A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).
1, 0, 1, 0, -3, 1, 0, 4, -5, 1, 0, -3, 13, -7, 1, 0, 6, -25, 26, -9, 1, 0, -12, 43, -70, 43, -11, 1, 0, 8, -70, 157, -147, 64, -13, 1, 0, -3, 109, -315, 408, -264, 89, -15, 1, 0, 13, -155, 582, -984, 872, -429, 118, -17, 1, 0, -18, 201, -1001, 2142, -2464, 1641, -650, 151, -19, 1
Offset: 0
Examples
The triangle T(n, m) begins: n\m 0 1 2 3 4 5 6 7 8 9 0: 1 1: 0 1 2: 0 -3 1 3: 0 4 -5 1 4: 0 -3 13 -7 1 5: 0 6 -25 26 -9 1 6: 0 -12 43 -70 43 -11 1 7: 0 8 -70 157 -147 64 -13 1 8: 0 -3 109 -315 408 -264 89 -15 1 9: 0 13 -155 582 -984 872 -429 118 -17 1 ... row n=10: 0 -18 201 -1001 2142 -2464 1641 -650 151 -19 1 ... n=4: the q^4 term of sn(u, k)/sin(v) is -3*(2*cos(v))^2 + 13*(2*cos(v))^4 - 7*(2*cos(v))^6 + (2*cos(v))^8. One can check the identity 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 sn function sn(1, sqrt(1/2)) = 0.8030018249 (10 digits). If one takes the expansion up to q^4 inclusive one obtains .8030012888 (10 digits).
References
- F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.
Links
- Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum
Formula
sn(u, k) = sin(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity when 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.
Comments