The listtoseries(l, x, gf) command accepts a list as input and returns a series.

By default, the listtoseries function creates a power series whose coefficients are exactly the elements of the list specified.

The listtolist(l, gf) command accepts a list as input and returns a list.

By default, the listtolist function returns the input list unchanged.

The seriestolist(s, gf) command accepts a series as input and returns a list.

By default, the seriestolist command returns a list whose entries are exactly the coefficients of the series specified.

The seriestoseries(s, gf) command accepts a series as input and returns a series.

By default, the seriestoseries function returns the input series unchanged.

Lists are viewed as lists of coefficients of power series and reciprocally series are viewed as generating series of lists of coefficients.

If gf is specified, it is considered to be a type of generating function. The coefficients of the output are those of the corresponding generating function of the input.

E.g., listtoseries(l, x, 'egf') resp. listtolist(l, 'egf') returns the series resp. the list whose coefficients are $\frac{l_{i+1}}{i!}$ for $0\le i$. Similarly, if s is a series with coefficients $s_i$ for $0\le i$, then seriestoseries(s, 'egf') resp. seriestolist(s, 'egf') returns the series resp. the list whose coefficients are $\frac{s_i}{i!}$.

$\mathrm{with}\left(\mathrm{gfun}\right)\:$

$l≔\left[1\,1\,2\,5\,14\,42\,132\,429\,1430\,4862\,16796\,58786\right]\:$

$\mathrm{listtoseries}\left(l\,x\,'\mathrm{egf}'\right)$

$1+x+x^2+\frac{5}{6}{x}^3+\frac{7}{12}{x}^4+\frac{7}{20}{x}^5+\frac{11}{60}{x}^6+\frac{143}{1680}{x}^7+\frac{143}{4032}{x}^8+\frac{2431}{181440}{x}^9+\frac{4199}{907200}{x}^{10}+\frac{4199}{2851200}{x}^{11}+O\left(x^{12}\right)$

$S≔\mathrm{series}\left(y\exp \left(-y\right)\,y\right)$

$S≔y-y^2+\frac{1}{2}{y}^3-\frac{1}{6}{y}^4+\frac{1}{24}{y}^5+O\left(y^6\right)$

$\mathrm{seriestoseries}\left(S\,'\mathrm{revogf}'\right)$

$y+y^2+\frac{3}{2}{y}^3+\frac{8}{3}{y}^4+\frac{125}{24}{y}^5+O\left(y^6\right)$

$L≔\mathrm{seriestolist}\left(\right)$

$L≔\left[0\,1\,1\,\frac{3}{2}\,\frac{8}{3}\,\frac{125}{24}\right]$

$\mathrm{listtolist}\left(L\,'\mathrm{Laplace}'\right)$

$\left[0\,1\,2\,9\,64\,625\right]$