In this note, we introduce the concept of a dual pair of recursive formulas for the sums of powers of integers

Central to this concept is the symmetry property exhibited by the power sum polynomial S_k(n). We illustrate the concept by some examples taken from the literature, and derive our own pair of dual recursive formulas for S_k(n).

sums of powers of integers, symmetry property of the power sum polynomials, dual recursive formulas

Received: September 16, 2024; Revised: October 16, 2024; Accepted: October 21, 2024; Published: October 24, 2024

How to cite this article: José L. Cereceda, Dual recursive formulas for the sums of powers of integers, Far East Journal of Mathematical Education 26(2) (2024), 111-121. http://dx.doi.org/10.17654/0973563124012

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] T. M. Apostol, A primer on Bernoulli numbers and polynomials, Mathematics Magazine 81(3) (2008), 178-190.
[2] J. L. Cereceda, On a recurrence relation for the sums of powers of integers, Far East Journal of Mathematical Education 24 (2023), 45-50.
[3] J. L. Cereceda, Sums of powers of integers and Stirling numbers, Resonance 27(5) (2022), 769-784.
[4] H. Eba, Sums of powers of integers, Missouri Journal of Mathematical Sciences 31(1) (2019), 66-78.
[5] E. G. Eigel, Jr., Sums of powers of integers, Pi Mu Epsilon Journal 4(1) (1964), 7 10.
[6] M. El-Mikkawy and F. Atlan, Notes on the power sum Annals of Pure and Applied Mathematics 9(2) (2015), 215-232.
[7] R. Farhadian, A probabilistic proof of a recursion formula for sums of integer powers, Integers 23 (2023), Article #A88.
[8] J. D. Greene, To compute sum of integers raised to powers, The Mathematical Gazette 68(446) (1984), 278-280.
[9] R. Hersh, Why the Faulhaber polynomials are sums of even or odd powers of The College Mathematics Journal 43(4) (2012), 322-324.
[10] K. Hirose, Computing sums of powers of integers with the hockey-stick identity and the Stirling numbers, The College Mathematics Journal 55(3) (2024), 247 249.
[11] F. T. Howard, Sums of powers of integers, Mathematical Spectrum 26(4) (1993/1994), 103-109.
[12] X. Hu and Y. Zhong, A probabilistic proof of a recursion formula for sums of powers, The American Mathematical Monthly 127(2) (2020), 166-168.
[13] K. Kanim, Proof without words: How did Archimedes sum squares in the sand?, Mathematics Magazine 74(4) (2001), 314-315.
[14] K. Kanim, Proof without words: The sum of cubes—An extension of Archimedes’ sum of squares, Mathematics Magazine 77(4) (2004), 298-299.
[15] C. Kelly, An algorithm for sums of integer powers, Mathematics Magazine 57(5) (1984), 296-297.
[16] T. C. T. Kotiah, Sums of powers of integers—A review, International Journal of Mathematical Education in Science and Technology 24(6) (1993), 863-874.
[17] R. S. Luthar, A simple way of evaluating Pi Mu Epsilon Journal 6(5) (1976), 282-284.
[18] N. J. Newsome, M. S. Nogin and A. H. Sabuwala, A proof of symmetry of the power sum polynomials using a novel Bernoulli number identity, Journal of Integer Sequences 20 (2017), Article 17.6.6.
[19] J. Quaintance and H. W. Gould, Combinatorial identities for Stirling numbers, The unpublished notes of H. W. Gould, World Scientific Publishing, Singapore, 2016.
[20] M. Z. Spivey, The Art of Proving Binomial Identities, CRC Press/Taylor & Francis Group, Boca Raton, 2019.
[21] B. Turner, Sums of powers of integers via the binomial theorem, Mathematics Magazine 53(2) (1980), 92-96.