Recurrent Sequences And Combinatorial Identities

Combinatorial counting is the foundation of combinatorics,which is one of more delightful branches of modern mathematics.There are many typical problems in the combinatorial enumeration, which are solved by the recursive relationship,but the Bell polynomial sequence is the basis of recursive relationship.In this paper, we discuss some recurrence relations of ordinary Bell polynomial, and obtain the factorization of the corresponding matrix.The connection between the ordinary Bell matrix and the Fibonacci matrix is studied, moreover, the relationship between the ordinary Bell polynomials,binomial coefficients and the Fibonacci numbers are also derived from the corresponding matrix representations.Furthermore, we provided a new combinatorial sequences-Jacobsthal sequences based on the former studies,and introduced their origin, definition, discussed their basic properties, combinatorial significance and the connection between Jacobsthal matrices and others(such as pascal matrix,Bell polynomial matrix,Stirling matrix and Lah matrix), moreover, the relationship between the binomial coefficients,Bell polynomials,the two kinds of Stirling numbers and Jacobsthal number are also derived from the corresponding matrix representations.At last, by use of theΩiterative matrix of the Bell polynomial,we extend the number of Jacobsthal identities to the more generally cases.