ON DIFFERENTIAL POLYNOMIALS, II

In Part II, we shall be concerned with applications of classical invar iant theory, to statistic physics and to theta functions. Main theorem in Chapter 2 is stated as follows: For a partition function xi(s) = ( l=1)Sigma(infinity) gamma l s(dl) satisfying gamma iota greater than o r equal to 0 (l greater than or equal to 1) and alpha > 0, the 2n-apol ar of xi(s) A(2n)(xi(s), xi(s)) = s(2n) (l=0)Sigma(2n) (-1)(k) ((2n)(k )) (d/ds)(2n-k) xi(s) (d/ds)(k) xi(s) has the expansion A(2n)(xi(s), x i(s) = (l=2)Sigma(infinity) beta(n,l)s(-alpha l) such that beta(n,l) g reater than or equal to 0 (l greater than or equal to 2). This means, for a given partition function xi(s) with nonnegative relative probabi lities, we construct a sequence of partition functions A(2n)(xi(s), xi (s))(n greater than or equal to 1) With the same properties. which may be considered a sequence of symbolical higher derivative of xi(s). Th e main theorem in Chapter 3 is stated as follows: For given theta func tions phi(1)(z) and phi(2)(z) of level n(1) and n(2) respectively, in g variables z = (z(1), z(2), ..., z(g)), then r = (r(1), r(2), ..., r( g))-apolar A(r)(phi(1)(z), phi(2)(z)) = Sigma(0 less than or equal to j less than or equal to r-j) (-1)(\j\)/n(1)(\j\)n(2)(\r-j\) ((r)(j)) ( partial derivative/partial derivative z)(j) phi(1)(z)(partial derivati ve/partial derivative z)(r-j) phi(2)(z) is a theta function of level n (1) + n(2), and [GRAPHICS]