dwumianowy

Dwumianowy is an adjective in Polish that denotes relation to the dwumian, a binomial or a two-term polynomial, and to the mathematical concepts and identities associated with it. In mathematics, the term is used in contexts such as binomial expansions, binomial coefficients and related theorems. The notion underpins a range of topics from algebra to combinatorics and analysis.

Key concepts related to the dwumian include the binomial coefficients, commonly written as binom(n, k) and defined as n!/(k!(n−k)!). The binomial theorem states that for a natural number n, (x + y)^n equals the sum from k = 0 to n of binom(n, k) x^{n−k} y^k. This is a foundational identity in algebra with wide applications.

There is also a generalized form often called the binomial series: (1 + x)^α = sum_{k=0}^∞ binom(α, k) x^k, where binom(α, k) = α(α−1)…(α−k+1)/k!. This extension allows the expansion to non-integer exponents under appropriate convergence conditions.

In Polish mathematical literature, dwumianowy appears in phrases such as wzór dwumianowy, twierdzenie dwumianowe and szereg dwumianowy, all referring to binomial-related results and series. The concept has applications in probability, combinatorics and analysis, and it provides both computational tools and combinatorial interpretations for choosing elements and forming polynomials with two terms.