Studies On Mathematical Properties Of Mina-type Matrices

The present thesis is devoted to the so called Mina-type matrix (A(n, k)) with entries given by As two kinds of main mathematical properties, the determinants and the inverse of such Mina-type are investigated mainly. The thesis is planned as follows.In Chapter one, by using of the classical Newton binomial formula, we find a LU decomposition of the Mina matrix (Dxn(fk(x))). Such decomposition is further generalized to the ring of formal power series. Based on such results, we obtain not only an elementary proof of the Mina determinant identity given by Wilf but also the inverse of Mina matrix.Chapter two is a further development of Chapter one, in which we resort to some known classical formulas such as the Lagrange interpolation formula as well as its special forms-Melzak formulas to establish some new results on the determinants.The last part of this thesis discusses Mina-type matrices which are subject to composition of two formal series. As one of main results, we obtain the following matrix inversion. It is obtained by using the classical Faa di Bruno formula and the Lagrange inversion formula.Theorem [Chapter 3, Theorem 3.2.1] Suppose Bn,k(a1,a2,…,an-k+1) is the Bell polynomial defined by f(x)=Σn≥1 anxn with a1≠0.Then whereIn the meantime, some basic applications are also discussed.