关于拉盖尔多项式卷积的正交性

On the Orthogonality of the Convolution of Laguerre Polynomials

设Ln(x)表示拉盖尔多项式,即L0(x)=1,L1(x)=-x+1,当n≥1时有递推关系式Ln+1(x)=(2n+1-x)Ln(x)-n2 Ln-1(x).运用初等方法以及幂级数的性质研究Ln(x)的一类卷积的计算问题,并给出该类卷积的一个有趣的计算公式.

For any integer n≥0,let Ln（x）denotes the Laguerre polynomials.That is,L0（x）=1,L1（x）=-x＋1,and Ln＋1（x）=（2n＋1-x）Ln（x）-n2Ln-1（x）for all positive integers n ≥1.By using the elementary method and the properties of power series,the computational problem of the convolution involving Laguerre polynomials is studied,and an interesting computational formula for it is proposed.