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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such, corresponding to Newton's generalized binomial theorem for negative powers, is ∏ k = 0 n − 1 1 1 − q k t = ∑ k = 0 ∞ ( n + k − 1 k ) q t k . {\displaystyle \prod _{k=0}^{n-1}{\frac {1}{1-q^{k}t}}=\sum _{k=0}^{\infty }{n+k-1 \choose k ...
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (). It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n; this coefficient can be ...
The usual argument to compute the sum of the binomial series goes as follows. Differentiating term-wise the binomial series within the disk of convergence | x | < 1 and using formula , one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x)u′(x) − αu(x) = 0 with initial condition u(0) = 1.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide ... Abel's binomial theorem, named after Niels Henrik Abel, ...
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
The expansion of the n th power uses the numbers n rows down from the top of the triangle. An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating Pythagorean triples: For m < n, let a = n 2 − m 2, b = 2mn, and c = n 2 + m 2; then a 2 + b 2 = c 2.
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials .