_{1}

^{*}

This paper investigates the log-concavity of the centered *m*-gonal figurate number sequences. The author proves that for
** m≥3**
, the sequence {

**C**(

_{n}**)}**

*m***of centered**

_{n≥1}*m*-gonal figurate numbers is a log-concave.

For

where

where

Some scholars have been studying the log-concavity (or log-convexity) of different numbers sequences such as Fibonacci & Hyperfibonacci numbers, Lucas & Hyperlucas numbers, Bell numbers, Hyperpell numbers, Motzkin numbers, Fine numbers, Franel numbers of order 3 & 4, Apéry numbers, Large Schröder numbers, Central Delannoy numbers, Catalan-Larcombe-French numbers sequences, and so on (see for instance [

To the best of the author’s knowledge, among all the aforementioned works on the log-concavity and log- convexity of number sequences, no one has studied the log-concavity (or log-convexity) of centered m-gonal figurate number sequences. In [

Definition 1. Let

Definition 2. Let

Definition 3. Let

Log-concavity and log-convexity are important properties of combinatorial sequences and they play a crucial role in many fields, for instance economics, probability, mathematical biology, quantum physics and white noise theory [

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

Theorem 4. For

with the initial conditions

with the initial condition

Proof. By (1), we have

It follows that

Rewriting (5) and (6) for

Multiplying (7) by

By denoting

and

one can obtain

with given initial conditions

By dividing (10) through by

with initial condition

Lemma 5. For the centered m-gonal figurate number sequence

Proof. Assume

It follows that

Assume that

For

Hence

Similarly, it is known that

Assume that

For

Hence

Thus, in general, from the above two cases it follows that

Lemma 6. For the centered m-gonal figurate number sequence

Proof. Let

By using (11), one can obtain

with initial condition

For

By Lemma 5 and induction assumption, one can get

Thus, the sequence

Theorem 7 For

Proof. Let

By Lemma 6, the quotient sequence

In this paper, we have discussed the log-behavior of centered m-gonal figurate number sequences. We have also proved that for

The author is grateful to the anonymous referees for their valuable comments and suggestions.

Fekadu Tolessa Gedefa, (2016) Log-Concavity of Centered Polygonal Figurate Number Sequences. Open Access Library Journal,03,1-5. doi: 10.4236/oalib.1102774