Editing Generating function (section)

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-order accurate]] power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the [[Jacobi-type continued fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab<sub>''i''</sub>}<nowiki/>}} and {{math|{''c''<sub>''i''</sub>}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>
<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
 & = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''j<sub>n</sub>'' ≔ [''z<sup>n</sup>''] ''J''<sup>[∞]</sup>(''z'')}}, in the previous equations correspond to matrix solutions of the equations:
<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
 \begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
 \begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''<sub>0</sub> ≡ ''k''<sub>0,0</sub> {{=}} 1}}, {{math|''j<sub>n</sub>'' {{=}} ''k''<sub>0,''n''</sub>}} for {{math|''n'' ≥ 1}}, {{math|''k''<sub>''r'',''s''</sub> {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by:
<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>