OFFSET
0,3
COMMENTS
Notice the unusual pattern in the signs of the coefficients of the series e(q)*e(-q), which, starting at q^3, appears to consist of a group of either 3 or 4 consecutive positive coefficients followed by a group of either 3 or 4 consecutive negative coefficients. Empirically, sign(a(n)) - sign(a(n+7)) is frequently zero. Similar behavior is seen in the coefficients of the power series e(-q) (A392020), 1/e(-q) (A392021) and e(q)/e(-q) (A392023).
LINKS
Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365v3 [math.QA], Oct 2019.
EXAMPLE
e(q)*e(-q) = 1 - q^2 - 2*q^4 + 5*q^6 + 14*q^8 + q^10 - 52*q^12 - 125*q^14 - 83*q^16 + 347*q^18 + 1217*q^20 + 1579*q^22 - ....
MAPLE
# uses 2 X 2 matrices to determine rational function approximations to e(q)*e(-q) - see Morier-Genoud and Ovsienko, Section 2.4
# calculate the partial quotients of the simple continued fraction expansion of e
c := proc(n) if n = 1 then 2 elif irem(n, 3) = 0 then (2/3)*n else 1 end if end proc:
# define the q-analog of the integer n
f(n, q) := simplify( (q^n - 1)/(q - 1) ):
with(LinearAlgebra):
Mat(n, q) := Matrix( [[f(c(2*n-1), q), q^(c(2*n-1))], [1, 0]] ).Matrix([[q*f(c(2*n), q), 1], [q^(c(2 n)), 0]] ):
Mat_prod := proc(n, q) option remember; if n = 1 then Mat(1, q) else simplify( Mat_prod(n-1, q).Mat(n, q) ) end if end proc:
# series expansion of convolution e(q)*e(-q)
S := series( simplify(Mat_prod(20, q)[1, 1])/simplify(Mat_prod(20, q)[2, 1]) * simplify(Mat_prod(20, -q)[1, 1])/simplify(Mat_prod(20, -q)[2, 1]), q, 202 ):
seq(coeftayl(S, q = 0, 2*n), n = 0..100);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Jan 05 2026
STATUS
approved