a war on f-notations (hypergeometric somnambulisms)
galathaea
i have always been suspicious of the hidden 1
or rather
the hidden (1)_i
in hypergeometric F notation
it seemed
not well hidden
and not very useful
if i want to represent
oo
--- i
\ x
/ -----
--- ( a )
i = 0 i
i must insert the numerator (1)_i
and the denominator (1)_i
to get
/ 1 \
F | ; x |
1 2 \ a, 1 /
and that disturbs me
-+-
i have recently been able to prove
to my satisfaction
that the (1)_i
is actually a goddess placed in the notation by gauss
to deceive and beguile the initiates
-Z-Z-Z-Z-z-z-z-
the differential equation form
revealed her for the obvious imposter
defining the point transformation operator
d
D = x --
dx
the product formula
D ( D + b0 - 1 ) ( D + b1 - 1 ) ... ( D + b(q-1) - 1 ) F
= x ( D + a0 ) ( D + a1 ) ... ( D + a(p-1) ) F
is satisfied by
_\
/ a \
F | _\ ; x |
p q \ b /
where now a and b are vectors
_\
a = ( a0, a1, ..., a(p-1) )
_\
b = ( b0, b1, ..., b(q-1) )
$^$*#$*^#(#()%)^(%^$*@@!
not only does she stand there in the product formula
for all eyes to see
but one discovers her name as well
D
the one on the leftend in the first line of the formula
D is (1)_i
so like parvati is kali
we have her pinned
~~~~~~~~~~~~~~~~~o
but i only truly understood
the nature of her beguiling
after i decided to transform my multisection results
to goddess-free notation
because i suspected her influence
in the beguiled notation
my multisection theorems relate
the (m, n)-multisection of a (p, q)-hypergeometric
to a ( p n + 1, q ( n + 1 ) )-hypergeometric
which can be reduced to ( p n, q ( n + 1 ) - 1 )
but the (p, q)-atheistic hypergeometric is defined as
_\
/ a \
H | _\ ; x | =
p q \ b /
p-1
oo --- oo
--- | | ( aj ) --- _\
\ j=0 i i \ / a \
/ ------------- x = / h | _\ ; x |
--- q-1 --- i \ b /
i = 0 --- i = 0
| | ( bk )
k=0 i
in this goddess-free formulation
the multisection is simply
_\
|m / a \
| H | _\ ; x | =
|n p q \ b /
_\ |m _\
/ a \ / |n a / x \ n \
h | _\ ; x | H | |m _\ ; | --- | |
m \ b / pn qn \ |n b \ q-p / /
n
where
p-1 n-1
|m _\ _ _ / ai + m + j \
|n a = (x) (x) | ---------- |
- - \ n /
i=0 j=0
and
q-1 n-1
|m _\ _ _ / bi + m + j \
|n b = (x) (x) | ---------- |
- - \ n /
i=0 j=0
this is radiantly clear
-+-+-+-+-
the key to the goddess' spell
is her obfuscation of the vector spaces involved
+X+X+X+X+X+X+X+X+X+X+X+X+X+X+X+
to demonstrate this atheistic notation
in the formulagenic uses of the multisection theorem
the fundament of circularre geometrie
x
y = e
manifests the goddess explicitly as
/ - \
H | ; x |
0 1 \ 1 /
the fundamental level-2 multisection theory
gives us the 3 relations
x |0 x |1 x
e = | e + | e
|2 |2
|0 x 1 x -x
| e = - ( e + e )
|2 2
|1 x 1 x -x
| e = - ( e - e )
|2 2
which hypergeometric multisection represents as
|0 / - \ / - / x \2 \
| H | ; x | = H | ; | - | |
|2 0 1 \ 1 / 0 2 \ 1/2, 1 \ 2 / /
|1 / - \ / - / x \2 \
| H | ; x | = x H | ; | - | |
|2 0 1 \ 1 / 0 2 \ 1, 3/2 \ 2 / /
and substituting into the above
provides the 3 projection relations of level 2
now using these in conjunction with
contiguous relations
allows us to derive a number
of famous hypergeometric relationships
involving the rational subspaces of the coefficient space
taking x at known values
provides a foundation for calculating
relationships for all unit-related values
scaled by the power term
we can apply the above to the classic gaussian
for instance
which atheistically is expressed as
/ a, b \
H | ; x |
2 2 \ c, 1 /
the level-2 multisection projections are
|0 / a, b \
| H | ; x | =
|2 2 2 \ c, 1 /
/ a/2, (a+1)/2, b/2, (b+1)/2 2 \
H | ; x |
4 4 \ c/2, (c+1)/2, 1/2, 1 /
and
|1 / a, b \
| H | ; x | =
|2 2 2 \ c, 1 /
ab / a/2, (a+1)/2, b/2, (b+1)/2 2 \
-- x H | ; x |
c 4 4 \ c/2, (c+1)/2, 1/2, 1 /
plugging into the basic level 2 forms gives
(*)
/ a/2, (a+1)/2, b/2, (b+1)/2 2 \
H | ; x | +
4 4 \ c/2, (c+1)/2, 1/2, 1 /
ab / a/2, (a+1)/2, b/2, (b+1)/2 2 \
-- x H | ; x | =
c 4 4 \ c/2, (c+1)/2, 1/2, 1 /
/ a, b \
H | ; x |
2 2 \ c, 1 /
(a)
/ a/2, (a+1)/2, b/2, (b+1)/2 2 \
H | ; x | =
4 4 \ c/2, (c+1)/2, 1/2, 1 /
1 / / a, b \ / a, b \ \
- | H | ; x | + H | ; -x | |
2 \ 2 2 \ c, 1 / 2 2 \ c, 1 / /
(b)
ab / a/2, (a+1)/2, b/2, (b+1)/2 2 \
-- x H | ; x | =
c 4 4 \ c/2, (c+1)/2, 1/2, 1 /
1 / / a, b \ / a, b \ \
- | H | ; x | - H | ; -x | |
2 \ 2 2 \ c, 1 / 2 2 \ c, 1 / /
and there are places gauss' form can be evaluated
such as at 1
providing explicit reflection formuli
^**^ ^**^ ^**^ ^**^ ^**^
by defeating the goddess
i had revealed the valuable secret
of clarity in form
and could safely banish it from the empire
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
galathaea
galathaea wrote:
> i have always been suspicious of the hidden 1
>
> or rather
> the hidden (1)_i
> in hypergeometric F notation
>
> it seemed
> not well hidden
>
> and not very useful
>
> if i want to represent
>
> oo
> --- i
> \ x
> / -----
> --- ( a )
> i = 0 i
>
> i must insert the numerator (1)_i
> and the denominator (1)_i
>
> to get
>
> / 1 \
> F | ; x |
> 1 2 \ a, 1 /
actually
her illusions had confused me
and i meant to hide her in the bottom
> / 1 \
> F | ; x |
> 1 1 \ a /
but was enraged until i had to reveal her form
but she mocks from above anyhow
Azure
Kewl Show
Sweet Sugir
Does your ass booty-juice?
--
-------------------------------------------
AJ - http://ClitIns.Com e In.
(800 folders. -- kiddie-filtered -- FREE,
Usenet Porn.)
"galathaea" <gala...@gmail.com> wrote in message news:1167539453.2...@v33g2000cwv.googlegroups.com...
galathaea
Sweet Sugir wrote:
> <fart>
>
> Does your ass booty-juice?
how else would she fit?
i hope your not suggesting i have trunk...
its present
but not blatant...
Sweet Sugir
"galathaea" <gala...@gmail.com> wrote in message news:1167642069.1...@42g2000cwt.googlegroups.com...
>
> Sweet Sugir wrote:
>> <fart>
>>
>> Does your ass booty-juice?
>
> how else would she fit?
O Damn, O Sartre, O Hell, patron of thieves...
Dancing on the gaping holes of the Fibonacci Series...
>
> i hope your not suggesting i have trunk...
> its present
> but not blatant...
I'm not requesting punk-goddess-sex on the first fjord.
Let's pace ourselves, love. Sto the trunk, I'm lubricated.
"So many Goddesses, so little time."(TM)
You know that Donald Knuth started out to write 12
volumes, got to #3 and wrote TeX.
He seems to have gone on to attempt one of his
previously unsolvable problems...
(that of parsing input tokens... noble)
He is a funny bald idiot that has numerous awards.
"I brake for bald idiots."(TM)
--
-------------------------------------------
AJ - http://ClitIns.Com e In.
(800 folders. -- kiddie-filtered -- FREE,
Usenet Porn.)
>