Available Types of Generating Functions
<Text-field style="Heading 2" layout="Heading 2" bookmark="info">Description</Text-field>
A generating function is an analytic encoding of numerical data. It is a formal power series which can be manipulated algebraically in ways which parallel the manipulation of the (often combinatorial) objects they represent. The gfun package recognizes several different ways to represent the information in a list l.
The following types of generating functions are accepted by the gfun package.
'ogf'
If type is NiMuSSRvZ2ZHNiI= (ordinary generating function), then the coefficients are the elements of l. For example, the NiNJJG9nZkc2Ig== which corresponds to the list, [1, 1, 2, 3, 5, 8], is NiMsLiokSSJ4RzYiIiImIiIpKiRGJSIiJUYnKiRGJSIiJEYsKiRGJSIiI0YuRiUiIiJGL0Yv.
'egf'
If type is NiMuSSRlZ2ZHNiI= (exponential generating function), then the ith coefficient is NiMqJi1JI29wR0kqcHJvdGVjdGVkR0YmNiRJImlHNiJJImxHRikiIiItSSpmYWN0b3JpYWxHRiY2I0YoISIi. For example, the NiNJJGVnZkc2Ig== which corresponds to to the list, [1, 1, 2, 3, 5, 8], is NiMsLiIiIkYkSSJ4RzYiRiQqJkYlIiIjLUkqZmFjdG9yaWFsR0kqcHJvdGVjdGVkR0YrNiNGKCEiIkYoKiZGJSIiJC1GKjYjRi9GLUYvKiZGJSIiJS1GKjYjRjNGLSIiJiomRiVGNi1GKjYjRjZGLSIiKQ==.
'revogf'
If type is NiMuSSdyZXZvZ2ZHNiI=, then the series is the reciprocal of the ordinary generating function.
'revegf'
If type is NiMuSSdyZXZlZ2ZHNiI=, then the series is the reciprocal of the exponential generating function.
'lgdogf'
If type is NiMuSSdsZ2RvZ2ZHNiI=, then the series is the logarithmic derivative of the ordinary generating function.
'lgdegf'
If type is NiMuSSdsZ2RlZ2ZHNiI=, then the series is the logarithmic derivative of the exponential generating function.
'Laplace'
If type is NiMuSShMYXBsYWNlRzYi, then the ith coefficient is NiMqJi1JI29wR0kqcHJvdGVjdGVkR0YmNiRJImlHNiJJImxHRikiIiItSSpmYWN0b3JpYWxHRiY2I0YoRis=.
You can define types by creating a procedure gfun[`listtoseries/mytypeofgf`], which accepts a list and a variable as input, and yields a series in this variable. This series must be of type taylor. In particular, it cannot have negative exponents.
See Alsogfunseries