Binomki

Binomki are a concept in algebraic combinatorics that describe a family of generalized binomial coefficients associated with binomial-type polynomials. For each nonnegative integer n there is a sequence of numbers binomki(n,k) with 0 ≤ k ≤ n, defined so that for a suitable polynomial sequence P_n the addition rule holds: P_n(x+y) = sum_{k=0}^n binomki(n,k) P_k(x) P_{n-k}(y). The numbers binomki(n,k) generalize ordinary binomial coefficients.

In the simplest case, where P_n(x) = x^n, the binomki(n,k) coincide with the ordinary binomial coefficients C(n,k). More generally, choosing P_n from a binomial-type family—polynomials that satisfy a similar addition rule with a different basis—yields other binomki sets. These can be generated by exponential generating functions G(t,x) = sum_{n≥0} P_n(x) t^n / n! which satisfy G(t, x+y) = G(t, x) G(t, y).

Applications and interpretations of binomki include providing a unified framework for binomial-type identities across polynomial bases, aiding combinatorial counting with weights and algebraic manipulations in computer algebra systems, and informing probabilistic models where sums of independent binomial-type variables are studied. They also relate to lattice-path models and to the broader theory of binomial-type polynomials.

See also: Binomial coefficient, Binomial theorem, Binomial-type polynomials.