sci.physics

Moa Adagodu and Doron Zeilberger of respectively Virginia Commonwealth
University Richmond Virginia USA and Rutgers University New Jersey
USA, in "Searching for strange hypergeometric identities by sheer
brute force," 6 pages, arXiv: v1 [] 26 Feb 2008,
obtain an arguably interesting but "strange" theorem by what they call
"brute force". However, simply searching for an equation of a related
type involving Probable Causation/Influence (PI) would give the same
result. The Theorem is, with (z)_k of their notation abbreviated to
(z)k here defined by:

1) (z)k = z(z + 1)(z + 2)...(z + k - 1), k positive integer

and with the Classical Hypergeometric Series defined by:

2) F(a, b, c, x) = sum (a)k (b)k x^k /[k! (c)k]

where sum is for k = 0 to infinity, given by:

Theorem 1. For all nonnegative integers r:

3) F(-2n, b, -2n + 2r - b, -1) = (A/B) C

where:

4) A = (1/2)n (b + 1 - r)n
5) B = (b/2 + 1 - r)n (b/2 + 1/2 - r)n
6) C = sum (D/E)G
7) D = 2^(2i) i! C(r + i - 1, 2i)
8) E = (b - r + 1)i
9) G = C(n, i)

where C(u, v) = u!/[(u - v)!v!], u > = v

and C(u, v) isn't related to the C of (6).

Notice that Probable Causation/Influence (PI) P(A-->B) is defined by:

10) P(A-->B) = 1 + y - x, y = P(AB), x = P(A)

or alternatively a second version is:

11) P ' (A-->B) = 1 + y - x, y = P(B), x = P(A), y < = x

were y < = x also in (10) automatically because P(AB) < = P(A) from
probability theory.

Looking at E of (8), it is just 1 + y - x with y = b, r = x, inside
the parentheses and i (a subscript) outside. So we can write up to a
constant of normalization:

12) E = (1 + y - x)i, y = b, x = r

Similarly for relevant factors of A and B of (4) and (5) respectively.

So a search for the appropriate PI equations would have yielded
Theorem 1 up to normalization constants except for the very common
C(u, v) in combinatorics and a few other factors.

Osher Doctorow