On The Log-Concavity of Polygonal Figurate Number Sequences

TL;DR

This paper proves that the sequences of m-gonal figurate numbers are log-concave for all m ≥ 3, providing recurrence formulas and analyzing quotient bounds to establish this property.

Contribution

It introduces and proves the log-concavity of m-gonal figurate number sequences, including recurrence relations and quotient sequence bounds.

Findings

m-gonal figurate number sequences are log-concave for m ≥ 3

Recurrence formulas for these sequences are established

Quotient sequences are shown to be bounded

Abstract

This paper presents the log-concavity of the $m$-gonal figurate number sequences. The author gives and proves the recurrence formula for $m$-gonal figurate number sequences and its corresponding quotient sequences which are found to be bounded. Finally, the author also show that for $m\geq 3$, the sequence $\big \{S_n(m)\big\}_{n\geq 1}$ of $m$-gonal figurate numbers is a log-concave.