Class Of Binomial Coefficients And The Sequence Of More Than Nature

It has nearly 700 year's histories about the binomial expression research. It's an important questions that is the recursion formula of the binomial coefficients and arrays in the theory of numbers research. In this dissertation, the congruence properties about several kinds of the binomial coefficients and arrays are mainly studied by using division and congruence theories.This paper is composed mainly of four parts. The fundamental concepts and the related knowledge are introduced and the congruence properties about the binomial coefficient and array bn(r) are mainly studied by using division, congruence theories and interrelated conclusions in the first chapter.The congruence properties about the binomial coefficient and array under module p2 are discussed in the second chapter. In the third chapter, the binomial array bn(r) is extended to Moreover, the following results are obtained that:①bkp(k,s)= 0(mod p) is true for every odd prime number p and k,s∈N;②bkp+t(r,s)= ak(r+s)bt(r,s)(mod p) is true for every odd prime number p and k,r,t∈N, t≤p-1.In the fourth chapter, the congruence properties of the binomial coefficient are discussed and the special index and the related results about the recurrence formula of polynomials which it satisfies with. Furthermore, the result that it doesn't exist three recurrence formulas is obtained.