Concept
Generating function
Related concepts (78)
Eulerian number
In combinatorics, the Eulerian number is the number of permutations of the numbers 1 to in which exactly elements are greater than the previous element (permutations with "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis. Other notations for are and . The Eulerian polynomials are defined by the exponential generating function The Eulerian numbers may be defined as the coefficients of the Eulerian polynomials: An explicit formula for is For fixed there is a single permutation which has 0 ascents: .
Linear recurrence with constant coefficients
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1.
Cauchy product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
Ramanujan's sum
In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula where (a, q) = 1 means that a only takes on values coprime to q. Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes.
Double factorial
In mathematics, the double factorial of a number n, denoted by n!!, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, Restated, this says that for even n, the double factorial is while for odd n it is For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0!! = 1 as an empty product. The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as The sequence of double factorials for odd n = 1, 3, 5, 7, 9,...
Jordan's totient function
Let be a positive integer. In number theory, the Jordan's totient function of a positive integer equals the number of -tuples of positive integers that are less than or equal to and that together with form a coprime set of integers. Jordan's totient function is a generalization of Euler's totient function, which is given by . The function is named after Camille Jordan. For each , Jordan's totient function is multiplicative and may be evaluated as where ranges through the prime divisors of .
Enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ...
Generalized Appell polynomials
In mathematics, a polynomial sequence has a generalized Appell representation if the generating function for the polynomials takes on a certain form: where the generating function or is composed of the series with and and all and with Given the above, it is not hard to show that is a polynomial of degree . Boas–Buck polynomials are a slightly more general class of polynomials. The choice of gives the class of Brenke polynomials. The choice of results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.
Difference polynomials
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases. The general difference polynomial sequence is given by where is the binomial coefficient. For , the generated polynomials are the Newton polynomials The case of generates Selberg's polynomials, and the case of generates Stirling's interpolation polynomials.