2 1/2
A conjectured good rational approximations to , -3 c - 3/2 - 3 c - 3 c ,
for positive intgers c,as good as from the continued fraction
By Shalosh B. Ekhad
2 1/2
Regarding the constant, -3 c - 3/2 - 3 (c + c)
Consider the recurrence satisfied by the contour-integral, around x=0, of th\
e function
/(c x + 1) (1 + (c + 1) x)\n
|-------------------------|
\ x /
----------------------------
1/2
x
let's call it X(n)
that in Maple notation is
((c*x+1)*(1+(c+1)*x)/x)^n/x^(1/2)
According to the famous Almkvist-Zeilberger algorithm, it satisfies the recu\
rrence
X(n) = 4*(2*c+1)*n/(2*n+1)*X(n-1)-4*n*(n-1)/(2*n-1)/(2*n+1)*X(n-2)
Let a(n), b(n), be the sequence that satisfy that recurrence with
with the initial conditions
a(0) = 0, a(1) = 1
b(0) = 1, b(1) = 0
We conjecture that the limit of a(n)/b(n) as n goes to infinity is the above\
2 1/2
-mentioned, -3 c - 3/2 - 3 (c + c)
Not only that, these seem to give irrationality measure 2, just like the one\
coming from the continued fraction.
The estimated irrationality measure for the case c=4 is
1.99915073193908718698254299933544062813603232407152470164117399101580021456\
0315642671109465789245188
Let's make c bigger and see how the effective irrationaliy measure shrinks
For c=, 11, the irrationality measure is estimated to be
1.99947328572055162849893869342398793095390713732342467397798302362273506450\
1946315449219698264109302
For c=, 101, the irrationality measure is estimated to be
1.99930513358741971399617960099126194140551487959795751388712743331141408519\
0520764095345856703052627
For c=, 1001, the irrationality measure is estimated to be
1.99922365115818617422396056985989868210486621832151800504192411476209369450\
8870902283979428907551897
For c=, 10001, the irrationality measure is estimated to be
1.99917722762411759657889501182835940437093644308950175600429207795684051522\
4100808748538153338085936
For c=, 100001, the irrationality measure is estimated to be
1.99914735812240415317130504063759387171912530025823994863811840697064988221\
9235368483911054561507306
as you can see, it seems to be 2