Research On Arithmetic Identities Of Several Famous Combination Numbers

Catalan number,Delannoy number and Schr(?)der number not only have many im-portant applications in number theory and combinatorics,but also are one of the hot issues in contemporary research.So far,many mathematicians have conducted exten-sive and in-depth research on their various properties,and found that some arithmetic equations of Catalan numbers,Delannoy numbers and Schr(?)der numbers play an im-portant role in the study of the counting problem of grid roads and some congruence topics.Therefore,this article uses some analytical methods and combination tech-niques to establish several identities related to these three numbers.It turns out that some known results at present are obtained in brief ways.The main work is as follows:(1)The Catalan number is further studied,and two new sum formulas for the product of any number of Catalan numbers are established by using some methods such as generating function,the properties of the Bell polynomial,Leibniz derivation,Faa di Bruno formula and the combined summation transformation.These equations can not only be regarded as a generalization of the Catalan number recursive formula,but also help to simplify some counting problems encountered in calculation and to study the estimation of upper and lower bounds of counting problems.(2)By applying the methods and techniques of the two Catalan number identities established above,we have studied Delannoy numbers,established two identities for the product sum of any number of Delannoy numbers,and conducted similar studies on Schr(?)der numbers,so that some results of some scholars on Delannoy numbers and Schr(?)der numbers are obtained as special cases.