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Melham's conjecture on odd power sums of fibonacci numbers
- Source :
- Quaestiones Mathematicae; Vol 39, No 7 (2016); 945-957
- Publication Year :
- 2016
- Publisher :
- Taylor & Francis, 2016.
-
Abstract
- Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at $1$, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.<br />Comment: 15pages
- Subjects :
- Discrete mathematics
Fibonacci number
Lucas sequence
010102 general mathematics
Reciprocal Fibonacci constant
11B39, 05A19
02 engineering and technology
Pisano period
01 natural sciences
Combinatorics
Mathematics (miscellaneous)
Fibonacci numbers, Lucas numbers, Fibonacci polynomials, Lucas polynomials, Melham's conjecture, the Ozeki-Prodinger formula
Integer
Lucas number
Fibonacci polynomials
0202 electrical engineering, electronic engineering, information engineering
FOS: Mathematics
Mathematics - Combinatorics
020201 artificial intelligence & image processing
Combinatorics (math.CO)
0101 mathematics
Fibonacci prime
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 16073606 and 1727933X
- Database :
- OpenAIRE
- Journal :
- Quaestiones Mathematicae
- Accession number :
- edsair.doi.dedup.....0a8ff956fef40cd9052f6f64b3da136e