En matemáticas, una función generadora o función generatriz es una serie formal de potencias cuyos coeficientes codifican información sobre una sucesión an cuyo índice corre sobre los enteros no negativos. Hay varios tipos de funciones generadoras: funciones generadoras ordinarias, funciones generadoras exponenciales, la serie de Lambert, la serie de Bell y la serie de Dirichlet; de las cuales abajo se ofrecen definiciones y ejemplos. Cada sucesión tiene una función generadora de cierto tipo. El tipo de función generadora que es apropiada en un contexto dado depende de la naturaleza de la sucesión y los detalles del problema que se analiza. Las funciones generadoras son expresiones cerradas en un argumento formal x. A veces, una función generadora se «evalúa» en un valor específico x=a pero hay que tener en cuenta que las funciones generadoras son series formales de potencias, por lo que no se considera ni se analiza el problema de la convergencia en todos los valores de x. Por lo mismo es importante observar que las funciones generadoras no son realmente funciones en en el sentido usual de ser mapeos entre un dominio y un codominio; el nombre es únicamente el resultado del desarrollo histórico de su estudio. Una función generadora es una cuerda de tender en la que colgamos una sucesión de números para mostrarla Herbert Wilf

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In mathematics, generating function is used to describe an infinite sequence of numbers (an) by treating them as the coefficients of a series expansion. The sum of this infinite series is the generating function. Unlike an ordinary series, this formal series is allowed to diverge, meaning that the generating function is not always a true function and the "variable" is actually an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series. 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, in that a series of terms can be said to be the generator of its sequence of term coefficients.

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