A292828 -id:A292828 - cp's OEIS frontend

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

Jean-François Alcover, Jun 19 2015

			0.954918789987674103751233978110291077632715373807805283148799191676094...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

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.

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

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)).
Equals 1/exp(A255695). - Hugo Pfoertner, May 28 2025

A292820 Decimal expansion of Product_{k>=1} (1 + exp(-Pi*k)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 24 2017

			1.045250214354711942547595012203562068003424782155586915500520985257117...

Cf. A259148, A292824, A292828, A292819, A292821, A292822.

RealDigits[E^(Pi/24) / 2^(1/8), 10, 120][[1]]
RealDigits[QPochhammer[-1, E^(-Pi)]/2, 10, 120][[1]]

exp(Pi/24)/sqrtn(2,8) \\ Charles R Greathouse IV, Mar 13 2018

Equals exp(Pi/24) / 2^(1/8).

Equals A259149 / A259148.

A292824 Decimal expansion of Product_{k>=1} (1 - exp(-Pi(2k-1))).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 24 2017

			0.956708725113587003449038717361890724715615702454393013400916466527611...

Cf. A259148, A292820, A292828.

Cf. A292823, A292825, A292826.

RealDigits[2^(1/8) * E^(-Pi/24), 10, 120][[1]]

Equals 2^(1/8) * exp(-Pi/24).

Equals A259148 / A259149.

A292827 Decimal expansion of Product_{k>=1} (1 + exp(-Pi(2k-1)/2)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 24 2017

			1.219224801998773577179427484349023235631844499700916838279952481630782...

Cf. A259147, A292819, A292823.

Cf. A292828, A292829.

RealDigits[2^(1/16) / (E^(Pi/48) * (Sqrt[2]-1)^(1/4)), 10, 120][[1]]

Equals 2^(1/16) / (exp(Pi/48) * (sqrt(2)-1)^(1/4)).

Equals A292819 / A292820.

A292829 Decimal expansion of Product_{k>=1} (1 + exp(-2Pi(2*k-1))).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 24 2017

			1.001867449256304435105432750510732150159885395483789405664498761179818...

Cf. A259149, A292821, A292825.

Cf. A292827, A292828.

RealDigits[2^(1/16) / (E^(Pi/12) * (Sqrt[2]-1)^(1/4)), 10, 120][[1]]

Equals 2^(1/16) / (exp(Pi/12) * (sqrt(2)-1)^(1/4)).

Equals A292821 / A292822.

A354296 Decimal expansion of Product_{k>=1} (1 - exp(-2kPi/sqrt(3))).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 23 2022

Note that Prudnikov incorrectly give this product as 3^(1/4)*exp(-Pi*sqrt(3)/18), which differs from the correct result by 0.0000182...

			0.972713586936242371513055024334538082849547588619101318683993472802594...

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 757, section 6.2.3, incorrect formula 4.

Eric Weisstein's World of Mathematics, Infinite Product, formula 52.

Cf. A292828.

evalf(Product(1 - exp(-2*k*Pi/sqrt(3)), k = 1..infinity), 105);

RealDigits[QPochhammer[E^(-2*Pi/Sqrt[3])], 10, 105][[1]]

prodinf(k=1, (1 - exp(-2*k*Pi/sqrt(3))))

cp's OEIS Frontend

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

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A292820 Decimal expansion of Product_{k>=1} (1 + exp(-Pi*k)).

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A292824 Decimal expansion of Product_{k>=1} (1 - exp(-Pi*(2*k-1))).

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A292827 Decimal expansion of Product_{k>=1} (1 + exp(-Pi*(2*k-1)/2)).

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A292829 Decimal expansion of Product_{k>=1} (1 + exp(-2*Pi*(2*k-1))).

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A354296 Decimal expansion of Product_{k>=1} (1 - exp(-2*k*Pi/sqrt(3))).

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A292824 Decimal expansion of Product_{k>=1} (1 - exp(-Pi(2k-1))).

A292827 Decimal expansion of Product_{k>=1} (1 + exp(-Pi(2k-1)/2)).

A292829 Decimal expansion of Product_{k>=1} (1 + exp(-2Pi(2*k-1))).

A354296 Decimal expansion of Product_{k>=1} (1 - exp(-2kPi/sqrt(3))).