Nonrenormalization vs Renormalization 10: Gamma Analyzed By PI

From Osher Doctorow mdoctorow@xxxxxxxxxxx

The gamma function already derives from Probable Influence/Causation
(PI) via PIA maximum entropy of the gamma family of distributions and
other asymmetric nonnegative real line distributions, but some further
analysis yields some additional combinatorics.

Recall that for positive integer n, the gamma function G(n) is:

1) G(n) = (n - 1)! = 1 * 2 * 3 *** * n (using * as times, that is to
say multiplication)

The multiplication in (1) can be "converted" to addition in PI in the
same way that conditional probability y/x converts to 1 + y - x in PI
(the 1 has the role of insuring that the result is a probability, that
is to say between 0 and 1, and although the 1 derives from the
mathematics the key thing is the change from division to subtraction,
which for multiplication corresponds to a change to addition):

2) Sum(i) = 1 + 2 + 3 + ... + n = n(n + 1)/2

where Sum(i) is the sum of consecutive positive integers i from 1 to n.


Equation (2) is well known in algebra and calculus, and retains the
idea of finite counting (which is a positive integer scenario) across
addition, not just multiplication as in (1). The important point is
the sum 1 + 2 + 3 + ... + n in (2), not so much the n(n + 1)/2
expression although it facilitates computation. Notice that n(n +
1)/2 only has one product of "variables", n times n + 1 or (n/2) times
n + 1.

For the sum of square consecutive positive integers from 1^2 to n^2 the
result is the well known n(n + 1)(2n +1)/6, and for cubes it is [n(n +
1)/2]^2, which again are well known equations.

Readers can try to see what happens with 1 - 2 + 3 - 4 + 5 ..., which
has the difficulty that depending on how terms are rearranged it yields
different sums. This is the well known rearrangement difficulty of
nonconvergent infinite series. Could one define convergent subseries
with appropriate associative rules?

Osher Doctorow

.

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