Some Properties Of Delannoy Numbers

The Delannoy numbers D(n,k)are the number of lattice paths from(0,0)to(n,k)using steps(1,0),(0,1)and(1,1).The object of this thesis is to study the congruence property of the Delannoy numbers and its related analysis properties.The thesis is organized as follows:In the first part,we present some basic concepts about the Delannoy numbers,in-cluding its origin,congruence properties,analytic properties.Besides,we also introduce the background and some researching methods about the Delannoy numbers.The second part is devoted to show the congruence properties of the Delannoy num-bers.Starting with the recurrence of the Delannoy numbers,we prove the self-similarity of the Delannoy array modulo 3 through the matrix.We also prove the symmetry of the principal cell by induction.Finally,we prove the distribution and number of zeros on the middle row and column of the principal cell.In the third part,we study the analytic properties of Delannoy numbers,including the total positivity of Delannoy square and Delannoy triangle,the Stieltjes moment prop-erty of the central Delannoy numbers.Furthermore,by generalizing the Galton knocked boards and simulating the distribution of beads,we verify the asymptotic normality of the sequence in the Delannoy triangle when ?=arctan1/3 and arctan1/2.