gfun[gftypes] - describes available types of generating functions
- 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 package gfun recognises several different ways to represent the information in a list
l. - The following types of generating functions are known in gfun:
ogf egf revogf revegf lgdogf lgdegf Laplace.
- If type is 'ogf' (ordinary generating function), then the coefficients are the elements of l. For example, the ogf which corresponds to the list [1,1,2,3,5,8] is
1+x+2*x^2+3*x^3+5*x^4+8*x^5.
If type is 'egf' (exponential generating function), then the i-th coefficient is op(i,l)/i!. For example, the egf which corresponds to to the list [1,1,2,3,5,8] is 1+x+2*x^2/2!+3*x^3/3!+5*x^4/4!+8*x^5/5!.
If type is 'revogf', then the series is the reciprocal of the ordinary generating function.
If type is 'revegf', then the series is the reciprocal of the exponential generating function.
If type is 'lgdogf', then the series is the logarithmic derivative of the ordinary generating function.
If type is 'lgdegf', then the series is the logarithmic derivative of the exponential generating function.
If type is 'Laplace', then the ith coefficient is op(i,l)*i!.
The user can define his own type by creating a procedure gfun[`listtoseries/mytypeofgf`], which takes 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.