【摘要】 <正> §1.引言 设P（x₁,…,x_k）与Q（x₁,…,x_k）是x₁,…,x_k的实系数多项式,其总的次数分别为p与q;又设Q（x_k,…,x_k）满足下列条件:i)除可能以（0,…,0）为零点外无其他实的零点,ii)其所有q次项之和为一正定的q次型。更多还原

【Abstract】 Let P(x1,…,xk) and Q(x1,…, xk) be polynomials in x1…, xk of total degree p and q respectively; and suppose that Q(x1,…,xk) satisfies the following two conditions:i) it has no real zeros, except possibly the point (0,…,0),ii) the sum of all the terms of the q-th degree is a positive definite form.Then we define the functionwhere "," denotes that (x1,…, xk)≠((0,…, 0).The aim of the present paper is to prove that D(s) can be continued analytically to the whole plane, except at the pointswhere it may, possibly, have simple poles.The function D(s) includes the Epstein Z function and the function Znk(s) considered by prof. Min Szu-hoa, as special cases. In fact, if P (x1,…,xk)=1 and Q(x1,…,xk) is a positive definite quadratic form, then D(s) becomes the Epstein Z function; if, on the other hand, P(x1,…,xk)=1 and Q(x1,…,xk)=x1+…+xkn (where n is an even integer), then D(s) becomes the functionZn1k(s).更多还原