A Symbolic Differentiation_Summation Framework for Power Sums of Arithmetic Sequences
Abstract
We give a self-contained derivation of the closed form of the power sums Sm(n)=∑i=1n(a0+(i−1)d)m in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope Sm'(0) of the polynomial at the origin and fixed by the value Sm(1)=a0m. This yields a closed form for Sm(n) in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as dmBm(a0/d); the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are Bm(1), so C1=B1(1)=+1/2, a value that reflects the summation range; a reflection identity shows, without generating functions, that the odd constants beyond the first vanish. The coefficients of the recurrence are scaled binomial coefficients from a single row of Pascal's triangle, and the same approach gives closed forms for sequences with polynomial differences. The integral recursion behind the method is known in the normalized case; each result is related to its classical counterpart, and a verified implementation is provided.
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BibTeX
@misc{abdlsahib2026symbolic,
title = {{A Symbolic Differentiation_Summation Framework for Power Sums of Arithmetic Sequences}},
author = {Ahmed Abdlmohsin Abdlsahib},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18898},
eprint = {2610.18898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18898},
}
RIS
TY - UNPB
TI - A Symbolic Differentiation_Summation Framework for Power Sums of Arithmetic Sequences
AU - Ahmed Abdlmohsin Abdlsahib
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.18898
UR - https://arxiv.org/abs/2610.18898
AB - We give a self-contained derivation of the closed form of the power sums S_m(n)=∑_i=1^n(a_0+(i−1)d)^m in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope S_m'(0) of the polynomial at the origin and fixed by the value S_m(1)=a_0^m. This yields a closed form for S_m(n) in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as d^mB_m(a_0/d); the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are B_m(1), so C_1=B_1(1)=+1/2, a value that reflects the summation range; a reflection identity shows, without generating functions, that the odd constants beyond the first vanish. The coefficients of the recurrence are scaled binomial coefficients from a single row of Pascal's triangle, and the same approach gives closed forms for sequences with polynomial differences. The integral recursion behind the method is known in the normalized case; each result is related to its classical counterpart, and a verified implementation is provided.
ER -