Divisors on Overlapped Intervals and Multiplicative Functions
José Manuel Rodríguez Caballero
Département de Mathématiques
Case Postale 8888, Succ. Centre-ville
Montréal, Québec H3C 3P8
Reutenauer and Kassel introduced a family
Pn(q) of polynomials
defined in terms of divisors of n on overlapped intervals. The
evaluation of Pn(q)
at roots of unity of order 2, 3, 4, 6 form
well-known integer sequences related to the number of integer solutions
of the equations x2 + y2 = n,
x2 + 2y2 = n,
and x2 + xy + y2
= n. Also, Pn(1) is the
sum of divisors of n.
In this paper we
define a new family Ln(q)
of polynomials defined in terms of
divisors of n on overlapped intervals,
slightly modifying the
definition of Pn(q).
The values of Ln(q)
at q = 1 and q = -1
are related to the sum of divisors of n and to the number of integer
solutions of the equations
x2 + xy + y2 = n
and x2 + 3 y2 = n.
Full version: pdf,
(Concerned with sequences
Received September 16 2017;
revised versions received October 13 2017; October 21 2017.
Published in Journal of Integer Sequences, October 29 2017.
Journal of Integer Sequences home page