Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas

Continuous generalizations [of additive rules]

Functions such as Binomial[t, n] and GegenbauerC[n, -t, -1/2] can immediately be evaluated for continuous t and n. The pictures on the right below show Sin[1/2 π a[t, n]]2 for these functions (equivalent to Mod[a[t, n], 2] for integer a[t, n]). The discrete results on the left can be obtained by sampling only where integer grid lines cross. Note that without further conditions the continuous forms cannot be considered unique extensions of the discrete ones. The presence of poles in quantities such as GegenbauerC[1/2, -t, -1/2] leads to essential singularities in the rightmost picture below. (Compare page 922.)

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]