Fermion quasi-spherical harmonics

Quasi-spherical harmonics, Y-l(m)(theta.phi) are derived and presented for half-odd-integer values of l and m. The form of the theta factor is identic al to that in the case of integer I and in: exp(im phi). However, the domai n of these functions in the half-odd-integer case is 0 less than or equal t o phi < 4 pi rather than the domain 0 less than or equal to phi < 2 pi in t he case of integer l and,m (the true spherical harmonics). The form of the a factor, p(l)(\m\)(theta) (an associated Legendre function) is las in the integer case) the factor (sin theta)(\m\) multiplied by a polynomial in cos theta of degree (l - \m\) tan associated Legendre polynomial). A substanti al difference between the spherical (integer l and m) and quasi-spherical ( half-odd-integer l and m) Legendre functions is that the latter have an irr ational factor of root sin theta whereas the factor of the truly spherical functions is an integer power of sin theta. The domain of both the true and quasi-spherical associated Legendre functions is the same: 0 less than or equal to theta < pi. A table of the associated Legendre functions is presen ted for both integer and half-odd-integer values of l and m, for \m\= 0, 1/ 2 , 1. . .11/2, (l - \m\) = 0. 1. 2. 3. 4. 5. The table displays the simila rity between the functions for integer l and,m (which are well known) and t hose for half-odd-integer l and m (which have not been recognized previousl y).