%0 Journal Article
%T Further Study of the Shape of the Numbers and More Calculation Formulas
%A Ji Peng
%J Open Access Library Journal
%V 8
%N 10
%P 1-27
%@ 2333-9721
%D 2021
%I Open Access Library
%R 10.4236/oalib.1107969
%X 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 ¡Æ<sub>n-0</sub><sup style="margin-left:-17px;">N-1</sup>¦°<sub>i-1</sub><sup style="margin-left:-15px;">M</sup> (K<sub>i</sub> D<sub>i</sub>q<sup>n</sup>) 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 ¡Æ<sub>n-0</sub><sup style="margin-left:-17px;">N-1</sup>q<sup>n</sup>(<sub>M</sub><sup style="margin-left:-10px;">N M</sup>) is obtained. By introducing a new number: A<sub>q</sub><sup style="margin-left:-10px;">M</sup>=¡Æ<sub>k-0</sub><sup style="margin-left:-12px;">M</sup>q<sup>n</sup>(1-q)<sup>M-k</sup>q<sup>K</sup>S<sub>2</sub>(M,k)k, this paper obtains the formula of ¡Æ<sub>n-0</sub><sup style="margin-left:-17px;">N-1</sup>q<sup>n</sup>n<sup>M</sup>, at the same time, find the other three expressions of A<sub>q</sub><sup style="margin-left:-10px;">M</sup>.
%K Shape of Numbers
%K Calculation Formula
%K Combinatorics
%K Congruence
%K Gaussian Coefficient
%K Stirling Number
%U http://www.oalib.com/paper/6763734