A392020 Expansion of the series e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of Euler's number e.

1, -1, 0, -1, 0, 1, 2, 3, 3, 1, -3, -9, -17, -25, -29, -23, 2, 54, 134, 232, 320, 347, 243, -71, -660, -1531, -2575, -3504, -3804, -2747, 488, 6537, 15395, 25819, 34716, 36780, 24771, -9096, -70197, -156811, -254178, -328753, -325431, -171870, 205308, 848411, 1719877, 2642829, 3247519, 2953230, 1027655, -3228913, -10138114

Offset

0,7

Comments

Morier-Genoud and Ovsienko associate to each positive real number alpha a formal power series (in the variable q) with integer coefficients, which we denote by alpha(q). They interpret this series as a q-analog or q-deformation of the real number alpha. The series alpha(q) is obtained from a q-analog of the simple contiued fraction expansion of alpha. For example, the g.f. of A337589 is the q-deformation of sqrt(2) and the coefficients in the q-deformation of the golden ratio phi are the terms of A004148 but with alternating signs.

The series e(q), associated with Euler's number e = exp(1), begins 1 + q + q^3 - q^5 + 2*q^6 - 3*q^7 + 3*q^8 - q^9 - 3*q^10 + 9*q^11 - 17*q^12 + 25*q^13 - 29*q^14 + 23*q^15 + ....

Here we consider the expansion of e(-q). Notice the unusual pattern in the signs of the coefficients of e(-q), which eventually appear to consist of a group of either 6 or 7 consecutive positive coefficients followed by a group of either 6 or 7 consecutive negative coefficients. Empirically, sign(a(n)) - sign(a(n+14)) is frequently zero. Similar behavior is seen in the coefficients of the power series 1/e(-q) (A392021), e(q)*e(-q) (A392022) and e(q)/e(-q) (A392023).

Links

Table of n, a(n) for n=0..52.

Peter Bala, The sign pattern of the terms of A392020.

Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365v3 [math.QA], Oct 2019.

Maple

# uses 2 X 2 matrices to determine rational function approximations to 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 e(-q)
S := series( simplify(Mat_prod(14, -q)[1, 1])/simplify(Mat_prod(14, -q)[2, 1]), q, 101):
seq(coeftayl(S, q = 0, n), n = 0..100);

Crossrefs

Cf. A001113, A003417, A004148, A337589, A392021 (1/e(-q)), A392022 (e(q)*e(-q)), A392023 (e(q)/e(-q)).

Sequence in context: A308734 A279004 A172528 * A087074 A136453 A114104

Adjacent sequences: A392017 A392018 A392019 * A392021 A392022 A392023

Keyword

sign,easy

Author

Peter Bala, Dec 27 2025

Status

approved