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 $S_m(n)={\sum }_{i=1}^n(a_0+(i-1)d{)}^m$ 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 $S_m^{\prime }(0)$ 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 $d^m{B}_m(a_0/d)$; the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are $B_m(1)$, so $C_1=B_1(1)=+\frac{1}{2}$, 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.

arXiv commentMajor revision: the previous version contained errors; fully rewritten with a rigorous derivation. 15 pages