Series of EllipticK[x]/Pi

[Wouter Meeussen]

hi,

if anyone knows how to write the n-th term of the Taylor series of the
function EllipticK[x]/Pi,
please point me to it.

I got no farther then the following:
****************************************************************************
*********************
the first dosen coefficients are:

{1/2, 1/8, 9/64, 75/256, 3675/4096, 59535/16384, 2401245/131072,
 57972915/524288, 13043905875/16777216, 418854310875/67108864,
 30241281245175/536870912, 1212400457192925/2147483648}

up to the 96th term, the denominators are 2^p, with p=

{1,3,6,8,12,14,17,19,24,26,29,31,35,37,40,42,48,50,53,55,59,61,64,66,71,73,76,
  78,82,84,87,89,96,98,101,103,107,109,112,114,119,121,124,126,130,132,135,
  137,143,145,148,150,154,156,159,161,166,168,171,173,177,179,182,184,192,194,
  197,199,203,205,208,210,215,217,220,222,226,228,231,233,239,241,244,246,250,
  252,255,257,262,264,267,269,273,275,278,280,287}

their differences are:


 {2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,7,2,3,2,4,
  2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,8,2,3,2,4,2,3,2,5,
  2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,7

a nicely regular thing, that can be recursively symbolised in Mathematica as:

In[91]:=
2+Flatten[Fold[{#1,#2,#1}&,0,Range[6]]]
Out[91]=
 {2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,7,2,3,2,4,
  2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,8,2,3,2,4,2,3,2,5,
  2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,7,
  2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2}

so that the denominators of the Taylor series of  EllipticK[x]/Pi can be written
as:

 2 ^ FoldList[Plus,1,2+Flatten[Fold[{#1,#2,#1}&,0,Range[6]]]]


pro memori:  "FoldList[f, x, {a, b, ... }] gives {x, f[x, a], f[f[x, a], b],
... }."

*** two remarks ***

for the numerators, I came up with ziltch, nada :
{1,1,9,75,3675,59535,2401245,57972915,13043905875,418854310875,30241281245175,
  1212400457192925}
                (* remark that they must be odd, all powers of 2 living in
the basement *)

they factor as enthousiasticly as the factorials do, but with some hidden
knack :

{{}}
{{}}
{{3^ 2}}
{{3^ 1}, {5, 2}}
{{3, 1}, {5, 2}, {7, 2}}
{{3, 5}, {5, 1}, {7, 2}}
{{3, 4}, {5, 1}, {7, 2}, {11, 2}}
{{3, 4}, {5, 1}, {7, 1}, {11, 2}, {13, 2}}
{{3, 6}, {5, 3}, {7, 1}, {11, 2}, {13, 2}}
{{3, 4}, {5, 3}, {7, 1}, {11, 2}, {13, 2}, {17, 2}}
{{3, 4}, {5, 2}, {7, 1}, {11, 2}, {13, 2}, {17, 2}, {19, 2}}
{{3, 6}, {5, 2}, {7, 3}, {11, 1}, {13, 2}, {17, 2}, {19, 2}}

and it is to be expected that a ratio of factorials underlies this regularity.
Anyone recognise this structure?

*** second remark ***

this may be far out, but any connection between Elliptic stuff (=functions &
integrals)
on one hand, and number theory on the other hand, could be inspiring to some
(? didn't some guys
connect these things recently in a gig 'round Fermat's last something ?)

The connection with number theory is vague, and has to do with the
Fold[{#1,#2,#1}&,0,Range[n]]
in the formula above. I met this structure before, when fiddling with the
divisibility of Catalan numbers ; J.H. Conway made a link between these and
the number of two's in the ternary expansion of 
these catalans.

Vague link, I know.


wouter.
****************************************************************************
***********************
References to structures build with 'Fold'-ing:


%I A019468
%S A019468 9,12,15,27,30,33,81,84,87,117,120,123,243,246,249,279,282,285,333,
%T A019468 336,339,351,354,357,729,732,735,765,768,771,819,822,825,837,840,
%U A019468 843,981,984,987,999,1002,1005,1053,1056,1059,1089,1092,1095,2187
%N A019468 (n-2)th Catalan number is congruent to 2n/3 mod n.
%O A019468 1,1
%K A019468 nonn
%A A019468 Wouter Meeussen (w.meeussen.vdmcc at vandemoortele.be)

%I A019467
%S A019467 3,6,36,39,42,90,93,96,108,111,114,252,255,258,270,273,276,324,327,
%T A019467 330,360,363,366,738,741,744,756,759,762,810,813,816,846,849,852,
%U A019467 972,975,978,1008,1011,1014,1062,1065,1068,1080,1083,1086,2196,2199
%N A019467 (n-2)th Catalan number is congruent to n/3 mod n.
%O A019467 1,1
%K A019467 nonn
%A A019467 Wouter Meeussen (w.meeussen.vdmcc at vandemoortele.be)

other sequences use a similar construct :


%I A028668
%S A028668 1,30,37800,1755432000,2946176634240000,178121125423535616000000,
%T A028668 387722609071165087097978880000000,
%U A028668 30383449623465746081582327522446540800000000
%N A028668 Pseudo Galois numbers for d=6.
%F A028668 a(n) = p^n Product (p^n - p^k) for k=0 to n-1
%O A028668 0,2
%K A028668 nonn
%A A028668 Olivier Gerard (ogerard at ext.jussieu.fr)
%t A028668 FoldList[#1*6^#2 (6^#2-1)&,1,Range[20]]
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be