A274662 -id:A274662 - cp's OEIS frontend

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2cos(v), with v = u/((2/Pi)K(k)).

1, 8, 1, -32, 11, 3, -736, -92, 9, 15, 2816, -593, -249, -65, 35, 48976, 6122, 1581, -970, -1295, 315, -951424, 61252, 67791, 46030, 18515, -21735, 3465, -1045952, -130744, -92082, -30445, 14455, 53928, -25179, 3003, 26933248, 1069361, -1666047, -634255, -1167740, -1258236, 1562253, -471471, 45045, 634836808, 79354601, 24881793, 17914550, 30289840, 12635028, -71064609, 42480438, -9594585, 765765

Offset: 0

Wolfdieter Lang, Aug 09 2016

The dimensionless scaled phase space coordinates of the plane pendulum are (qtilde(tau, k), ptilde(tau, k)) with tau = omega_0*t, omega^2 = g/L (L is the  length of the pendulum, g the acceleration), and the energy variable E = 2*k^2 = 2*sin^2(Theta_0/2), with the maximal deflection angle Theta_0 (from [0, Pi/2]). qtilde = Theta/(2*k)) with the deflection angle Theta. Similarly ptilde = (d(Theta)/d(tau))/(2*k).
The exact solution is qtilde(tau, k) = (1/k)*arcsin(k*sn(tau, k)) with Jacobi's elliptic sn function, and ptilde(tau,k) = cn(tau, k) with the elliptic cn function.
Here the expansion in new variables v and q is used where v = tau/((2/Pi)*K(k)) and q = exp(-Pi*K'(k)/K(k)) with the real and imaginary quarter periods K and K'. This leads to qhat(v, q) = qtilde(tau(v, q), k(q)) with tau(v, q) = theta_3^2(0, q)*v. (For theta_3^2(0, q) see A004018.) Because k is actually a function of k^2 one uses the q expansion of (k/4)^2 given in A005798.
Using the result for the sn expansion in q and v from A274662 one obtains qhat(v, q) = sin(v)*Sum_{n >= 0} q^n/L(n)*Sum_{m=0..n} T(n, m)*(2*cos(v))^(2*m) with L(n) = A025547(n+1) = lcm{1, 3, ..., (2*n+1)}.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506). Thanks for discussions via e-mail go to him.

			The triangle T(n, m) begins:
n\m    0    1    2    3     4   5 ...
0:     1
1:     8    1
2:   -32   11    3
3:  -736  -92    9   15
4:  2816 -593 -249  -65    35
5: 48976 6122 1581 -970 -1295 315
...
row n=6: -951424 61252 67791 46030 18515 -21735 3465,
row n=7: -1045952 -130744 -92082 -30445 14455 53928 -25179 3003,
row n=8: 26933248 1069361 -1666047 -634255 -1167740 -1258236 1562253 -471471 45045,
row n=9: 634836808 79354601 24881793 17914550 30289840 12635028 -71064609 42480438 -9594585 765765.
...
The corresponding L(n) = A025547(n+1) numbers are 1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535,...
n=4: the contribution to qhat(v, q) of order q^4 is (q^4/315)*(2816 - 593*(2*cos(v))^2 - 249*(2*cos(v))^4 - 65*(2*cos(v))^6 + 35*(2*cos(v))^8).

Cf. A004018, A005798, A025547, A274662.

T(n, m)*(2*cos(v))^(2*m)), n >= 0, m = 0, 1, ..., n, gives the contribution to q^n/L(n) (L(n) = A025547(n+1)) in the rescaled phase space coordinate qhat(v, q) expansion of the plane pendulum. See a comment above for details.

A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2cos(v) with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, -4, 1, 4, -5, 1, 0, 12, -7, 1, 4, -21, 25, -9, 1, -8, 30, -63, 42, -11, 1, 0, -44, 131, -138, 63, -13, 1, 0, 72, -246, 365, -253, 88, -15, 1, 4, -85, 425, -837, 808, -416, 117, -17, 1, -4, 85, -685, 1734, -2200, 1552, -635, 150, -19, 1, 8, -134, 1053, -3319, 5326, -4888, 2705, -918, 187, -21, 1

Offset: 0

Wolfdieter Lang, Aug 10 2016

The representation of Jacobi's elliptic cn(u, k) function in terms of quotients of theta functions of the variables q (Jacobi nome) and v = u/((2/Pi)*K(k)) with the real quarter period K is
  cn(u, k) = (theta_4(0, q)/theta_2(0, q)) * (theta_2(v, q)/theta_4(v, q)).
  This can be written either in terms of infinite sums or products. (see e.g. Tricomi, p. 176, eq. (3.87), p. 156, eq. (3.51), p. 167, eq. (3.71) with (3.71'), p. 173, eq. (3.81)).
The sums representation involves cos((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev T polynomial (A053120) one can write cn(u, k)/cos(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m).
The product representation involves directly (2*cos(v))^2 powers in the q expansion:
  cn(u, k)/cos(v) = Product_{n >= 1} ((1 - q^(2*n-1))^2 *((1 - q^(2*n))^2 + q^(2*n)*(2*cos(v))^2) / ((1 + q^(2*n))^2*((1 + q^(2*n-1))^2 - q^(2*n-1)*(2*cos(v))^2))) = Sum_{n >=0} q^n*Sum_{m = 1..n} T(n, m) * (2*cos(v))^(2*m).
For another version of this cn expansion see A274661.
For the sn(u, k)/sin(v) analog see A274662.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506).
See also the W. Lang link, equations (59) and (60).

			The triangle T(n, m) begins:
n\m 0   1    2    3    4    5    6   7   8 9
0:   1
1:  -4   1
2:   4  -5    1
3:   0  12   -7    1
4:   4 -21   25   -9     1
5:  -8  30  -63   42   -11    1
6:   0 -44  131 -138    63  -13    1
7:   0  72 -246  365  -253   88  -15   1
8:   4 -85  425 -837   808 -416  117 -17   1
9:  -4  85 -685 1734 -2200 1552 -635 150 -19 1
...
Row n=10: 8 -134 1053 -3319 5326 -4888 2705 -918 187 -21 1.
...
n=4: q^4 term of cn(u, k)/cos(v) is  4 - 21*(2*cos(v))^2 + 25*(2*cos(v))^4 - 9*(2*cos(v))^6 + (2*cos(v))^8.
One can check the identity for cn(u, k), for example for u = 1 and k = sqrt(1/2), belonging to v = 0.8472130848 and q = 0.04321391815 (Maple 10 digits), with the result from Maple's cn function cn(1, sqrt(1/2)) = 0.5959765676 (10 digits). If one takes the expansion up to q^4 inclusive one obtains 0.5959776092 (10 digits). If one goes up to q^6 inclusive one gets 0.5959765640 (10 digits).

F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.

Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A005797, A002103, A038534/A056982, A053120, A274661, A274662.

cn(u, k) = cos(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity if q, the Jacobi nome, is replaced by exp(-Pi*K'(k)/K(k)) and v by u/((2/Pi)*K(k)) with the real and imaginary quarter periods K' and K, respectively. For the expansions of q = q(k) see A005797 or better A002103 for q = q((1-k^2)^(1/4)), and for (2/Pi)*K(k) see A038534 / A056982.

cp's OEIS Frontend

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2cos(v), with v = u/((2/Pi)K(k)).

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A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2cos(v) with v = u/((2/Pi)K(k)).

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cp's OEIS Frontend

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).

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Author

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Formula

A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2*cos(v) with v = u/((2/Pi)*K(k)).

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Author

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Examples

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Crossrefs

Formula

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2cos(v), with v = u/((2/Pi)K(k)).

A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2cos(v) with v = u/((2/Pi)K(k)).