Sumk

Sumk is a generalized summation operator used in mathematics and computer science. Parameterized by a nonnegative integer k, it defines a weighted sum of the terms of a finite sequence and is employed to study moments, weighted aggregates, and discrete transforms. The term appears in discussions of numerical methods, discrete calculus, and algorithmic data processing, where controlled weighting of terms can affect convergence and stability.

Definition and variants: Let a = (a_1, ..., a_N) be a finite sequence. For an integer k ≥ 0, sumk(a; N) is defined as the weighted sum sum_{i=1}^N w_k(i) a_i, where w_k is a weight function depending on i and k. A common choice is w_k(i) = i^k, yielding sum0(a; N) = sum a_i, sum1(a; N) = sum i a_i, and so on. More generally, sumk can involve alternative kernels such as binomial, exponential, or piecewise weights, enabling customized emphasis on early or late terms.

Properties and connections: Sumk is linear in the input sequence. If all a_i are nonnegative and w_k(i) are nonnegative, sumk is nondecreasing in N. Sumk relates to moments in probability theory when a_i represent probability masses and i encodes discrete outcomes. It also connects to generating functions: the ordinary generating function of {a_i} can be combined with w_k to form weighted transforms.

Applications include computing discrete moments, performing weighted sums in numerical integration and quadrature, and accelerating the convergence of slowly convergent series through weighting schemes. In data analysis, sumk-like operators appear in feature extraction, where higher-k weights emphasize tail terms.

Implementation considerations: sumk can be computed in a single pass with O(N) time and O(1) extra space, and is amenable to vectorization on modern hardware. Variants with different kernels can be precomputed or implemented via convolution-like techniques.