Editing Generating function (section)

==== Examples ====

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''j<sub>n</sub>''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.