SI'onist Pig, o};-)
Kewl Show
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 /
>
> 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) )
>
> $^$*#$*^#(#()%)^(%^$*@[EMAIL PROTECTED]
>
> 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 relation****ps
> involving the rational subspaces of the coefficient space
>
> taking x at known values
> provides a foundation for calculating
> relation****ps 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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