Combinatorial Sequences And Their Applications

This dissertation studies some structure properties and their applications to combinatorial sequences. The content is as follows:1. By the first columns of Aigner matrix and its inverse, we characterize the concept of Catalan-like number pair and discuss their properties. Based on these, the applications are displayed specially on combinatorial identities involving various combinatorial sequences. Interesting that the Hankel determinants of the Catalan-like numbers of the first kind may be expressed in the Catalan-like numbers of the second kind. Finally, we give a result similar to the Talor's series expression for a class of special Catalan-like numbers of the first kind.2. Based on the (q, h)-deformed quantum plane by Benaoum, we establish the transformation formulae of arbitrary degree power of two variables on the (q, h)-deformed quantum plane. Furthermore, we give the (q, h)-analogues of multinomial theorem, binomial reciprocal formula, Chu- Vandermonde identities and a pair of new double-index series inverse formula.3. By considering the enumerative problem of animals resulting from the stacking of labeled squares on a single staircase, we introduce the new concept of the Fibonacci-like numbers and Lucas-like numbers, and discuss their combinatorial properties and the computed method of mean values and mean-square for this enumerative problem. We consider also the enumerative problem of animals resulting from the stacking of labeled squares on double staircase.4. Introducing a class of generalized Fibonacci sequences with double variables, we establish the relation with Aitken, Secant, New-Raphson, Halley transformation etc., generalize the results of many authors. Furthermore, we give a more general generalization of Q-matrix. We also obtain some identities involving two kinds of Chebyshev polynomials which generalized a result of Grabner and Prodinger in more general setting containing Melham's conjecture as a special case. Finally, we discuss the convergence and emanative properties involving the summations for second-order recurrence sequences and a generalization of generalized Jacobsthal polynomials.5. By applying our recurrence method of decreasing order, we obtain the close formulas of convoluted summations for Generalized Fibonacci-Lucas numbers, Euler numbers and Genocchi numbers etc. Furthermore, we obtain the computed formulae of the higher-order cumulants for a class of the lattice animal and some identities for Riemann Zeta functions and Beta functions.6. We define three kinds of generalized Pascal matrices (left, right and symmetry), and discuss the their relations and the Cholesky factorial of generalized symmetry Pascal matrix. Finally, a close relation is obtained between the diagonal of generalized right Pascal matrix and a class of recurrence sequences.7. We establish a class of combinatorial identity involving two sequences and a partialsum of the binomial coefficients, which contain a lot of new and curious combinatorial identities as its special cases.