TITLE:
Further Study of the Shape of the Numbers and More Calculation Formulas
AUTHORS:
Ji Peng
KEYWORDS:
Shape of Numbers, Calculation Formula, Combinatorics, Congruence, Gaussian Coefficient, Stirling Number
JOURNAL NAME:
Open Access Library Journal,
Vol.8 No.10,
October
20,
2021
ABSTRACT: The core of Shape of numbers is formal calculation, which has three forms. This paper proves the equivalence of these forms and extends the formula to the general case. Some properties of the coefficients are summarized and some new conclusions are drawn. The coefficient matrix is studied and the corresponding results are obtained. Using the formal method, the calculation formula of ∑n-0N-1Πi-1M (Ki+Diqn) is obtained. The key is to use the Gaussian coefficient, which shows its new scope of application. Using the derivation in this paper, the calculation formula of ∑n-0N-1qn(MN+M) is obtained. By introducing a new number: AqM=∑k-0Mqn(1-q)M-kqKS2(M,k)k, this paper obtains the formula of ∑n-0N-1qnnM, at the same time, find the other three expressions of AqM.