Paritylike

Paritylike is a term used to describe a generalized symmetry in sequences or functions that resembles parity but does not require the strict even or odd condition. In many mathematical contexts, parity refers to functions satisfying f(-x) = f(x) (even) or f(-x) = -f(x) (odd). Paritylike broadens this idea to accommodate sign changes that depend on the index, rather than a universal rule.

A common formalization is: a sequence a_n indexed by all integers is paritylike if there exists a sign function s(n) taking values in {+1, -1} such that a_{-n} = s(n) a_n for every integer n. Equivalently, one can write a_n = s(n) b_{|n|} for some nonnegative or nonzero sequence b_k. In this view, even and odd sequences are special cases where s(n) is identically +1 or -1, respectively.

Properties of paritylike patterns include their behavior under scaling and shifts. If s(n) depends only on |n|, the relation a_{-n} = s(n) a_n defines a symmetry that is stable under certain algebraic operations. Paritylike sequences can be characterized by a two-way symmetry between positive and negative indices, often simplifying analysis in problems with symmetric domains or Fourier-like decompositions.

Examples commonly cited include sequences where a_{-n} equals a_n (paritylike with s ≡ +1) and those where a_{-n} equals -a_n (paritylike with s ≡ -1). More complex paritylike patterns arise when s(n) varies with n in a prescribed manner.

See also parity, even and odd functions, symmetry, and sign patterns. The term paritylike is not universally standardized and may appear with context-dependent definitions in mathematical literature.