Geodesic
Summation on the basis of combinatorial representation of equal powers
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 149-157

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In the paper the conclusion of combinatorial expressions for the sums of members of several sequences is considered. Conclusion is made on the basis of combinatorial representation of the sum of the weighted equal powers. The weighted members of a geometrical progression, the simple arithmetic-geometrical and combined progressions are subject to summation. One of principal places in the given conclusion occupies representation of members of each of the specified progressions in the form of matrix elements. The row of this matrix is formed with use of a gang of equal powers with the set weight factor. Besides, in work formulas of combinatorial identities with participation of free components of the sums of equal powers, and also separate power-member of sequence of equal powers or a geometrical progression are presented. All presented formulas have the general basis-components of the sums of equal powers.
Keywords: sum of equal powers, power, geometrical progression, simple arithmetic-geometrical progression, simple combined progression
Mots-clés : matrix. 
A. I. Nikonov. Summation on the basis of combinatorial representation of equal powers. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 149-157. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a10/
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[1] Coolidge J. L., “The story of the Binomial Theorem”, Amer. Math. Monthly, 56:3 (1949), 147–157 | DOI | Zbl

[2] Lin Cong, On Bernoulli Numbers and Its Properties, 2004, 7 pp., arXiv: [math.HO] math/0408082

[3] Benjamin A. T., Plott S. S., “A combinatorial approach to Fibonomial coefcients”, Fibonacci Quarterly, 46/47:1 (2008/2009), 7–9 Retrieved from (August 08, 2015) http://www.fq.math.ca/Papers1/46_47-1/Benjamin_11-08.pdf

[4] Benjamin A. T., Greg O. P., Quinn J. J., “A Stirling Encounter with Harmonic Numbers”, Mathematics Magazine, 75:2 (2002), 95–103 | DOI | Zbl

[5] Edwards A. W. F., “Sums of Powers of Integers: A Little of the History”, The Mathematical Gazette, 66:435 (1982), 22–28 | DOI | Zbl

[6] Beery J., Sums of Powers of Positive Integers - Introduction, Convergence (July 2010), 2010 | DOI

[7] Nikonov A. I., “Converting the Sum of Weighted Degrees of Natural Numbers with the Same Parameters”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2010, no. 1(20), 258–262 | DOI

[8] Nikonov A. I., “Reduction of the sum of the weight equal powers to explicit combinatorial representation”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 3(28), 163–169 | DOI | Zbl

[9] Arithmetic and Geometric Series in Ken Ward's Mathematics Pages Retrieved from (August 08, 2015) http://www.trans4mind.com/personal_development/mathematics/series/airthmeticGeometricSeries.htm

[10] Riley K. F., Hobson M. P., Bence S. Y., Mathematical Methods for Physics and Engineering, Cambridge University Press, Cambridge, 2006, xxiv+1232 pp. | DOI

[11] Nikonov A. I., “Rising of Efficiency of Enciphering on the Basis of Summation of Products”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, no. 2(35), 199–207 | DOI | Zbl

[12] Graham R. L., Knuth D. E., Patashnik O., Concrete mathematics. A foundation for computer science, Addison-Wesley Publishing Company, Reading, MA, 1994, xiv+657 pp. | Zbl