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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
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