Some Identities On Laguerre Polynomial

The study on the algebra about all kinds of number theory's functions in the elementary theory is always one of the important issues of many number theorists, including the orthogonality of the Laguerre polynomial, the irreducibility of the generalized Laguerre polynomial, the Galois groups of the generalized Laguerre polynomial, the algebraic properties of the 2D-Laguerre polynomial etc have the extremely important significance. So far, many profound and important conclu-sions have been given on these issues, which make a significant contribution to number theory's farther development and application, and inspire young people make progress in the number theory's research simultaneously, we make farther research based on the interest on it and the predecessor's research achievements. In particular, the main conclusions that the convolution and its orthogonality of the Laguerre polynomial, the orthogonality of the generalized Laguerre polynomial are as follows:The above problem in this paper, we research the convolution and its or-thogonality of the Laguerre polynomial and the orthogonality of the generalized Laguerre polynomial mainly based on elementary methods and the properties of power series, the definition, the algebraic properties and orthogonality of the La-guerre polynomial, and give a strong general term formula for the orthogonality and the formulas of the orthogonality.