A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

9, 5, 4, 9, 1, 8, 7, 8, 9, 9, 8, 7, 6, 7, 4, 1, 0, 3, 7, 5, 1, 2, 3, 3, 9, 7, 8, 1, 1, 0, 2, 9, 1, 0, 7, 7, 6, 3, 2, 7, 1, 5, 3, 7, 3, 8, 0, 7, 8, 0, 5, 2, 8, 3, 1, 4, 8, 7, 9, 9, 1, 9, 1, 6, 7, 6, 0, 9, 4, 0, 3, 5, 6, 8, 6, 7, 1, 4, 5, 3, 9, 5, 3, 4, 9, 8, 1, 5, 1, 8, 6, 7, 4, 4, 6, 1, 0, 9, 8, 7, 6, 7, 4, 9

Offset

0,1

Links

Table of n, a(n) for n=0..103.

Istvan Mezo, Several special values of Jacobi theta functions arXiv:1106.2703v3 [math.CA] 24 Sep 2013

Eric Weisstein's MathWorld, Infinite Product

Eric Weisstein's MathWorld, Jacobi Theta Functions

Eric Weisstein's MathWorld, q-Pochhammer Symbol

Wikipedia, Euler function

Formula

phi(q) = QPochhammer(q,q) = (q;q)_infinity.

phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.

phi(exp(-Pi)) = exp(Pi/24)*Gamma(1/4)/(2^(7/8)*Pi^(3/4)).

Example

0.954918789987674103751233978110291077632715373807805283148799191676094...

Mathematica

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi]], 10, 104] // First

Crossrefs

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292820, A292824, A292828.

Sequence in context: A359282 A198933 A353772 * A089491 A199792 A193960

Adjacent sequences: A259145 A259146 A259147 * A259149 A259150 A259151

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jun 19 2015

Status

approved