"Generating functions were first introduced by Abraham de Moivre in 1730..."
A formal power series used to encode sequences of numbers by representing them as coefficients of the series. Used in combinatorial analysis to study discrete structures.
Power series: A power series is a polynomial with infinitely many terms. The coefficients represent the terms in the series and can be used to create generating functions.
Formal power series: A formal power series is a power series where the coefficients are treated as abstract symbols rather than numbers. These can be used to define generating functions for combinatorial problems.
Binomial theorem: The binomial theorem is a formula that provides a way to expand powers of a binomial expression. It is used to derive many of the generating functions used in combinatorics.
Recurrence relations: A recurrence relation is a mathematical relationship that describes a sequence of numbers in terms of the previous terms in the sequence. These relationships can be used to construct generating functions for combinatorial problems.
Combinatorial classes: Combinatorial classes are sets of mathematical objects that share similar properties. These classes can be used to define generating functions that count the number of objects in these sets.
Exponential generating functions: Exponential generating functions are a type of generating function that counts the number of objects with respect to the size of their set. They are particularly useful in combinatorial problems involving permutations.
Ordinary generating functions: Ordinary generating functions are a type of generating function that counts the number of objects with respect to their integer values. They are particularly useful in combinatorial problems involving partitions.
Generating functions for partitions: Generating functions for partitions are generating functions that are used to count the number of ways in which an integer or a set of integers can be partitioned.
Generating functions for permutations: Generating functions for permutations are generating functions that are used to count the number of permutations of a set with certain properties.
Lagrange inversion formula: The Lagrange inversion formula is a formula that can be used to extract coefficients from generating functions. This formula is particularly useful in combinatorics as it allows for the counting of the number of objects in a set with certain properties.
Ordinary Generating Functions (OGF): These are generated functions used to count the number of items in a set by expressing it as a series of numbers. Each number in the series represents the number of items of a particular type in the set.
Exponential Generating Functions (EGF): Exponential generating functions count the number of items in a set with respect to their order. For example, an EGF can be used to count the number of permutations of a set.
Dirichlet Generating Functions (DGF): These generating functions are used to find the sum of numbers which are products of certain prime factors.
Bell Generating Functions (BGF): Bell generating functions count the number of partitions of a set into non-empty subsets.
Lambert Series (LS): Lambert series are used to represent a function as a series of infinite summands. They are useful for finding values of certain arithmetic functions.
Laguerre Generating Functions (LGF): Laguerre generating functions are used to find the orthogonal polynomials for sequences.
Jacobi-Gauss Generating Functions (JGF): JGFs are used to find the orthogonal polynomials for sequences of real numbers.
Legendre Generating Functions (LGF): Legendre generating functions are used to find the orthogonal polynomials for sequences of real numbers.
Chebyshev Generating Functions (CGF): These are generating functions used to find the Chebyshev polynomials for sequences of real numbers.
Zeta Functions: Zeta functions are used to count the number of ways in which a number can be expressed as a sum of powers of other numbers.
Riemann Zeta Functions (RZF): These are zeta functions used to find the distribution of prime numbers.
Hurwitz Zeta Functions (HZF): These zeta functions are closely related to RZF and help in finding values of the Dirichlet character.
Mellin Generating Functions: These generating functions are used to express the multiplicative properties of certain functions as an infinite series.
Theta Functions: Theta functions are used to study the properties of elliptic curves, modular forms and quadratic forms.
Partition Functions: Partition functions are used to count the number of ways in which a set can be partitioned into disjoint sets.
Bivariate Generating Functions: Bivariate generating functions are used to study the joint properties of two sequences.
"In order to solve the general linear recurrence problem."
"...ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series..."
"Every sequence in principle has a generating function of each type..."
"Generating functions are often expressed in closed form..."
"These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with other generating functions..."
"Yes, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x..."
"No, formal series are not required to give a convergent series..."
"No, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series..."
"No, generating functions are not functions in the formal sense of a mapping from a domain to a codomain..."
"Generating functions are sometimes called generating series..."
"...by treating them as the coefficients of a formal power series."
"No, the formal power series is not required to converge..."
"The 'variable' remains an indeterminate."
"To encode information about infinite multi-dimensional arrays of numbers."
"The nature of the sequence and the details of the problem being addressed."
"Lambert and Dirichlet series require indices to start at 1 rather than 0."
"The ease with which they can be handled may differ considerably."
"Yes, definitions and examples are given below."
"Every sequence in principle has a generating function of each type...the generating function of the sequence."