Note on Orthogonal Polynomials which are ``Invariant in Form'' to Rotations of Axes

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Abstract
The precedure of Bhatia and Wolf for constructing orthogonal sets whose elements are ``invariant in form'' with respect to rotations of axes, is extended to include sets which are defined over the entire two‐dimensional plane. Use was made of the Gram‐Schmidt process to derive general expressions for generating elements of many unique and complete sets corresponding to different circularly symmetric weight functions for two cases, where in the first case the elements are only functions of the real variables x and y while in the other case they are also functions of the real variable r = (x2 + y2)½. These general expressions were used to obtain two new, unique and complete sets corresponding to Gaussian and exponential weight functions, respectively. The radial polynomials for these two sets were found to be closely related to the Laguerre polynomials. The generating functions for these radial polynomials are also given.