An Euler-Like Product with Fibonacci Exponents

The pentagonal theorem for partitions is a consequence of the expansion of Euler’s famous product ( 1 - y ) ( 1 - y 2 ) ( 1 - y 3 ) ( 1 - y 4 ) ( 1 - y 5 ) ⋯ We investigate the nature of the coefficients of the series expansion of ( 1 - y ) ( 1 - y 2 ) ( 1 - y 3 ) ( 1 - y 5 ) ( 1 - y 8 ) ⋯ , in whic...

Full description

Saved in:
Bibliographic Details
Published inLa matematica Vol. 3; no. 2; pp. 677 - 703
Main Author Eğecioğlu, Ömer
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2024
Subjects
Mathematics
Mathematics and Statistics
Original Research Article
Fibonacci exponents
Euler’s product
series expansion
Fibonacci number
Online AccessGet full text
ISSN2730-9657
2730-9657
DOI10.1007/s44007-024-00099-w

Cover

More Information
Summary:The pentagonal theorem for partitions is a consequence of the expansion of Euler’s famous product ( 1 - y ) ( 1 - y 2 ) ( 1 - y 3 ) ( 1 - y 4 ) ( 1 - y 5 ) ⋯ We investigate the nature of the coefficients of the series expansion of ( 1 - y ) ( 1 - y 2 ) ( 1 - y 3 ) ( 1 - y 5 ) ( 1 - y 8 ) ⋯ , in which the sequence of exponents is the Fibonacci numbers. As a part of the study of the combinatorial properties of the development of this product, we show that the series expansion coefficients are from { - 1 , 0 , 1 } , and their behavior is determined by a monoid of twenty-five 2 × 2 matrices.
ISSN:2730-9657
2730-9657
DOI:10.1007/s44007-024-00099-w