Continuous generalizations [of additive rules]

Functions such asBinomial[t, n] andGegenbauerC[n, -t, -1/2] can immediately be evaluated for continuoust andn. The pictures on the right below showSin[1/2π a[t, n]]² for these functions (equivalent toMod[a[t, n], 2] for integera[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 asGegenbauerC[1/2, -t, -1/2] leads to essential singularities in the rightmost picture below. (Compare page922.)