On sums of powers of complex numbers

1964 ◽  
Vol 12 (1-2) ◽  
pp. 185-188 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

Modified Newton identities

2020 ◽  
Vol 20 (3) ◽  
pp. 999-1015
Author(s):  
M. Mestechkin
Keyword(s):  
Algebraic Equation ◽  
Polynomial Coefficient ◽  
Complex Numbers ◽  
Numerical Factor ◽  
Sums Of Powers

The sums of powers with identical exponents of natural, real, or complex numbers, considered as roots of algebraic equation, are expressed directly through the products of coefficients of that equation, starting from the well-known Newton identities. The final Eq. (6) includes the same power of sum of all numbers ± a sum over all partitions of the exponent. Each term of the last sum is the equation coefficients product with the net power keeping the “dimensionality” of the exponent and having a numerical factor, equal to a proper polynomial coefficient, built of exponents of equation coefficients entering the product. The revers Eq. (43) for equation coefficients is also a sum over all partitions of the same exponent with known numerical coefficients. The entering products are built of “commutators-anticommutators of power of sum and sum of powers” (C-A) of the initial sum addends. The numerous identities Eq. (44) for a C-A with an exponent, exceeding the number of C-A sum terms by 2, and similar C-A-s with lower exponents are established.

On a problem of Turán concerning sums of powers of complex numbers

1994 ◽  
Vol 65 (3) ◽  
pp. 209-216 ◽  
Author(s):  
A. Biró
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

Some further estimates concerning sums of powers of complex numbers

1969 ◽  
Vol 20 (1-2) ◽  
pp. 193-210 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

A note on sums of powers of complex numbers

1993 ◽  
Vol 62 (3-4) ◽  
pp. 209-210 ◽  
Author(s):  
J. Fabrykowski
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

scholarly journals Turán's second theorem on sums of powers of complex numbers.

1979 ◽  
Vol 26 (2) ◽  
pp. 205-211
Author(s):  
T. W. Cusick ◽  
G. Kolesnik
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

scholarly journals On algorithms for computing the sums of powers of arithmetic progressions

2020 ◽  
Vol 26 (4) ◽  
pp. 113-121
Author(s):  
Peter J. Shiue ◽  
◽  
Shen C. Huang ◽  
Eric Jameson ◽  
◽  
...  
Keyword(s):  
Elementary Proof ◽  
Stirling Numbers ◽  
Arithmetic Progressions ◽  
Experimental Results ◽  
Negative Integer ◽  
Complex Numbers ◽  
Sums Of Powers

This paper is concerned with sums of powers of arithmetic progressions of the form a^p+(a+d)^p+(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This paper gives an elementary proof to a theorem presented by Laissaoui and Rahmani [9] as well as an algorithm based on their formula. Additionally, this paper presents a simplification to Laissaoui and Rahmani’s formula that is better suited to computation, and a second algorithm based on this simplification. Both formulas use Stirling numbers of the second kind, which are the number of ways to partition p labelled objects into k nonempty unlabelled subsets [4]. An analysis of both algorithms is presented to show the theoretical time complexities. Finally, this paper conducts experiments with varying values of p. The experimental results show the proposed algorithm remains around 10% faster as p increases.

scholarly journals Sums of powers of complex numbers

1967 ◽  
Vol 17 (2) ◽  
pp. 269-279 ◽  
Author(s):  
J.D Buckholtz
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

On the sums of powers of complex numbers

1956 ◽  
Vol 7 (3-4) ◽  
pp. 283-289 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Complex Numbers ◽  
Sums Of Powers

Extremal problems for sums of powers of complex numbers

1966 ◽  
Vol 17 (1-2) ◽  
pp. 147-153 ◽  
Author(s):  
J. D. Buckholtz
Keyword(s):  
Extremal Problems ◽  
Complex Numbers ◽  
Sums Of Powers

scholarly journals Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers

2021 ◽  
Vol 27 (4) ◽  
pp. 140-148
Author(s):  
Peter J. Shiue ◽  
◽  
Shen C. Huang ◽  
Jorge E. Reyes ◽  
◽  
...  
Keyword(s):  
Arithmetic Progressions ◽  
Experimental Results ◽  
Negative Integer ◽  
New Method ◽  
Complex Numbers ◽  
Eulerian Numbers ◽  
Sums Of Powers ◽  
Novel Algorithms

The sums of powers of arithmetic progressions is of the form a^p+(a+d)^p +(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This sum can be computed using classical Eulerian numbers \cite{worpitzky1883studien} and general Eulerian numbers \cite{xiong2013general}. This paper gives a new method using classical Eulerian numbers to compute this sum. The existing formula that uses general Eulerian numbers are more algorithmically complex due to more numbers to compute. This paper presents and focuses on two novel algorithms involving both types of Eulerian numbers. This paper gives a comparison to Xiong \textit{et al.}’s result with general Eulerian numbers \cite{xiong2013general}. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of p are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around 70\% faster as p increases.

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