MonomialIn mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty product and to for any variable . If only a single variable is considered, this means that a monomial is either or a power of , with a positive integer.
Triangular numberA triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is The triangular numbers are given by the following explicit formulas: where , does not mean division, but is the notation for a binomial coefficient.
Beta functionIn mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral for complex number inputs such that . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta. The beta function is symmetric, meaning that for all inputs and .
Binomial (polynomial)In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials. A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable.
Digamma functionIn mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , and it asymptotically behaves as for large arguments () in the sector with some infinitesimally small positive constant . The digamma function is often denoted as or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
Pascal's ruleIn 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 xk term in the expansion of (1 + x)n. There is no restriction on the relative sizes of n and k, since, if n < k the value of the binomial coefficient is zero and the identity remains valid. Pascal's rule can also be viewed as a statement that the formula solves the linear two-dimensional difference equation over the natural numbers.
Binomial transformIn combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined by Formally, one may write for the transformation, where T is an infinite-dimensional operator with matrix elements Tnk.
Multi-index notationMulti-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. An n-dimensional multi-index is an n-tuple of non-negative integers (i.e. an element of the n-dimensional set of natural numbers, denoted ). For multi-indices and one defines: Componentwise sum and difference Partial order Sum of components (absolute value) Factorial Binomial coefficient Multinomial coefficient where .
Binomial seriesIn mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor series for the function centered at , where and . Explicitly, where the power series on the right-hand side of () is expressed in terms of the (generalized) binomial coefficients If α is a nonnegative integer n, then the (n + 2)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formula.
Narayana numberIn combinatorics, the Narayana numbers form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T. V. Narayana (1930–1987). The Narayana numbers can be expressed in terms of binomial coefficients: The first eight rows of the Narayana triangle read: An example of a counting problem whose solution can be given in terms of the Narayana numbers , is the number of words containing n pairs of parentheses, which are correctly matched (known as Dyck words) and which contain k distinct nestings.
