Konrad J. Heuvers

THE FUNCTIONAL EQUATION OF THE SQUARE ROOT SPIRAL

The square roots of the positive integers can be placed on a well known square root spiral. In order to characterize it, polar coordinates are introduced with θ = g(r). Then we find the general solution to the functional equation

A characterization of the permanent function by the Binet-Cauchy theorem

g\left( {\sqrt {r^2 + 1} } \right) = g(r) + \arctan \left( {\frac{1} {r}} \right)

The Binet-Pexider functional equation for rectangular matrices

where g(1) = 0 and g(r) is monotone increasing for r > 0. The resulting curve θ = g(r) gives a continuous square root spiral

An inversion relation of multinomial type

SummaryIf Φ is a function of one variable, itsnth order Cauchy kernel is defined by

The Binet-Cauchy functional equation and nonsingular multiindexed matrices

\mathop K\limits_n \Phi (x_1 ,...,x_n ) = \sum\limits_{r = 1}^n {( - 1)^{n - r} \sum\limits_{\left| J \right| = r} {\Phi (x_J )} }

Functional equations which characterizen-forms and homogeneous functions of degreen

where ∅ ≠J