若干包含Laguerre多项式的等式和同余式

On Some Identities and Congruences Involving the Laguerre Polynomials

令L_n(x)为Laguerre多项式,即L_0(x)=1,L_1(x)=-x+1,且对所有整数n≥1,有递推公式L_(n+1)(x)=(2n+1-x)L_n(x)-n^2L_(n-1)(x).主要使用组合及初等方法研究一类包含L_n(x)的卷积和式,给出其有趣的计算公式,并得到一些包含Laguerre多项式的等式和同余式,这些结果均有着重要的应用.

Let L_n（x） denotes the Laguerre polynomials.That is,L_0（x） = 1,L_1（x） =- x ＋1,and the recurrence formula L_（n＋1）（x） =（2n ＋ 1- x）L_n（x）- n^2L_（n-1）（x） for all integers n ≥1.In this paper,we using some combinational skill and elementary methods to study the calculating problem of one kind convolution sums involving L_n（x）,and give some interesting computational formulae for it.As some applications of our results,we obtained some new identities and congruences involving the Laguerre polynomials.