Generating function

In various areas of mathematics is meant by the generating function of a sequence, the formal power series

For example, the generating function of the constant sequence is the geometric series

The series converges for all and has the value

Because of the use of formal power series, however, play no role in general convergence issues - is merely a symbol. This explicit representation as a power series often allows conclusions to be the result.

Examples

There are the following identities:

• ( generating function of the sequence)
• ( generating function of the sequence)
• ( generating function of the sequence)

Manipulation of generating functions

Assuming a row as generating function is, meet certain manipulations of the sequence corresponding manipulations of the generating function.

For example, consider the sequence with the generating function. deriving results

This corresponds to the result of multiplying by yields

Thus we obtain the result

Deriving a generating function that is equivalent to the multiplication of the n-th element of the sequence and anschließdender index shift to the left, multiplied by z corresponds to a shift of the indices to the right.

Consider a further consequence of the generating function. You multipy with the generating function from above according to the Cauchy product formula is obtained:

The n- th coefficient of the product is thus of the form. This is exactly the partial sum of the first coefficients of the original generating function. Multiplying a generating function thus provides a the Paritialsummen.

An overview of other possible manipulation yields the following table:

Is the generating function of the sequence of the sequence.

Application

Generating functions are an important tool for solving recurrences and difference equations, as well as for the counting of number partitions. The pointwise multiplication of a generating function with the identity corresponding to the displacement of the modeled sequence elements one position to the rear, front, as a new member to the index, a is added. Suppose we have the recursion to solve, then, and it applies to the generating function

So

Solving for F yields

But we know from the previous section that this corresponds to the series, so true by comparing coefficients.

Various types of generating functions

There are in addition to the ordinary generating function or other types of generating functions. Sometimes it proves to be convenient to consider sequences by means of the following two types of generating functions.

Exponential generating function

The exponential generating function (or generating function of exponential type ) a sequence is the series.

For example, the exponential function is the exponential generating function of the sequence

Dirichlet generating function

The Dirichlet generating function of a sequence is the number. It is named after Peter Gustav Lejeune Dirichlet.

For example, the Riemann zeta function is the Dirichlet generating function of the sequence

The " partition function " as a generating function in thermodynamics

In statistical physics, a theoretical- physical discipline in which mainly the thermodynamics is treated, is called the partition function ( partition function ) the formal power series in which β is essentially the reciprocal temperature 1 / T and En the energy values ​​of the treated is assumed to be discrete system are. The " partition function " is the " generating function" of a large number of so-called thermodynamic potentials, in particular the internal energy of the free energy and entropy.

By differentiation with respect to a parameter β generated relations, analogous to the relationship between and occur in connection with the concept of path integral in other areas of theoretical physics. Also the term is " generating function " is used, even if you did not do it with power series for these relationships.