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The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
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 ...
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.
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that It states that ( 1 + x ) α ≈ 1 + α x . {\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}
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).
Thus many identities on binomial coefficients carry over to the falling and rising factorials. The rising and falling factorials are well defined in any unital ring, and therefore can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
Lucas's theorem can be generalized to give an expression for the remainder when () is divided by a prime power p k.However, the formulas become more complicated. If the modulo is the square of a prime p, the following congruence relation holds for all 0 ≤ s ≤ r ≤ p − 1, a ≥ 0, and b ≥ 0.