Genererinsfunksjonen

Genererinsfunksjonen is a mathematical concept referring to a generating function, a tool used to encode a sequence of numbers into a formal power series. In this context, a sequence {a_n} is represented by a function F(x) = sum_{n>=0} a_n x^n, called the ordinary generating function. Variants exist, including the exponential generating function F_e(x) = sum_{n>=0} a_n x^n / n!, and probability generating functions where the coefficients represent probabilities.

A generating function serves as an algebraic device that makes it easier to study properties of the underlying sequence. By manipulating the power series, one can derive recurrences, find closed forms, or count combinatorial objects. Basic operations follow standard algebra: linearity allows combining sequences, and the product of generating functions corresponds to the convolution of sequences. The derivative and integral of a generating function relate to shifts and weighted sums of the sequence; for example, differentiating F(x) = sum a_n x^n yields F'(x) = sum n a_n x^{n-1}, linking to n-th terms.

Common examples illustrate the idea. If a_n = 1 for all n, the ordinary generating function is F(x) = 1/(1-x). For the Fibonacci sequence, F(x) = x/(1 - x - x^2). The sequence a_n = n has generating function F(x) = x/(1-x)^2, while the partition function has a product-form generating function.

Generating functions have wide applications in combinatorics, probability, number theory, and computer science, where they help solve counting problems, analyze distributions, and study the growth of sequences. They can be treated formally (without convergence) or analytically, depending on context.