On hyperspherical associated Legendre functions: the extension of\n spherical harmonics to $N$ dimensions

The solution in hyperspherical coordinates for $N$ dimensions is given for a\ngeneral class of partial differential equations of mathematical physics\nincluding the Laplace, wave, heat and Helmholtz, Schr\\"{o}dinger, Klein-Gordon\nand telegraph equations and their combinations. The starting point is the\nLaplacian operator specified by the scale factors of hyperspherical\ncoordinates. The general equation of mathematical physics is solved by\nseparation of variables leading to the dependencies: (i) on time by the usual\nexponential function; (ii) on longitude by the usual sinusoidal function; (iii)\non radius by Bessel functions of order generally distinct from cylindrical or\nspherical Bessel functions; (iv) on one latitude by associated Legendre\nfunctions; (v) on the remaining latitudes by an extension, namely the\nhyperspherical associated Legendre functions. The original associated Legendre\nfunctions are a particular case of the Gaussian hypergeometric functions, and\nthe hyperspherical associated Legendre functions are also a more general\nparticular case of the Gaussian hypergeometric functions so that it is not\nnecessary to consider extended Gaussian hypergeometric functions.\n