Editing Generating function (section)

=== Other generating functions ===
Other sequences generated by more complex generating functions include:

* Double exponential generating functions e.g. the [[Bell numbers#Generating function|Bell numbers]]
* Hadamard products of generating functions and diagonal generating functions, and their corresponding [[generating function transformation#Hadamard products and diagonal generating functions|integral transformations]]

==== Convolution polynomials ====
Knuth's article titled "''Convolution Polynomials''"<ref>{{cite journal |last1=Knuth |first1=D. E. |date=1992 |title=Convolution Polynomials |journal=Mathematica J. |volume=2 |pages=67–78 |arxiv=math/9207221 |bibcode=1992math......7221K}}</ref> defines a generalized class of ''convolution polynomial'' sequences by their special generating functions of the form
<math display="block">F(z)^x = \exp\bigl(x \log F(z)\bigr) = \sum_{n = 0}^\infty f_n(x) z^n,</math>
for some analytic function {{mvar|F}} with a power series expansion such that {{math|''F''(0) {{=}} 1}}.

We say that a family of polynomials, {{math|''f''<sub>0</sub>, ''f''<sub>1</sub>, ''f''<sub>2</sub>, ...}}, forms a ''convolution family'' if {{math|[[Degree of a polynomial|deg]] ''f<sub>n</sub>'' ≤ ''n''}} and if the following convolution condition holds for all {{mvar|x}}, {{mvar|y}} and for all {{math|''n'' ≥ 0}}:
<math display="block">f_n(x+y) = f_n(x) f_0(y) + f_{n-1}(x) f_1(y) + \cdots + f_1(x) f_{n-1}(y) + f_0(x) f_n(y). </math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

* The sequence {{math|''n''! · ''f<sub>n</sub>''(''x'')}} is of [[binomial type]]
* Special values of the sequence include {{math|''f<sub>n</sub>''(1) {{=}} [''z<sup>n</sup>''] ''F''(''z'')}} and {{math|''f<sub>n</sub>''(0) {{=}} ''δ''<sub>''n'',0</sub>}}, and
* For arbitrary (fixed) <math>x, y, t \isin \mathbb{C}</math>, these polynomials satisfy convolution formulas of the form
<math display="block">\begin{align}
f_n(x+y) & = \sum_{k=0}^n f_k(x) f_{n-k}(y) \\
f_n(2x) & = \sum_{k=0}^n f_k(x) f_{n-k}(x) \\
xn f_n(x+y) & = (x+y) \sum_{k=0}^n k f_k(x) f_{n-k}(y) \\
\frac{(x+y) f_n(x+y+tn)}{x+y+tn} & = \sum_{k=0}^n \frac{x f_k(x+tk)}{x+tk} \frac{y f_{n-k}(y+t(n-k))}{y+t(n-k)}.
\end{align}</math>

For a fixed non-zero parameter <math>t \isin \mathbb{C}</math>, we have modified generating functions for these convolution polynomial sequences given by
<math display="block">\frac{z F_n(x+tn)}{(x+tn)} = \left[z^n\right] \mathcal{F}_t(z)^x, </math>
where {{math|𝓕<sub>''t''</sub>(''z'')}} is implicitly defined by a [[functional equation]] of the form {{math|𝓕<sub>''t''</sub>(''z'') {{=}} ''F''(''x''𝓕<sub>''t''</sub>(''z'')<sup>''t''</sup>)}}. Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, {{math|⟨ ''f<sub>n</sub>''(''x'') ⟩}} and {{math|⟨ ''g<sub>n</sub>''(''x'') ⟩}}, with respective corresponding generating functions, {{math|''F''(''z'')<sup>''x''</sup>}} and {{math|''G''(''z'')<sup>''x''</sup>}}, then for arbitrary {{mvar|t}} we have the identity
<math display="block">\left[z^n\right] \left(G(z) F\left(z G(z)^t\right)\right)^x = \sum_{k=0}^n F_k(x) G_{n-k}(x+tk). </math>

Examples of convolution polynomial sequences include the ''binomial power series'', {{math|𝓑<sub>''t''</sub>(''z'') {{=}} 1 + ''z''𝓑<sub>''t''</sub>(''z'')<sup>''t''</sup>}}, so-termed ''tree polynomials'', the [[Bell numbers]], {{math|''B''(''n'')}}, the [[Laguerre polynomials]], and the [[Stirling polynomial|Stirling convolution polynomials]].