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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)).

%I A274662 #20 Aug 26 2016 17:38:50
%S A274662 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,
%T A274662 -11,1,0,8,-70,157,-147,64,-13,1,0,-3,109,-315,408,-264,89,-15,1,0,13,
%U A274662 -155,582,-984,872,-429,118,-17,1,0,-18,201,-1001,2142,-2464,1641,-650,151,-19,1
%N 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)).
%C A274662 The representation of Jacobi's elliptic sn(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
%C A274662   sn(u, k) = (theta_3(0, q)/theta_2(0, q)) * (theta_1(v, q)/theta_4(v, q)).
%C A274662   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)).
%C A274662 The sums representation involves sin((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev S and T polynomial (A049310 and A053120) one can write sn(u, k)/sin(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m) * (2*cos(v))^(2*m).
%C A274662 The product representation involves directly (2*cos(v))^2 powers in the q expansion:
%C A274662   sn(u, k)/sin(v) = Product_{n >= 1} (1 - (q^(2*n)/(1 + q^(2*n))^2)*(2*cos(v))^2) / (1 - (q^(2*n-1)/(1 + q^(2*n-1))^2)*(2*cos(v))^2) = Sum_{n >=0} q^n * Sum_{m = 1..n} T(n, m)*(2*cos(v))^(2*m).
%C A274662 This sn expansion in the v and q variables is used in the scaled phase space coordinate qhat(v, q) of the plane pendulum. See A275790.
%C A274662 An alternative expansion of sn in the variables v and q is given in A274659.
%C A274662 See also the W. Lang link, equations (52) and (53).
%D A274662 F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.
%H A274662 Wolfdieter Lang, <a href="/A273506/a273506_5.pdf">Expansions for phase space coordinates for the plane pendulum</a>
%F A274662 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.
%e A274662 The triangle T(n, m) begins:
%e A274662 n\m 0   1    2    3    4    5    6   7   8 9
%e A274662 0:  1
%e A274662 1:  0   1
%e A274662 2:  0  -3    1
%e A274662 3:  0   4   -5    1
%e A274662 4:  0  -3   13   -7    1
%e A274662 5:  0   6  -25   26   -9    1
%e A274662 6:  0 -12   43  -70   43  -11    1
%e A274662 7:  0   8  -70  157 -147   64  -13   1
%e A274662 8:  0  -3  109 -315  408 -264   89 -15   1
%e A274662 9:  0  13 -155  582 -984  872 -429 118 -17 1
%e A274662 ...
%e A274662 row n=10: 0 -18 201 -1001 2142 -2464 1641 -650 151 -19 1
%e A274662 ...
%e A274662 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.
%e A274662 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).
%Y A274662 Cf. A002103, A005797, A038534/A056982, A049310, A053120, A274659, A275790.
%K A274662 sign,tabl,easy
%O A274662 0,5
%A A274662 _Wolfdieter Lang_, Aug 08 2016