Class Of Sequences Of Multiple Convolution Formula

Convolution and multiple convolution are the basic content of combinational analysis. They are also the basic ways to solve some problems on combinatorial enumeration, proof of combinatorial identity and generating function for sequence. We utilize the knowledge of recurrence relation and generating function to inves-tigate multiple convolution sum problems on several types of sequence:(1) We give the simple convolution formulae for natural numbers and square numbers. In the general case, we make use of Stirling numbers of the second kind to get an explicit formula. Furthermore, we also solve multiple convolution sum problems on positive odd numbers, positive even numbers and their square numbers.(2) We investigate some problems of multiple convolution of Fibonacci se-quence and give 1-fold,2-fold and 3-fold multiple convolution formulae on Fi-bonacci sequence. For the issues of multiple convolutions of odd terms and even terms of Fibonacci sequence, we give 1-fold and 2-fold multiple convolutions.(3) We investigate some problems on Lucas sequence associated with Fi-bonacci sequence and give multiple convolution formula on Lucas sequence.(4) Multiple convolution of general second-order recursive sequence is rather complex. We only investigate the case of Pell sequence and give multiple convolu-tion formula on Pell sequence similar to multiple convolution formula on Fibonacci sequence.