Natural Science Forum / Physics / General Physics / February 2008

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: 0802.3832 v1 [math.CO] 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