A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.

1, 0, -1, 0, 4, -3, 0, -16, 60, -45, 0, 64, -1008, 2520, -1575, 0, -256, 16320, -105840, 189000, -99225, 0, 1024, -261888, 4055040, -15800400, 21829500, -9823275, 0, -4096, 4193280, -149909760, 1153152000, -3178375200, 3575672100, -1404728325, 0, 16384, -67104768, 5459650560

Offset: 1

Author

R. J. Mathar, Jun 26 2006

Keywords

• sign
• tabl
• easy

Comments

Table has only rows for odd h because all coefficients for even h are zero:

=====|=======================================================================

h \ s| 0 1 2 3 4 5 6

-----|-----------------------------------------------------------------------

1 | 1

3 | 0 -1

5 | 0 4 -3

7 | 0 -16 60 -45

9 | 0 64 -1008 2520 -1575

11 | 0 -256 16320 -105840 189000 -99225

13 | 0 1024 -261888 4055040 -15800400 21829500 -9823275

15 | 0 -4096 4193280 -149909760 1153152000 -3178375200 3575672100

17 | 0 16384 -67104768 5459650560 -79048569600 390486096000 -829555927200

...

From Francesco Franco, Jan 12 2016: (Start)

Conjecture:

If t(h,s) is any term of the previous table after the first column (s>0), then:

t(h,s) = -( 4*s^2*t(h-2,s) + Sum_{j=0..s-1} (t(h-2,j) + t(h,j)) ), with t(1,0) = 1, t(h,0) = 0 for h>1 and t(h,s) = 0 for odd h = 1..2*s-1.

Version without the summation:

t(h,s) = -( 4*s^2*t(h-2,s) - (4*(s-1)^2-1)*t(h-2,s-1) ).

Some example (starting from j=1 in the summation):

t(11,3) = -( 4*t(9,3)*3^2 + Sum_{j=1..2} (t(9,j) + t(11,j)) ) = -( 4*2520*9 + (64-256) + (-1008+16320) ) = -105840; second version:

t(17,5) = -( 4*5^2*t(15,5) - (4*4^2-1)*t(15,4) ) = -( 4*25*(-3178375200) - 63*1153152000 ) = 390486096000.

Also:

t(h,1) = (-1)^(h/2-1/2)*A000302(h/2-3/2) for h>1;

t(h,2) = (-1)^(h/2-3/2)*A115490(h/2-3/2) for h>3;

a(A000124(n)) = 0.

(End)

Examples

			E(m,phi) = phi - m*phi^3/3! + (4*m-3*m^2)*phi^5/5! + (-16*m+60*m^2-45*m^3)*phi^7/7! + ...
so the first row (order phi^1) is a(1,1)=1 for the coefficient of phi,
the second row (order phi^3) is a(2,0)=0 for the missing coefficient of m^0*phi^3, and a(2,1)=-1 for the coefficient of m^1*phi^3/3!.
		

Links

• R. J. Mathar, Chebyshev series expansion of the Elliptic Integral of the Second Kind

Crossrefs

Cf. A010370, A079484 (diagonal).

Programs

Maple

an := proc(m,n,s) local f: f := coeftayl(EllipticE(sin(phi),m^(1/2)),phi=0,n); coeftayl(f*n!,m=0,s) ; end: nmax := 27 ; for n from 1 to nmax by 2 do for s from 0 to (n-1)/2 do printf("%d,",an(m,n,s)) ; od ; od;

Mathematica

a[n_, s_] := SeriesCoefficient[EllipticE[phi, m], {phi, 0, n}, {m, 0, s}]*n!; Table[a[n, s], {n, 1, 17, 2}, {s, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 06 2014 *)

PARI

{T(n, k) = my(m = 2*n+1); if( k<0 || nMichael Somos, May 04 2017 */

Formula

E(m,phi) = Int_{theta=0..phi} sqrt(1-m*sin^2 theta) d theta.

E(m,phi) = Sum_{n=1,3,5,7,9,...} ( Sum_{s=0..(n-1)/2} a( (n+1)/2,s ) * m^s )*phi^n/n!.