and ∆( n ) > for all n ≥ N , where N is a positive integer.If there exist f n and g n such that for all n ≥ N ,( C ) f n ≤ S n S n − < g n ;( C ) f n ≥ − a ( n ) − √ ∆( n )6 a ( n ) ;( C ) a ( n ) g n + a ( n ) g n + a ( n ) g n + a ( n ) > ,then { S n } ∞ n = N is strictly 2-log-convex, that is, for n ≥ N , (cid:0) S n − S n +1 − S n (cid:1) (cid:0) S n +1 S n +3 − S n +2 (cid:1) > (cid:0) S n S n +2 − S n +1 (cid:1) . (2.2) Proof.
By the recurrence relation (2.1), we have (cid:0) S n − S n +1 − S n (cid:1) (cid:0) S n +1 S n +3 − S n +2 (cid:1) − (cid:0) S n S n +2 − S n +1 (cid:1) = S n +1 (cid:0) S n S n +1 S n +2 + S n − S n +1 S n +3 − S n +1 − S n S n +3 − S n − S n +2 (cid:1) = S n +1 (cid:0) a ( n ) S n + a ( n ) S n S n − + a ( n ) S n S n − + a ( n ) S n − (cid:1) . Since { S n } ∞ n =0 is a positive sequence, in order to prove (2.2), it suffices to showthat for all n ≥ N , a ( n ) (cid:18) S n S n − (cid:19) + a ( n ) (cid:18) S n S n − (cid:19) + a ( n ) S n S n − + a ( n ) > . (2.3)Consider the polynomial f ( x ) = a ( n ) x + a ( n ) x + a ( n ) x + a ( n ). Note that f ′ ( x ) = 3 a ( n ) x + 2 a ( n ) x + a ( n ) . Since a ( n ) < n ) > n ≥ N , we see that the quadratic function f ′ ( x ) is negative for x > − a ( n ) − √ ∆( n )6 a ( n ) . Thus, f ( x ) is strictly decreasing on theinterval [ − a ( n ) − √ ∆( n )6 a ( n ) , + ∞ ). From the assumption g n > f n ≥ − a ( n ) − √ ∆( n )6 a ( n ) , itfollows that f ( x ) is strictly decreasing on the interval [ f n , g n ]. Since S n S n − ∈ [ f n , g n ],it remains to show that f ( g n ) > n ≥ N , which is equivalent to condition( C ), that is, a ( n ) g n + a ( n ) g n + a ( n ) g n + a ( n ) > n ≥ N . This completes the proof. (cid:3) WILLIAM Y. C. CHEN AND ERNEST X. W. XIA A heuristic approach to computing the bounds
In this section, we present a procedure to derive a lower bound f n and an upperbound g n for the ratio S n /S n − based on a three-term recurrence relation of S n .We first describe how to obtain an upper bound g n as required in Theorem 2.1. Aswill be seen, this procedure is not guaranteed to give an upper bound g n , but it ispractically valid for many cases.Assume that lim n →∞ b ( n ) = b and lim n →∞ c ( n ) = c , where b and c are two constantsand b + 4 c >
0. All sequences considered in this paper satisfy this condition. Let x = b + √ b + 4 c . (3.1)We begin with the case c ( n ) <
0, and we shall try to construct g n which satisfiesthe condition ( C ) together with the following inequality: g n +1 − (cid:18) b ( n + 1) + c ( n + 1) g n (cid:19) > . (3.2)In fact, the condition (3.2) is essential to find an upper bound g n for S n /S n − .As will be seen in the following lemma, if we find a function g n satisfying (3.2)and S n /S n − < g n for small n , then we can deduce that g n is an upper bound for S n /S n − for any n . Lemma 3.1.
Let S n be the sequence defined by the recurrence relation (2.1) . As-sume that N is a positive integer such that c ( n ) < for n ≥ N . If S N S N − ≤ g N andthe condition (3.2) holds for n ≥ N , then we have for n ≥ N , S n S n − ≤ g n . (3.3) Proof.
We use induction on n . Obviously, the lemma holds for n = N . We assumethat it is true for n = m ≥ N , that is, S m S m − < g m . Since c ( m ) < m ≥ N , wesee that c ( m + 1) S m − S m < c ( m + 1) g m . (3.4)We now consider the case n = m + 1. From (2.1) and (3.4) it follows that S m +1 S m = b ( m + 1) + c ( m + 1) S m − S m ≤ b ( m + 1) + c ( m + 1) g m . (3.5)From (3.2) and (3.5) we deduce that for m ≥ N , g m +1 − S m +1 S m ≥ g m +1 − (cid:18) b ( m + 1) + c ( m + 1) g m (cid:19) > , which is the statement of the lemma for n = m + 1. This completes the proof. (cid:3) Now we present a heuristic procedure to find the desired upper bound g n . Let g n = x as given by (3.1). If g n satisfies the conditions ( C ) and (3.2), then g n is thedesired choice. Otherwise, let g n = x + xn . Substitute g n into (3.2) and let Y ( n )denote the numerator of the left hand side of (3.2), which is often a polynomialin n and x . Setting the coefficient of the highest degree in n of Y ( n ) to be 0, weobtain an equation in x . If x is the unique solution of this equation, then we set g n = x + x n . If g n = x + x n satisfies the conditions ( C ) and (3.2), then g n isthe desired choice. Otherwise, set g n = x + x n + xn and repeat the above process HE 2-LOG-CONVEXITY OF THE AP´ERY NUMBERS 5 to find a solution x of the equation. By iteration, we may find x , x , . . . , x i suchthat g n = x + x n + x n + · · · + x i n i satisfies the conditions ( C ) and (3.2).For example, let S n = A n , where A n is Ap´ery number defined by (1.1). Sincelim n →∞ b ( n ) = 34 and lim n →∞ c ( n ) = −
1, by the definition of A n , we have x = 17 +12 √
2. Since g n = 17 + 12 √ C ) in Theorem 2.1,we further consider g n = 17 + 12 √ xn . Let Y ( n ) denote the numerator of the lefthand side of (3.2). It is easy to see that Y ( n ) is a cubic polynomial in n with theleading coefficient equal to E = − (17 √ − x + 864 √ . Setting E = 0 gives x = − − √
2. Again, g n = x + x n does not satisfy (3.2).So we continue to consider g n = x + x n + xn and we find that x = √ .Now, g n = x + x n + x n does not satisfy the condition ( C ). Repeating the aboveprocedure, we find that x = − √ − and g n = x + x n + x n + x n satisfies(3.2) and the condition ( C ).For the case c ( n ) >
0, we aim to construct an upper bound g n which satisfiescondition ( C ) and the following inequality g n − b ( n ) + c ( n ) b ( n −
1) + c ( n − g n − > . (3.6)Similarly, if we find a function g n satisfying (3.6) and S n /S n − < g n for certain n ,then we can deduce that g n is an upper bound for any n . To be precise, we havethe following lemma. Lemma 3.2.
Let S n be defined by (2.1) . If there exists a positive integer N suchthat the inequality (3.6) holds, S N S N − ≤ g N , S N +1 S N ≤ g N +1 and c ( n ) > for n ≥ N ,then we have for n ≥ N , S n S n − ≤ g n . (3.7) Proof.
We conduct induction on n . Clearly, the lemma holds for n = N and n = N + 1. Assume that it is true for n = m − ≥ N , that is, S m − S m − ≤ g m − . (3.8)We shall show that the lemma is true for n = m , that is, S m S m − ≤ g m . (3.9)Since c ( n ) > n ≥ N , from (2.1) and (3.8) it follows that S m S m − = b ( m ) + c ( m ) S m − S m − = b ( m ) + c ( m ) b ( m −
1) + c ( m − S m − S m − (3.10) ≤ b ( m ) + c ( m ) b ( m −
1) + c ( m − g m − . WILLIAM Y. C. CHEN AND ERNEST X. W. XIA
In view of (3.6) and (3.10), we find that g m − S m S m − ≥ g m − b ( m ) + c ( m ) b ( m −
1) + c ( m − g m − > , which yields (3.9). This completes the proof. (cid:3) Now we can use the same approach as in the case c ( n ) < g n . Moreover, if we have obtain an approximation g n that does not simultaneouslysatisfy (3.2) ((3.6)) and the condition ( C ), instead of going further to update theestimation of g n , we may try to adjust some coefficients to find a desired bound.For example, let S n = B n , where B n is defined by (1.2). At some point, we get g n = 112 + 5 √ −
112 + 5 √ ! n (3.11) + (cid:18) √ (cid:19) n + 125 n +
150 + 23 √ ! n . Here g n satisfies the condition ( C ) in Theorem 2.1, but it fails to satisfy (3.6). Ifwe replace the coefficient in (3.11) by , then the adjusted bound g ′ n satisfiesboth conditions ( C ) and (3.6).To conclude this section, we need to mention that it is much easier to find alower bound f n for the ratio S n /S n − . In many cases, we have f ( n ) = b ( n ) when b ( n ) and c ( n ) are positive for n ≥ N and f n = b ( n ) + c ( n ) when c ( n ) is negativeand S n ≥ S n − for n ≥ N .4. Upper bounds for A n /A n − , B n /B n − and U n /U n − In this section, we shall use the heuristic approach described in the previoussection to find upper bounds for the ratios A n /A n − , B n /B n − and U n /U n − . Lemma 4.1.
Let P ( n ) =17 + 12 √ − (cid:18)
512 + 18 √ (cid:19) n (4.1) + (cid:18)
272 + 60964 √ (cid:19) n − √ ! n . For n ≥ , we have A n A n − < P ( n ) . Proof.
For the Ap´ery numbers A n , we use Lemma 3.1 by setting N = 2 and g n = P ( n ). Evidently, A A < P (2). Also, it is easily checked that P ( n + 1) − (cid:18) (2 n + 1)(17 n + 17 n + 5)( n + 1) − n ( n + 1) P ( n ) (cid:19) = 9(17 − √ n − √ n + 1225)256(256 n − n − √ n + 288 n + 90 √ − n + 1) , which is positive for n ≥
2. By lemma 3.1, we see that P ( n ) is an upper bound for A n /A n − when n ≥
2. This completes the proof. (cid:3)
HE 2-LOG-CONVEXITY OF THE AP´ERY NUMBERS 7
Lemma 4.2.
Let T ( n ) = 112 + 5 √ −
112 + 5 √ ! n (4.2) + (cid:18) √ (cid:19) n + 125 n +
125 + 23 √ ! n . For n ≥ , we have B n B n − < T ( n ) . Proof.
Set N = 20 and g n = T ( n ) in Lemma 3.2. It is easy to check that B B 20. Hence we have11 n − n + 3 n + ( n − n (cid:16) n − n +25( n − + ( n − ( n − T ( n − (cid:17) < T ( n ) . (4.3)In view of Lemma 3.2, we deduce that T ( n ) is an upper bound for B n /B n − when n ≥ (cid:3) Using the same procedure, we find the following upper bound for U n /U n − . Theproof is omitted. Lemma 4.3. Let Q ( n ) =135 + 78 √ − (cid:18) √ (cid:19) n + (cid:18) √ (cid:19) n (4.4) − (cid:18) √ (cid:19) n + (cid:18) √ (cid:19) n . For n ≥ , we have U n U n − < Q ( n ) . WILLIAM Y. C. CHEN AND ERNEST X. W. XIA The 2-log-convexity Based on the criterion given in Theorem 2.1 and the upper bounds obtained inthe previous section, we shall give the proofs of the 2-log-convexity of the sequencesof Ap´ery numbers and other aforementioned combinatorial numbers. Theorem 5.1. The sequence { A n } ∞ n =0 is strictly -log-convex.Proof. We first consider the case n ≥ 2. To apply Theorem 2.1, let b ( n ) = 34 n − n + 27 n − n and c ( n ) = − ( n − n . It is straightforward to check that a ( n ) < n ) > n ≥ 2. Since (cid:18) n − k (cid:19) (cid:18) n − kk (cid:19) ≥ (cid:18) n − k (cid:19) (cid:18) n − kk (cid:19) , we have A n − ≥ A n − . Let f n = 33 n − n + 24 n − n . Thus, by the recurrence relation (1.3), we see that A n A n − = 34 n − n + 27 n − n − ( n − n A n − A n − (5.1) ≥ n − n + 27 n − − ( n − n = f n . Set g n = P ( n ), where P ( n ) is given by (4.1). We proceed to verify the conditions( C ), ( C ) and ( C ) in Theorem 2.1. By (5.1) and Lemma 4.1, we find that f n ≤ A n A n − < g n , which is the condition ( C ). Define R ( n ) = 6 a ( n ) f n + 2 a ( n ). It iseasily checked that R ( n ) = − H ( n ) L ( n ) , where H ( n ) and L ( n ) are polynomials in n and the leading coefficients of H ( n ) and L ( n ) are positive. Hence we deducethat R ( n ) < n ≥ 2. Similarly, define R ( n ) = ∆( n ) − R ( n ), which can berewritten as − H ( n ) L ( n ) where H ( n ) and L ( n ) are polynomials in n and the leadingcoefficients of H ( n ) and L ( n ) are positive. Consequently, we deduce R ( n ) < n ≥ 2. It follows that for n ≥ a ( n ) f n + 2 a ( n ) < − p ∆( n ) , which is equivalent to the following inequality for n ≥ f n > − a ( n ) − p ∆( n )6 a ( n ) . This is exactly the condition ( C ). Finally, it remains to verify the condition ( C ).To this end, we find that a ( n ) g n + a ( n ) g n + a ( n ) g n + a ( n )(5.2) = 9 (cid:16) √ (cid:17) H ( n ) L ( n ) , HE 2-LOG-CONVEXITY OF THE AP´ERY NUMBERS 9 where H ( n ) and L ( n ) are polynomials in n . Observe that the leading coefficientsof H ( n ) and L ( n ) are both positive. This implies that the right hand side of (5.2)is positive for n ≥ 2. Now we are left with the case n = 1, that is( A A − A )( A A − A ) > ( A A − A ) , which can be easily checked. This completes the proof. (cid:3) Theorem 5.2. The sequence { B n } ∞ n =0 is strictly -log-convex.Proof. For n ≥ 20, apply Theorem 2.1 with f n = 11 n − n + 3 n , and g n = T ( n ), where T ( n ) is given by (4.2). Using the argument in the proofof Theorem 5.1, we find that f n and g n satisfy all the conditions in Theorem 2.1.Finally, it is easy to verify that for 1 ≤ n ≤ (cid:0) B n − B n +1 − B n (cid:1) (cid:0) B n +1 B n +3 − B n +2 (cid:1) > (cid:0) B n B n +2 − B n +1 (cid:1) . This completes the proof. (cid:3) Theorem 5.3. The sequence { U n } ∞ n =0 is strictly -log-convex. The above theorem follows from Theorem 2.1 by setting f n = 3(2 n − n − n + 1)(15 n − n + 4) n and setting g n = Q ( n ), where Q ( n ) is given by (4.4). The proof is similar to thatof Theorem 5.1, and it is omitted.Doˇsli´c [7, 8] has proved the log-convexity of several well-known sequences ofcombinatorial numbers such as the Motzkin numbers M n , the Fine numbers F n ,the Franel numbers F (3) n and F (4) n of order 3 and 4, and the large Schr¨oder numbers s n . Based on the recurrence relations satisfied by these numbers, we utilize Theorem2.1 to deduce that these sequences are all strictly 2-log-convex possibly except fora fixed number of terms at the beginning.We conclude this paper with a conjecture concerning the infinite log-convexityof the A´ery numbers. The notion of infinite log-convexity is analogous to that ofinfinite log-concavity introduced by Moll [12]. Given a sequence A = { a i } ≤ i ≤∞ ,define the operator L by L ( A ) = { b i } ≤ i ≤∞ , where b i = a i − a i +1 − a i for i ≥ 1. We say that { a i } ≤ i ≤∞ is k -log-convex if L j ( { a i } ≤ i ≤∞ ) is log-convex for j = 0 , , . . . , k − 1, and that { a i } ≤ i ≤∞ is ∞ -log-convex if L k ( { a i } ≤ i ≤∞ ) is log-convex for any k ≥ Conjecture 5.4. The sequences { A n } ∞ n =0 , { B n } ∞ n =0 , { U n } ∞ n =0 and { s n } ∞ n =0 areinfinitely log-convex. The sequences { M n } ∞ n =0 , { F n } ∞ n =0 , { F (3) n } ∞ n =0 and { F (4) n } ∞ n =0 are k -log-convex for any k ≥ k ) ofterms at the beginning. References 1. S. Ahlgren, S.B. Ekhad, K. Ono and D. Zeilberger, A binomial coefficient identity associatedto a conjecture of Beukers, Electron. J. Combin. 5 (1998), R10.2. R. Ap´ery, Irrationalit´e de ζ (2) et ζ (3). Ast´erisque 61 (1979) 11–13.3. F. Beukers, Some congruences for Ap´ery numbers, J. Number Theory 21 (1985) 141–155.4. F. Beukers, Another congruence for the Ap´ery numbers, J. Number Theory 25 (1987) 201–210.5. S. Chowla, J. Cowles and M. Cowles, Congruence properties of Ap´ery numbers, J. 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