Gegenbauer functions Introduced by Leopold Gegenbauer in 1893 GegenbauerC[n, m, z] is a polynomial in z with integer coefficients for all integer n and m . … The GegenbauerC[n, d/2 - 1, z] form a set of orthogonal functions on a d -dimensional sphere. The GegenbauerC[n, 1/2, z] obtained for d = 3 are LegendreP[n, z] .

Note (c) for 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 presence of poles in quantities such as GegenbauerC[1/2, -t, -1/2] leads to essential singularities in the rightmost picture below.

Trinomial coefficients The coefficient of x n in the expansion of (1 + x + x 2 ) t is Sum[Binomial[n + t - 1 - 3k, n - 3k] Binomial[t, k] (-1) k , {k, 0, t}] which can be evaluated as Binomial[2t, n] Hypergeometric2F1[-n, n - 2t, 1/2 - t, 1/4] or finally GegenbauerC[n, -t, -1/2] . This result follows directly from the generating function formula (1 - 2 x z + x 2 ) -m  Sum[GegenbauerC[n, m, z] x n , {n, 0, ∞ }]

These coefficients can also be obtained from the formulas in terms of Binomial and GegenbauerC given. … GegenbauerC is a so-called orthogonal polynomial—a higher mathematical function.

Note (c) for More Cellular Automata…The cell at position n on row t turns out to be given by Mod[GegenbauerC[n, -t, -1/2], 2] , as discussed on page 612 .

Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing r i Exp[  n θ i ] , and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions. … (Sums of squares of moments of given order in general provide rotationally invariant measures of anisotropy—equal to pair correlations weighted with LegendreP or GegenbauerC functions.)