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Further Study of the Shape of the Numbers and More Calculation Formulas

DOI: 10.4236/oalib.1107969, PP. 1-27

Subject Areas: Discrete Mathematics

Keywords: Shape of Numbers, Calculation Formula, Combinatorics, Congruence, Gaussian Coefficient, Stirling Number

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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.

Cite this paper

Peng, J. (2021). Further Study of the Shape of the Numbers and More Calculation Formulas. Open Access Library Journal, 8, e7969. doi: http://dx.doi.org/10.4236/oalib.1107969.

References

[1]  Peng, J. (2021) Redefining the Shape of Numbers and Three Forms of Calculation. Open Access Library Journal, 8, 1-22. https://doi.org/10.4236/oalib.1107277
[2]  Peng, J. (2020) Shape of Numbers and Calculation Formula of Stirling Numbers. Open Access Library Journal, 7, 1-11. https://doi.org/10.4236/oalib.1106081
[3]  Peng, J. (2020) Subdivide the Shape of Numbers and a Theorem of Ring. Open Access Library Journal, 7, 1-14. https://doi.org/10.4236/oalib.1106719
[4]  Peng, J. (2020) Subset of the Shape of Numbers. Open Access Library Journal, 7, 1-15. https://doi.org/10.4236/oalib.1107040
[5]  Peng, J. (2021) Expansion of the Shape of Numbers. Open Access Library Journal, 8, 1-18. https://doi.org/10.4236/oalib.1107120
[6]  MacMahon, P.A. (1913) The Indices of Permutations and the Derivation Therefrom of Functions of a Single Variable Associated with the Permutations of Any Assemblage of Objects. American Journal of Mathematics, 35, 281-322. https://doi.org/10.2307/2370312

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