A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.

1, 1, 1, 5, 6, 1, 61, 107, 47, 1, 1385, 3116, 2142, 412, 1, 50521, 138933, 130250, 45530, 3693, 1, 2702765, 8783986, 10430983, 5353260, 1036715, 33218, 1, 199360981, 747603679, 1074680289, 728130163, 226132303

Offset: 0

Author

R. J. Mathar, Jan 30 2011

Keywords

• nonn
• tabl

Examples

			The triangle starts in row n=0 as
1;
1, 1;
5, 6, 1;
61, 107, 47, 1;
1385, 3116, 2142, 412, 1;
50521, 138933, 130250, 45530, 3693, 1;
		

References

• M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
• D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052v2 [math.CA].
• NIST Digital Library of Mathematical Functions, NIST Handbook of Mathematical Functions, Chapter 22.
• Eric W. Weisstein, Jacobi Elliptic Functions.

Crossrefs

Cf. A060627, A060628, A181613, A000364 (apparently the column m=0).

Programs

Maple

A181612 := proc(n,m) JacobiDC(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^m*coeftayl(%,k=0,2*m)*(2*n)! ; end proc:
seq( seq(A181612(n,m),m=0..n),n=0..10) ;

Mathematica

nmax = 8; se = Series[JacobiDC[x, y], {x, 0, 2*nmax}]; t[n_, m_] := Coefficient[se, x, 2*n]*(2*n)! // Coefficient[#, y, m]& // Abs; Table[t[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

Formula

From Peter Bala, Aug 23 2011: (Start)

The elliptic function dc(x,k) (JacobiDC(x,k) in Maple notation) is defined as dn(x,k)/cn(x,k) where dn(x,k) and cn(x,k) are the Jacobian elliptic functions of modulus k. The Taylor expansions begin

dn(x,k) = 1-k^2*x^2/2!+k^2*(4+k^2)*x^4/4!-k^2*(16+44*k^2+k^4)*x^6/6!+...

cn(x,k) = 1-x^2/2!+(1+4*k^2)*x^4/4!-(1+44*k^2+16*k^4)*x^6/6!+... and hence

dc(x,k) = 1+(1-k^2)*x^2/2!+(5-6*k^2+k^4)*x^4/4!+(61-107*k^2+47*k^4-k^6)*x^6/6!+....

The coefficients for cn(x,k) are in A060627. The coefficients of dn(x,k) may be obtained by row reversal of A060627.

The expansion for dc(x,k) can also be obtained directly from that of dn(x,k) since by Jacobi's imaginary transformations we have dc(x,k) = dn(i*x,k'), where the complementary modulus k' is given by k' = sqrt(1-k^2).

By Jacobi's real transformation the reciprocal of dc(x,k) is given by 1/dc(x,k) = dc(x*k,1/k).

The row polynomials of this table can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):

Let f(x) = sqrt(1-(1-k^2)*sin^2(x)). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.

See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).

Then the coefficient of x^(2*n)/(2*n)! in the expansion of dc(x,k) is given by (-1)^n*D^(2*n)[f](0).

(End)