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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A274659 Triangle entry T(n, m) gives the m-th contribution T(n, m)*sin((2*m+1)*v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic sn(u|k) function when expressed in the variables v = u/(2*K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, -1, -2, 0, 1, 2, 1, -2, 0, 1, 2, 3, 0, -2, 0, 1, -4, -2, 3, 0, -2, 0, 1, -4, -5, 1, 3, 0, -2, 0, 1, 7, 3, -6, 0, 3, 0, -2, 0, 1, 7, 9, -2, -6, 0, 3, 0, -2, 0, 1, -11, -5, 11, 1, -6, 0, 3, 0, -2, 0, 1, -11, -15, 3, 11, 0, -6, 0, 3, 0, -2, 0, 1, 17, 9, -17, -2, 11, 0, -6, 0, 3, 0, -2, 0, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jul 18 2016

Keywords

Comments

If one takes the row polynomials as R(n, x) = Sum_{m=0..n} T(n, m)*x^(2*m+1), n >= 0, Jacobi's elliptic sn(u|k) function in terms of the new variables v and q becomes sn(u|k) = Sum_{n>=0} R(n, x)*q^n, if one replaces in R(n, x) x^j by sin(j*v).
v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.
A test for sn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.8030018002 to be compared with sn(1|sqrt(1/2)) approximately 0.8030018249.
For the derivation of the Fourier series formula of sn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.
For the cn expansion see A274661.
See also the W. Lang link, equations (34) and (35).

Examples

			The triangle T(n, m) begins:
      m  0   1  2  3  4  5  6  7  8  9 10 11
n\ 2m+1  1   3  5  7  9 11 13 15 17 19 21 23
0:       1
1:       1   1
2:      -1   0  1
3:      -1  -2  0  1
4:       2   1 -2  0  1
5:       2   3  0 -2  0  1
6:      -4  -2  3  0 -2  0  1
7:      -4  -5  1  3  0 -2  0  1
8:       7   3 -6  0  3  0 -2  0  1
9:       7   9 -2 -6  0  3  0 -2  0  1
10:    -11  -5 11  1 -6  0  3  0 -2  0  1
11:    -11 -15  3 11  0 -6  0  3  0 -2  0  1
...
T(4, 0) = 2 from the x^1 term in b(0, x)*a(4) + b(2, x)*a(2) + b(4, x)*a(0), that is x^1*3 + x^1*(-2) + x^1*1 = +2*x^1.
n=4: R(4, x) = 2*x^1 + 1*x^3 - 2*x^5 + 0*x^7 + 1*x^9, that is the sn(u|k) contribution of order q^4 in the new variables v and q is (2*sin(1*v) + 1*sin(3*v) - 2*sin(5*v) + 1*sin(9*v))*q^4.

References

  • J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

Links

Crossrefs

Cf. A002103, A038534/A056982, A099774, A274621, A274658, A274661.

Formula

T(n, m) = [x^(2*m+1)]Sum_{j=0..n} b(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and b(j, x) = Sum_{r | 2*j+1} x^r = Sum_{k=1..A099774(j+1)} x^(A274658(j, k)), for j >= 0.

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)).

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

Views

Author

Wolfdieter Lang, Aug 10 2016

Keywords

Comments

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).

Examples

			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).

References

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

Links

Crossrefs

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

Formula

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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