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Research Paper | Mathematics | Volume 15 Issue 5, May 2026 | Pages: 1739 - 1740 | India
A Generalized Finite Difference Structure on Arithmetic Power Arrays
Abstract: We present a finite-difference framework, kind of built from a two-dimensional arithmetic power table. Fix integers a and d and also a nonnegative integer n; form a table so that the (k, j)-entry equals (a + (j-1)n, meaning the kth row just consists of the nth powers of an arithmetic progression or something close to that. Let Sn(k) be the sum of the entries in the kth row. Elementary algebra shows that Sn(k) ends up as a polynomial in k, with degree n. Then, using classical finite-difference theory and a bunch of combinatorial identities, we show that the (n+1)th forward difference Δ(n+1) Sn(k) vanishes identically for all integer k, and we also write down explicit formulas for the lower-order differences using binomial sums. The paper then supplies precise definitions, plus illustrative examples for small n, and some structural observations about how the progression parameters a and d influence the coefficients. Finally, there is a complete self-contained proof of the main theorem, with everything laid out in a fairly direct way, not too hand-wavy.
Keywords: Arithmetic Progression, Finite Difference, Polynomial Sequence, Difference Table
How to Cite?: Subham Dutta, "A Generalized Finite Difference Structure on Arithmetic Power Arrays", Volume 15 Issue 5, May 2026, International Journal of Science and Research (IJSR), Pages: 1739-1740, https://www.ijsr.net/getabstract.php?paperid=SR26527124530, DOI: https://dx.dx.doi.org/10.21275/SR26527124530