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  2. Binomial theorem - Wikipedia

    en.wikipedia.org/wiki/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 ...

  3. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    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 ...

  4. Falling and rising factorials - Wikipedia

    en.wikipedia.org/wiki/Falling_and_rising_factorials

    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.

  5. Faulhaber's formula - Wikipedia

    en.wikipedia.org/wiki/Faulhaber's_formula

    A derivation of Faulhaber's formula using the umbral form is available in The Book of Numbers by John Horton Conway and Richard K. Guy. [17] Classically, this umbral form was considered as a notational convenience. In the modern umbral calculus, on the other hand, this is given a formal mathematical underpinning.

  6. Negative binomial distribution - Wikipedia

    en.wikipedia.org/wiki/Negative_binomial_distribution

    Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, [1] so identifying the specific parametrization used is crucial in any ...

  7. Negative binomial theorem - Wikipedia

    en.wikipedia.org/?title=Negative_binomial...

    This page was last edited on 23 November 2021, at 17:07 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.

  8. Binomial approximation - Wikipedia

    en.wikipedia.org/wiki/Binomial_approximation

    The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.

  9. Gaussian binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Gaussian_binomial_coefficient

    The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian (,).