On Zeros of Polynomials and Infinite Series With Some Bounds

. In this paper, we consider a given in(cid:12)nite series in x of the form y ( x ) = ∑ 1 k =0 b k x k expressed formally also by an in(cid:12)nite product as y ( x ) = (cid:5) 1 k =1 (1 (cid:0) xa k ) into real positive zeros a i ; i = 1 ; 2 ; : : : ; 1 forming a strictly increasing sequence. For consideration of polynomials of degree n , we replace suitably 1 by n . Using the known formal solution of a second linear differential y " = f ( x ) y; y (0) = y 0 ; y ′ (0) = y ′ 0 in the form y ( x ) = ∑ 1 k =0 d k x k , we demonstrate that the above in(cid:12)nite product form of y ( x ) yields the set of in(cid:12)nite equations of the form for a suitable f ( x ). ∑ 1 k =1 ( a k ) (cid:0) p = c p , p = 1 ; 2 ; : : : ; 1 with c ′ k s depending on f ( x ), its derivarives at x = 0 and b ′ k s. Recognizing the in(cid:12)nite matrix as the in(cid:12)nite Vandermonde matrix, some bounds for the zeros are given.