Ϝ, The Digamma Function
To begin in the most informative way, I present the following example, which produces successive approximations of Φ (Phi) with sufficient recursions:
Digamma as Processor Instructions:
If we choose any number other than 0 or -1, we may add 1 to it, and then divide it by its original value. Thus, if we choose 1 as the first value, the result of the first iteration will be 2. [(1 + 1) / 1 = 2] Then we obtain 1.5, then 1.667, then 1.6, finally settling down nicely after two dozen or so iterations, to approximately 1.618033989 in ten decimal places. Thus, we have created a mathematical formula which reproduces the behavior of a “for … next loop” within a computer program, or successive calculations using a hand calculator with a memory function.
Digamma as Contents in Memory:
If we list the successive values of and the values of <X sub i> as a sequence of pairs, then we have an Equivalence Class, or a “look-up table”, depending on how we propose to use such a sequence. This can be made into the basis of a new Digamma Function in its own right.
Digamma as Sum Function (Σ) and Product Function (Π):
It seems fairly obvious that the Sum and Product Functions can both be expressed fairly easily as Digamma Functions. In fact, a Digamma Function could express any mixture of sums and products simultaneously, without having to choose one form of notation or the other. I would think that this would make certain concepts easier to express in a single form of notation.
Other Examples of the Use of Digamma:
My rather uneasy speculation, “Infinite Root of omega”:
This is a recursive expression of exponentiation. I’d love for someone to verify this, as it seems rather intuitive, but I don’t have the body of experience in formal notation necessary to prove it.
A somewhat more confident speculation, “Infinite Root of Unity”:
Again, an unproven conjecture that is rather elegant in appearance.
Concerns and Further Development:
Can the Digamma Function be made to reproduce the various mathematical functions, and in a way that avoids expressions which are too difficult or cumbersome to be worthwhile? I haven’t explored the use of Digamma in representing any form of continued fraction, but success in this effort would create another valuable connection between mathematical notation and the procedures familiar to data processing. I’m sure that this presentation raises many more questions than it answers, but overall I see that as much a strength as a weakness, in that anyone can contribute their understanding and efforts, and that such contributions will be accepted more on the basis of sound reasoning, rather than some less helpful standing upon credentialed or proprietary attitudes and approaches.