Summation free expression for some special Clebsch-Gordan coefficients

The known general formula for the Clebsch-Gordan coefficients of the 3-dimensional rotation group involves one summation which results in explicit summation-free expressions for the coefficients where either one of the angular momenta is the sum of the other two or the magnetic quantum number corresponding to one of the angular momenta takes its maximum value in magnitude. By using very different techniques, explicit expressions for the coefficients , are also obtained, where the integral or half-integral nature of the j's is indicated by the magnetic quantum number involved. Here the expressions depend upon whether j1+j2+j is an even or an odd integer. For these coefficients, the magnetic quantum numbers involved take their minimum value in magnitude. By using the recursion relation for the coefficients of the form , we can calculate these coefficients in terms of the above known ones provided we also have the explicit value of the coefficient where j1+j2+j is an odd integer. (The recursion relation for these coefficients in terms of becomes a triviality since vanishes when j1+j2+j is an odd integer.) Our main purpose in this paper is to give an explicit expression for the coefficients where j1+j2+j is an odd integer. We are able to arrive at our expression by using a complicated transformation between hypergeometric functions which seems to have been neglected so far. For the coefficients where the magnetic quantum numbers have their minimum value in magnitude, this transformed expression becomes summation free and we obtain explicit values of the three already known coefficients and the fourth so far unknown. We believe that further study of this transformation may be useful on its own because it provides a link between very different types of expressions. (author)