Analytic solution of transcendental equations

According to the assumption that si = sj for i = j, the matrix (5) has an inverse and the system (2) and (4) can be solved. We denote by V the Vandermonde determinant of the matrix (5) and by Vk the Vandermonde determinant of order (n − 1) of the variables s1, . . . , sk−1, sk+1, . . . , sn. We also denote by Φ r the fundamental symmetric function of the r-th order of (n − 1) variables s1, . . . , sk−1, sk+1, . . . , sn for r = 0, 1, . . . , n− 1: Φ 0 = 1, Φ 1 = s1 + s2 + · · ·+ sk−1 + sk+1 · · ·+ sn = −a1 − sk, Φ 2 = s1s2 + s1s3 + · · ·+ s1sk−1 + s1sk+1 · · ·+ s2s3 + · · ·+ s2sk−1 + s2sk+1 · · ·+ s2sn + . . . = a2 − s1sk − s2sk − · · · − snsk, Φ n = n ∏