Analytical Relations and Statistical Estimations for Sums of Powered Integers
Finding analytical closed-form solutions for the sums of powers of the first <i>n</i> positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via the f...
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MDPI AG
2025-01-01
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| author | Stan Lipovetsky |
| author_facet | Stan Lipovetsky |
| author_sort | Stan Lipovetsky |
| collection | DOAJ |
| description | Finding analytical closed-form solutions for the sums of powers of the first <i>n</i> positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via the first two power sums. The current paper describes new presentations of the power sums and their extensions from polynomial to algebraic functions. Particularly, it shows that power sums of any higher order can be expressed just by a value of the arithmetic progression of the first power sum, or by the second power sum, or approximately by any another power sum. Regression modeling for the estimation of the powered sums is also considered, which is helpful for finding approximate values of long sums for big powers. Several problems based on the relations between sums of different powers in explicit forms are suggested for educational purposes. |
| format | Article |
| id | doaj-art-f7e7d4cdc2b84dad81a4f9786d629094 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-f7e7d4cdc2b84dad81a4f9786d6290942025-01-24T13:22:12ZengMDPI AGAxioms2075-16802025-01-011413010.3390/axioms14010030Analytical Relations and Statistical Estimations for Sums of Powered IntegersStan Lipovetsky0Independent Researcher, Minneapolis, MN 55305, USAFinding analytical closed-form solutions for the sums of powers of the first <i>n</i> positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via the first two power sums. The current paper describes new presentations of the power sums and their extensions from polynomial to algebraic functions. Particularly, it shows that power sums of any higher order can be expressed just by a value of the arithmetic progression of the first power sum, or by the second power sum, or approximately by any another power sum. Regression modeling for the estimation of the powered sums is also considered, which is helpful for finding approximate values of long sums for big powers. Several problems based on the relations between sums of different powers in explicit forms are suggested for educational purposes.https://www.mdpi.com/2075-1680/14/1/30power sums of integersNicomachus theoremFaulhaber polynomialsalgebraic functionsrelations between sums of different powersregression models for sums |
| spellingShingle | Stan Lipovetsky Analytical Relations and Statistical Estimations for Sums of Powered Integers Axioms power sums of integers Nicomachus theorem Faulhaber polynomials algebraic functions relations between sums of different powers regression models for sums |
| title | Analytical Relations and Statistical Estimations for Sums of Powered Integers |
| title_full | Analytical Relations and Statistical Estimations for Sums of Powered Integers |
| title_fullStr | Analytical Relations and Statistical Estimations for Sums of Powered Integers |
| title_full_unstemmed | Analytical Relations and Statistical Estimations for Sums of Powered Integers |
| title_short | Analytical Relations and Statistical Estimations for Sums of Powered Integers |
| title_sort | analytical relations and statistical estimations for sums of powered integers |
| topic | power sums of integers Nicomachus theorem Faulhaber polynomials algebraic functions relations between sums of different powers regression models for sums |
| url | https://www.mdpi.com/2075-1680/14/1/30 |
| work_keys_str_mv | AT stanlipovetsky analyticalrelationsandstatisticalestimationsforsumsofpoweredintegers |