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A275791
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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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1
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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
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OFFSET
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0,2
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COMMENTS
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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).
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REFERENCES
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F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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