binomiske

Binomiske is a term used in some mathematical discussions to describe a generalized form of the binomial expansion that incorporates a weighting function across the terms. The concept is not standardized and appears mainly in niche texts and online discussions, where it is used to illustrate how binomial coefficients can be adapted to nonuniform contributions of each term.

Let n be a nonnegative integer and w: {0,...,n} -> R be a weight function. The binomiske polynomial of degree n is defined by B_n(x,y; w) = sum_{k=0}^n binom(n,k) w(k) x^{n-k} y^k. The standard binomial theorem is recovered when w(k)=1 for all k.

If w is symmetric with respect to k and n-k (i.e., w(k)=w(n-k)), B_n(x,y; w) is invariant under interchange of x and y. If w is chosen polynomially, the coefficients become polynomial functions in w and expand accordingly. Special case: w is constant c, then B_n = c (x+y)^n; scaling aside, you recover standard form up to a constant factor.

Examples include w(k)=k+1, giving B_n(x,y; w)= sum binom(n,k) (k+1) x^{n-k} y^k. Applications are primarily educational and theoretical, used to illustrate how binomial identities adapt under weighting, and to explore connections with generating functions and biased sampling models.

See also: Binomial theorem, Weighted binomial coefficients. Note: Binomiske is a niche or exploratory term and not part of a widely standardized mathematical vocabulary.