Editing Generating function (section)

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== 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>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''<sup>[∞]</sup>(''z'')}}, expanded by:
<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''P<sub>h</sub>''(''z'')}} and {{math|''Q<sub>h</sub>''(''z'')}}, defined recursively by:
<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv<sub>''h''</sub>(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''j<sub>n</sub>''}}, ''and'' for {{math|''M<sub>h</sub>'' ≔ ab<sub>2</sub> ⋯ ab<sub>''h'' + 1</sub>}} if {{math|''h'' ‖  ''M''<sub>''h''</sub>}} then we have the congruence
<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab<sub>''i''</sub>}<nowiki/>}} and {{math|{''c''<sub>''i''</sub>}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} as in the examples contained in the table below.

==== 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.