On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers

Li-Na Zheng and Rui Liu School of Mathematical Sciences Dalian University of Technology Dalian 116024 P. R. China

              linazheng2007@yahoo.com.cn

Feng-Zhen Zhao Department of Mathematics Shanghai University Shanghai 200444 P. R. China

                fengzhenzhao@shu.edu.cn

Abstract

In this paper, we discuss the properties of the hyperfibonacci numbers F_{n}^{[r]} and hyperlucas numbers L_{n}^{[r]}. We investigate the log-concavity (log-convexity) of hyperfibonacci numbers and hyperlucas numbers. For example, we prove that {F_{n}^{[r]}}_{n≥1} is log-concave. In addition, we also study the log-concavity (log-convexity) of generalized hyperfibonacci numbers and hyperlucas numbers.

1 Introduction

Let {F_{n}}_{n≥0} and {L_{n}}_{n≥0} denote the Fibonacci and Lucas sequence, respectively. It is well known that the Binet forms of F_{n} and L_{n} are

F_{n} = \frac{α^{n} − (−1)^{n}α^{−n}}{\sqrt{5}} , L_{n} = α^{n} + (−1)^{n}α^{−n},  (1)

where α = (1 + \sqrt{5})/2. It is evident that \{F_{n}\}_{n≥0} and \{L_{n}\}_{n≥0} satisfy

W_{n} = W_{n−1} + W_{n−2}, n ≥ 2.  (2)

For positive integers r, hyperfibonacci numbers F_{n}^{[r]} and hyperlucas numbers L_{n}^{[r]} are defined as follows [5]:

F_{n}^{[r]} = ∑_{k=0}^{n} F_{k}^{[r−1]}, L_{n}^{[r]} = ∑_{k=0}^{n} L_{k}^{[r−1]},

where F_{n}^{[0]} = F_{n} and L_{n}^{[0]} = L_{n}. Initial values of {F_{n}^{[1]}} and {L_{n}^{[1]}} are as follows:

\begin{matrix}n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ F_{n}^{[1]} & 0 & 1 & 2 & 4 & 7 & 12 & 20 & 33 & 54 & 88 & 143 & 232 & 376 & 609 & 986 \\ L_{n}^{[1]} & 2 & 3 & 6 & 10 & 17 & 28 & 46 & 75 & 122 & 198 & 321 & 520 & 842 & 1363 & 2206 \end{matrix}

In [3, 5, 9], some properties of hyperfibonacci numbers F_{n}^{[r]} and hyperlucas numbers L_{n}^{[r]} are given. In this paper, we continue discussing the properties of F_{n}^{[r]} and L_{n}^{[r]}. Now we recall some other definitions involved in this paper.

Definition 1. Let {a_{n}}_{n≥0} be a sequence of positive numbers. If for all j ≥ 1, a_{j}^{2} ≥ a_{j−1}a_{j+1} (respectively a_{j−1}a_{j+1} ≥ a_{j}^{2}), the sequence {a_{n}}_{n≥0} is called log-concave (respectively logconvex).

Definition 2. [6] Let {a_{n}}_{n≥0} be a sequence of positive real numbers. We say that {a_{n}}_{n≥0} is log-balanced if \{a_{n}\}_{n≥0} is log-convex and \{\frac{a_{n}}{n!} \}_{n≥0} is log-concave.

The log-convexity and log-concavity are important properties of combinatorial sequences, and they play an important role in many fields such as quantum physics, white noise theory, probability, economics and mathematical biology. See for instance [1, 2, 6, 7, 8, 10, 11]. Clearly, log-balancedness implies log-convexity. It is well known that {a_{n}}_{n≥0} is log-convex (log-concave) if and only if its quotient sequence {a_{n+1}/a_{n}}_{n≥0} is nondecreasing (nonincreasing). Naturally, the quotient sequence of a log-balanced sequence does not grow too fast. For the Fibonacci sequence {F_{n}} and Lucas sequence {L_{n}}, their log-concavity (log-convexity) are related to the parity of n. It is well known that {F_{2n+1}} and {L_{2n}} are log-convex and {F_{2n}} and {L_{2n+1}} are log-concave. In this paper, we discuss the log-concavity (logconvexity) of hyperfibonacci numbers F_{n}^{[r]} and hyperlucas numbers L_{n}^{[r]}. In Section 2, we show that {F_{n}^{[r]}}_{n≥1} and {L_{n}^{[r]}}_{n≥3} are log-concave for r ≥ 1. In addition, we also consider the log-concavity ( log-convexity) of generalized hyperfibonacci numbers and generalized hyperlucas numbers.

2 The log-concavity of hyperfibonacci numbers and hyperlucas numbers

In this section, we state and prove the main results of this paper.

Lemma 3. [4, 12] If the sequences {x_{n}} and {y_{n}} are log-concave, then so is their ordinary convolution z_{n} = ∑_{k=0}^{n} x_{k}y_{n−k}, n = 0, 1, 2, ….

We note that above {F_{n}^{[r]}}_{n≥1} is the convolution of {F_{n}^{[r]}}_{n≥1} and {1}_{n≥0}.

Lemma 4. For n ≥ 0, the following equalities hold:

L_{n+2}^{2} − L_{n+1}L_{n+3} = 5(−1)^{n}, F_{n+2}^{2} − F_{n+1}F_{n+3} = (−1)^{n+1}

Proof. From (1), we can immediately prove that this lemma holds.

Theorem 5. For r ≥ 1, the sequences {F_{n}^{[r]}}_{n≥1}, {L_{n}^{[1]}}_{n≥3}, and {L_{n}^{[r]}}_{n≥0}(r ≥ 2) are logconcave.

Proof. A simple induction using the defining recurrence (2) shows that

F_{n}^{[1]} = F_{n+2} − 1, L_{n}^{[1]} = L_{n+2} − 1.  (3)

By using (3) and (2), we can verify that {F_{n}^{[1]}} and {L_{n}^{[1]}} satisfy the following recurrence relation

W_{n+1} = 2W_{n} − W_{n−2}, n ≥ 2.  (4)

For n ≥ 1, let x_{n} = F_{n+1}^{[1]}/F_{n}^{[1]}. We prove by induction that {x_{n}}_{n≥1} is decreasing. Clearly, x_{1} = x_{2} = 2 > x_{3} = 7/4. For n ≥ 3, assume that x_{k−1} ≥ x_{k} for all 1≤ k ≤ n. It follows from (4) that

x_{n} = 2 − \frac{1}{x_{n−1}x_{n−2}} .

Then we have

x_{n} − x_{n+1} = \frac{x_{n−2} − x_{n}}{x_{n−2}x_{n−1}x_{n}} .

Since x_{n−2} ≥ x_{n−1} ≥ x_{n}, it follows that x_{n} ≥ x_{n+1}. Then {x_{n}}_{n≥1} is decreasing and {F_{n}^{[1]}}_{n≥1} is log-concave. For {L_{n}^{[1]}}_{n≥0}, using a similar method, we can prove that {L_{n}^{[1]}}_{n≥3} is logconcave. It is clear that the initial cases (0 ≤ n ≤ 2) of {L_{n}^{[1]}}_{n≥0} are not log-concave. According to Lemma 3, we know that {F_{n}^{[r]}}_{n≥1} is log-concave. Now we show that {L_{n}^{[2]}}_{n≥0} is log-concave. We can verify that

L_{n}^{[2]} = L_{n+4} − n − 5.  (5)

By using (5), (2) and Lemma 4, we get

(L_{n}^{[2]})^{2} − L_{n−1}^{[2]}L_{n+1}^{[2]} = (L_{n+4} − n − 5)^{2} − (L_{n+3} − n − 4)(L_{n+5} − n − 6)
= L_{n+4}^{2} − L_{n+3}L_{n+5} + nL_{n+1} + 2L_{n−1} + 1
= 5(−1)^{n} + nL_{n+1} + 2L_{n−1} + 1.

When n ≥ 1,

(L_{n}^{[2]})^{2} − L_{n−1}^{[2]}L_{n+1}^{[2]} ≥ 3n − 2 > 0.

Then {L_{n}^{[2]}}_{n≥0} is log-concave. According to Lemma 3, we know that {L_{n}^{[r]}}_{n≥0}(r ≥ 3) is log-concave.

Theorem 6. The sequences {n!F_{n}^{[1]}}_{n≥1} and {n!L_{n}^{[1]}}_{n≥3} are log-balanced.

Proof. By Theorem 5, in order to prove the log-balancedness of {n!F_{n}^{[1]}}_{n≥1} and {n!L_{n}^{[1]}}_{n≥3}, we only need to show that they are log-convex. It follows from (3), Lemma 4 and (2) that

(F_{n}^{[1]})^{2} − F_{n−1}^{[1]}F_{n+1}^{[1]} = (−1)^{n+1} + F_{n−1},  (6)
(L_{n}^{[1]})^{2} − L_{n−1}^{[1]}L_{n+1}^{[1]} = 5(−1)^{n} + L_{n−1}.  (7)

From (3), (6) and (7), we have

n(F_{n}^{[1]})^{2} − (n + 1)F_{n−1}^{[1]}F_{n+1}^{[1]} = (n + 1)[(−1)^{n+1} + F_{n−1}] − (F_{n+2} − 1)^{2},
n(L_{n}^{[1]})^{2} − (n + 1)L_{n−1}^{[1]}L_{n+1}^{[1]} = (n + 1)[5(−1)^{n} + L_{n−1}] − (L_{n+2} − 1)^{2}.

Let S_{n} = (n + 1)[(−1)^{n+1} + F_{n−1}] − (F_{n+2} − 1)^{2}, T_{n} = (n + 1)[5(−1)^{n} + L_{n−1}] − (L_{n+2} − 1)^{2}. Clearly, S_{k} < 0 for 2 ≤ k ≤ 5 and T_{k} < 0 for k = 4 or k = 5. We can prove by induction that

F_{n} ≥ n, n ≥ 5,  (8)
L_{n} ≥ n, n ≥ 0.  (9)

For n ≥ 6, by applying (8), (9) and (2), we obtain

S_{n} ≤ (n + 1)(1 + F_{n−1}) − (n + 1)(F_{n+2} − 1)
= (n + 1)(2 − 2F_{n})
< 0,
T_{n} ≤ (n + 1)(6 − 2L_{n})
< 0.

Noting that

(n!F_{n}^{[1]})^{2} − (n − 1)!(n + 1)!F_{n−1}^{[1]}F_{n+1}^{[1]} = (n − 1)!n!S_{n},
(n!L_{n}^{[1]})^{2} − (n − 1)!(n + 1)!L_{n−1}^{[1]}L_{n+1}^{[1]} = (n − 1)!n!T_{n},

we have

(n!F_{n}^{[1]})^{2} − F_{n−1}^{[1]}F_{n+1}^{[1]} < 0 for n ≥ 2; (n!L_{n}^{[1]})^{2} − L_{n−1}^{[1]}L_{n+1}^{[1]} < 0 for n ≥ 4.

Hence {n!F_{n}^{[1]}}_{n≥1} and {n!L_{n}^{[1]}}_{n≥3} are log-convex. As the sequences {F_{n}^{[1]}}_{n≥1} and {L_{n}^{[1]}}_{n≥3} are log-concave, the sequences {n!F_{n}^{[1]}}_{n≥1} and {n!L_{n}^{[1]}}_{n≥3} are log-balanced.

In the final part of this section, we discuss the log-concavity (log-convexity) of generalized hyperfibonacci numbers and hyperlucas numbers. Let {U_{n}}_{n≥0} and {V_{n}}_{n≥0} denote the generalized Fibonacci and Lucas sequence, respectively. Their Binet forms of U_{n} and V_{n} are

U_{n} = \frac{τ^{n} − (−1)^{n}τ^{−n}}{\sqrt{Δ}} , V_{n} = τ^{n} + (−1)^{n}τ^{−n},  (10)

where Δ = p^{2} + 4, τ = (p + \sqrt{Δ})/2, and p ≥ 1. It is clear that (10) is a generalization of (1), and {U_{n}}_{n≥0} and {V_{n}}_{n≥0} satisfy the recurrence

W_{n+1} = pW_{n} + W_{n−1} (n ≥ 1), with U_{0} = 0, U_{1} = 1, V_{0} = 2, V_{1} = p.  (11)

For positive integers r, the generalized hyperfibonacci numbers U_{n}^{[r]} and generalized hyperlucas numbers V_{n}^{[r]} are defined as follows:

U_{n}^{[r]} = ∑_{k=0}^{n} U_{k}^{[r−1]}, V_{n}^{[r]} = ∑_{k=0}^{n} V_{k}^{[r−1]},

where U_{n}^{[0]} = U_{n} and V_{n}^{[0]} = V_{n}. For some properties of {U_{n}^{[r]}} and {V_{n}^{[r]}}, see [3, 9]. Now we prove the log-concavity of {U_{n}^{[r]}} and {V_{n}^{[r]}}.

Theorem 7. For r ≥ 1 and p ≥ 1, the sequences {U_{n}^{[r]}}_{n≥1} and {V_{n}^{[1]}}_{n≥3} are log-concave.

Proof. From the definitions of {U_{n}^{[1]}} and {V_{n}^{[1]}} and (11), we can verify that

W_{n+1} = (1 + p)W_{n} + (1 − p)W_{n−1} − W_{n−2}, n ≥ 2.  (12)

We first show that {U_{n}^{[1]}}_{n≥1} is log-concave. Let x_{n} = U_{n+1}^{[1]}/U_{n}^{[1]} for n ≥ 1. It follows from (12) that

x_{n} = 1 + p + \frac{1 − p}{x_{n−1}} − \frac{1}{x_{n−1}x_{n−2}} , n ≥ 3.

Then we have

x_{n+1} − x_{n} = \frac{(p − 1)(x_{n} − x_{n−1})}{x_{n−1}x_{n}} + \frac{x_{n} − x_{n−2}}{x_{n−2}x_{n−1}x_{n}} , n ≥ 3.  (13)

Now we prove by induction that {x_{n}}_{n≥1} is decreasing. It is clear that

U_{1}^{[1]} = 1, U_{2}^{[1]} = p + 1, U_{3}^{[1]} = p^{2} + p + 2, U_{4}^{[1]} = p^{3} + p^{2} + 3p + 2,

x_{1} = p + 1 ≥ x_{2} = p + \frac{2}{p + 1} > x_{3} = p + \frac{p + 2}{p^{2} + p + 2}.

Assume that x_{k−1} ≥ x_{k} for all 2 ≤ k ≤ n. It follows from this assumption and (13) that x_{n} ≥ x_{n+1} for n ≥ 1. Hence {x_{n}}_{n≥1} is decreasing, and {U_{n}^{[1]}}_{n≥1} is log-concave. Using this method, we can also prove that {V_{n}^{[1]}}_{n≥3} is log-concave. It follows from Lemma 3 that {U_{n}^{[r]}}_{n≥1} is log-concave for r ≥ 2.

Theorem 8. For p ≥ 1, the sequences {n!U_{n}^{[1]}}_{n≥5} and {n!V_{n}^{[1]}}_{n≥5} are log-balanced.

Proof. By (10), we derive

U_{n}^{[1]} = \frac{U_{n} + U_{n+1} − 1}{p}, V_{n}^{[1]} = \frac{V_{n} + V_{n+1} + p − 2}{p}.  (14)

By (14) and (11), we can verify that

n(U_{n}^{[1]})^{2} − (n + 1)U_{n−1}^{[1]}U_{n+1}^{[1]} = \frac{(n + 1)[(−1)^{n+1} + U_{n+1} − U_{n}]}{p} − \frac{(U_{n} + U_{n+1} − 1)U_{n}^{[1]}}{p} ,
n(V_{n}^{[1]})^{2} − (n + 1)V_{n−1}^{[1]}V_{n+1}^{[1]} = \frac{(n + 1)[(−1)^{n}Δ − (p − 2)(V_{n+1} − V_{n})]}{p}

−\frac{(V_{n+1} + V_{n} + p − 2)V_{n}^{[1]}}{p} .

We can prove that

U_{n}^{[1]} ≥ n + 1, n ≥ 4,
V_{n}^{[1]} ≥ n + 1, n ≥ 4.

On the other hand,

U_{n} ≥ 1, n ≥ 1,
V_{n} ≥ p^{2} + 2, n ≥ 2.

Then we get

n(U_{n}^{[1]})^{2} − (n + 1)U_{n−1}^{[1]}U_{n+1}^{[1]} ≤ \frac{2(n + 1)(1 − U_{n})}{p}
≤ 0,
n(V_{n}^{[1]})^{2} − (n + 1)V_{n−1}^{[1]}V_{n+1}^{[1]} ≤ \frac{(n + 1)[Δ − (p − 2)(V_{n+1} − V_{n}) − (V_{n+1} + V_{n} + p − 2)]}{p}
= \frac{(n + 1)[p^{2} − p + 6 − (p − 1)(V_{n+1} − V_{n}) − 2V_{n}]}{p}
≤ \frac{(n + 1)[−p^{2} − p + 2 − (p − 1)(V_{n+1} − V_{n})]}{p}
≤ 0.

Then the sequences {n!U_{n}^{[1]}}_{n≥5} and {n!V_{n}^{[1]}}_{n≥5} are log-convex for p ≥ 1. Hence the sequences {n!U_{n}^{[1]}}_{n≥5} and {n!V_{n}^{[1]}}_{n≥5} are log-balanced.

3 Conclusions

We have discussed the log-concavity of hyperfibonacci numbers and hyperlucas numbers. Our next work is to study the log-concavity (log-convexity) of various recurrence sequences appearing in combinatorics.

4 Acknowledgment

The authors are grateful to the anonymous referees for their valuable comments and suggestions.

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2010 Mathematics Subject Classification: Primary 05A20; Secondary 11B37, 11B39. Keywords: log-convexity, log-concavity, Fibonacci number, Lucas number.

(Concerned with sequences \underline{A000071} and \underline{A001610}.)

Received March 17 2013; revised versions received November 9 2013; November 24 2013. Published in Journal of Integer Sequences, December 16 2013.

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