Fractals, recursions, divisibility

The self similar (fractal) structure of the Pascal triangle of binomial coefficients modulo some fixed prime has been known for some time and explored in detail. It turns out that such a fractal structure is the common property arithmetical functions of two variables possessing a more general recursive relation (R) f(i,j) = af(i-l,j) + bf(i 1,j-1) where a and b are fixed integers. 'When a, b are functions of i and j, as for in the of Gaussian binomials, or Stirling numbers of first or second kind, or when the recursion of f(i,j)-1,j)-1,j-1) + cf(i-1,j 1) + df(i,j-1) more complexity is shown in the modulo p Functions defined by the recursion R, two dnnel[lSllon;al and Stirling numbers produce p which described