Found an recursive identity (involving a continued fraction) for which some simplification is needed.

This is my second question in this forum; as I previously explained it, I am a “hobbyst” mathematician and not a professional one; I apologize by advance if something is wrong in my question.

I enjoy doing numerical computations on my leisure time, and at the end of year 2015, I was working on some personal routines related to the world of ISC. With the help of these pieces of code, I detected algorithmically several identities; one was already described here and solved (see this question regarding a continued fraction for tanh). After having spent some time on another one a year ago, I would like to get some help for simplifying what I found. This new continued fraction is:

$$
\mathcal{K}\left(k,x\right)=\operatorname*{K}_{n=1}^{\infty} \frac{
\left(n+1\right)\left(\left(k-1\right)x-n\right)k}{\left(n+1\right)\left(k+1\right)} \tag{1} \label{1}
$$

The previous notation is the one I use; I find it convenient and it can be found for instance in Continued Fractions with Applications by Lorentzen & Waadeland, but I know that some people don’t like it; it has to be read the following way:
$$
a_0 + \operatorname*{K}_{n=1}^{\infty} \frac{b_n}{a_n} = a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \dotsb}}}
$$

I found many partial formulas, some of them being very nice, involving hypergeometric functions or the Lerch Phi function. But of course I was rather interested by a fully general identity. I finally found something but that would need to be simplified now and this is what I am asking here. I would be very happy to finally see a nice identity for this continued fraction. Maybe such an identity could be published somewhere (if it happens to be interesting) and I would happily leave it to anyone who would have taken some time on it.

Is it true that:
$$\mathcal{K}\left(k,x\right)=
\frac{\Gamma\left(x+1\right)\Gamma\big((k-1)x\big)}{\Gamma\left(kx\right)}
\times\mathcal{L}\left(k,x\right)$$

where,
$$\mathcal{L}\left(k,x\right)=\frac{k^{kx}}{2\left(k-1\right)^{\left(k-1\right)x-1}}+\sum_{i=0}^\infty
\big(
\alpha\;\mathcal{K}(k, x+i)

\beta\;\mathcal{K}(k, x+i+1)
\big)$$

and,
$$\alpha=\frac{\left(k-1\right)^{\left(k-1\right)i}\Gamma\left(k\left(i+x\right)\right)}{k^{ki}\,\Gamma\left(1+i+x\right)\Gamma\left(\left(k-1\right)\left(x+i\right)\right)}\\
\beta=\frac{\left(k-1\right)^{\left(k-1\right)i+k}\Gamma\left(k\left(1+i+x\right)\right)}{k^{k\left(i+1\right)}\Gamma\left(1+i+x\right)\Gamma\left(\left(k-1\right)\left(x+i\right)+k\right)}$$

Since I work with empirical computations, I have to say that this formula is rather difficult to check because convergence is rather slow, but it is a result I managed to get by gathering various other materials.


In case someone would wonder whether it is worth spending some time on it or not, I can provide some partial results like nice special values:
$$
\mathcal{K}\left(2, 1/2\right)\;=\;\pi/2-2
\mathrm{,}
\qquad
\mathcal{K}\left(4, 1/2\right)\;=\;4\pi\sqrt{3}/9-2
\qquad\mathrm{etc.}
$$

The case $k$ being an integer

When $k$ is an integer, $k\geq3$, the continued fraction follows a functional identity:

$$
\mathcal{K}\left(k,x\right)\;=\;
% \displaystyle\frac{k^{kx} \Gamma{\left(x+1\right)}\Gamma{\left(\left(k-1\right)x\right)} }{2\left(k-1\right)^{\left(k-1\right)x-1}\Gamma{\left(kx\right)}} + \mathcal{K}\left(k, x+1\right)g_k\left(x\right)
g_k\left(x\right) + \left(\displaystyle\frac{k-1}{k}\right)^k \displaystyle\frac{\Gamma{\left(\left(k-1\right)x\right)}\Gamma{\left(k\left(x+1\right)\right)}}{\Gamma{\left(\left(k-1\right)x+k\right)}\Gamma{\left(kx\right)}}\mathcal{K}\left(k, x+1\right)
$$

where $g$ is a sequence of rational functions. Unfortunately I couldn’t find a general form for it but I could easely compute about 30 of them with the help of numerical pieces of software. The best I could do here was to build a triangle of integer coefficients needed for building a given $g_k$ function.

Let’s call $m_{(a,b)}$ the coefficient in the $a^\textrm{th}$ row and $b^\textrm{th}$ column from the following triangle:

$$
\begin{array}{rrrr@{\qquad}l}
-8&&&&\textrm{for $k=3$}\\
-20&2&&&\textrm{for $k=4$}\\
-40&12&-24&&\textrm{for $k=5$}\\
-70&42&-202&624&\textrm{for $k=6$}\\
\textrm{etc.}
\end{array}
$$

(I can provide about 30 rows of these coefficients, see below); then,

$$
g_k\left(x\right)\;=\;\displaystyle\sum_{i=1}^{k-2}\displaystyle\frac{m_{(k-2,i)}x
\Gamma{\left(\left(k-1\right)x\right)}}{\left(x+1\right)k^i\Gamma{\left(\left(k-1\right)x+i\right)}}
$$

A direct formula for $\mathcal{K}\left(k,x\right)$ involves the same rational function $g_k$:

\begin{equation}
\begin{array}{lcl}
\mathcal{K}\left(k,x\right)&=&
\displaystyle\frac{ \Gamma{\left(x+1\right)}\Gamma{\left(\left(k-1\right)x\right)} }{ \Gamma{\left(kx\right)} }\\[24pt]
&&\times\quad\left( \displaystyle\frac{k^{kx} }{2\left(k-1\right)^{\left(k-1\right)x-1}}
+\displaystyle\sum_{i=0}^\infty
\displaystyle\frac{ \left(k-1\right)^{\left(k-1\right)i}
\Gamma{\left(k\left(i+x\right)\right)}}{k^{ki}\Gamma{\left(1+i+x\right)}\Gamma{\left(\left(k-1\right)\left(x+i\right)\right)}}g_k\left(x+i\right)
\right)
\end{array}
\label{directform}
\end{equation}

The first formula I gave in my question was found by substituting $g$ from these last expressions. Of course, finding some formula for the $m$ coefficients would be nice, but I couldn’t figure out any despite the time I spent on it.

It looks also like the infinite sum above can be turned into a finite sum of $k-2$ hypergeometric functions as a direct formula for $\mathcal{K}\left(k,x\right)$. The initial cases are the simplest, like for $k=3$:

$$
\begin{array}{lcl}
\displaystyle\mathcal{K}\left(3,x\right)&=&
\displaystyle\operatorname*{K}_{n=1}^{\infty}\displaystyle\frac{\left(n+1\right)\left(6x-3n\right)}{4n+4}\\
&=&
\cfrac{12x-6}{8+\cfrac{18x-18}{12+\cfrac{24x-36}{16+\cfrac{30x-60}{20+\ddots}}}}\\
&=&
\displaystyle\frac{27^x\, \Gamma\left(x+1\right)\Gamma\left(2x\right)}
{4^x\, \Gamma\left(3x\right)}

\displaystyle\frac{4\,\displaystyle{}_3F_2\left(1,x+\frac{1}{3},x+\frac{2}{3};\;x+\frac{1}{2},x+2;\;1\right)}{ 3\left(x+1\right) }
\end{array}
$$

The general case being:

$$
\begin{array}{lcl}
\mathcal{K}\left(k,x\right)&=&
\displaystyle\frac{k^{kx} \Gamma{\left(x+1\right)}\Gamma{\left(\left(k-1\right)x\right)} }{2\left(k-1\right)^{\left(k-1\right)x-1}\Gamma{\left(kx\right)}}\\[24pt]
&+&
\displaystyle\sum_{i=1}^{k-2}
\displaystyle\frac{m_{(k-2,i)} x
\;\;\displaystyle
{}_{k+1}F_{k}
\left(
\begin{array}{l}
1,x+\frac{1}{k},x+\frac{2}{k},\dots,x+\frac{k-1}{k},x+1\\[4pt]
x+\frac{i}{k-1}, x+\frac{i+1}{k-1}, \dots,x+\frac{i+k-2}{k-1},x+2
\end{array}
{1}
\right)
}
{k^i\left(x+1\right) \Gamma{\left(\left(k-1\right)x+i \right)}/\Gamma{\left(\left(k-1\right)x\right)} }
\end{array}
$$

where obviously the $x+1$ term can always be cancelled (since one of the $x+(i+\dots)/(k-1)$ terms is always equal to $x+1$ leading to a ${}_{k}F_{k-1}$ function.

Unfortunately, I couldn’t figure out how to make some substitution between this last formula with $m$ coefficients and previous formulae above involving the $g$ function.

The case $kx$ being an integer

In order to study consecutive integer values of the~$kx$ product, I will now use another notation:

$$
\mathcal{K}’\left(k’, x’\right)\;=\;\mathcal{K}\left(x’, k’/x’\right)
$$

Starting from here, $k’$ is assumed to be an integer (according to the previous notation, it means that~$kx$ is an integer).

New identities, involving different expressions can be found (leading to different kinds of special values when possible):

$$
\begin{array}{lcl}
\mathcal{K}’\left(k’,x’\right)&=&
h_{k’}\left(x’\right) –
\left(\displaystyle\frac{x’}{x’-1}\right)^{k’-1}
\displaystyle\frac
{k’\;\Phi\left(-(x’-1)^{-1}, 1, k’/x’\right)-x’+1}
{k’/x’\; \beta\left(k’, -k’/x’\right)}\\[24pt]
&=& h_{k’}\left(x’\right) –
\left(\displaystyle\frac{x’}{x’-1}\right)^{k’-1}
\displaystyle\frac{
\displaystyle\int_{t=0}^1
\left(
\displaystyle\frac{\left(1-t\right)\left(x’-1\right)}{x’-1+t}
\right)^{k’/x’}\textrm{d}t}
{k’/x’\; \beta\left(k’, -k’/x’\right)}\\[24pt]
&=& h_{k’}\left(x’\right) –
\left(\displaystyle\frac{x’}{x’-1}\right)^{k’-1}
\;
\displaystyle\frac
{
\displaystyle{}_{2}F_{1}\left(1,k’/x’\;; 1+k’/x’\;; \left(1-x’\right)^{-1}\right) x’
-x’ + 1 }
{k’/x’\; \beta\left(k’, -k’/x’\right)}\\[24pt]
\end{array}
$$

where $h$ is (again) a sequence of rational functions. Let’s call $m’$ the sequence of following polynomial functions (I computed about 20 of them):

$$
\begin{array}{l@{\qquad}l}
1,&\textrm{(for $k’=3$)}\\
-6 + 4x,&\textrm{(for $k’=4$)}\\
41 – 53x + 18x^2,&\textrm{(for $k’=5$)}\\
-348 + 648x – 420x^2 + 96x^3,&\textrm{(for $k’=6$)}\\
\textrm{etc.}
\end{array}
$$

(I have about 20 rows like that, see below); then,

$$
\left\{\begin{array}{l@{\qquad}l}
h_1\left(x’\right)\;=\;-1&\textrm{if $k’=1$}\\[12pt]
h_2\left(x’\right)\;=\;0&\textrm{if $k’=2$}\\[12pt]
h_{k’}\left(x’\right)\;=\;
\displaystyle\frac{ m’_{k’-2} \left(k’\left(x’-1\right)-x’\right)\left(x’-1\right) }{ \left(k’-1\right)!\left(x’-1\right)^{k’-1} }
&\textrm{if $k’\geq3$}
\end{array}\right.
$$

Again, finding some formula for these $m’$ coefficients would lead to a beautiful formula for $\mathcal{K}$.

Materials (data)

Finding a formula for one of these two triangles would lead to a direct formula; I put my data below in case someone would have a glance at them. I used the Mathematica format but it is very easy to convert it to anything else.

Two pieces of data are attaches:

  • the triangle of $m$ coefficients;
  • the triangle of $m’$ polynomials.

First, the triangle of $m$ coefficients:

m = {
    {-8},
    {-20,2},
    {-40,12,-24},
    {-70,42,-202,624},
    {-112,112,-944,6800,-28160},
    {-168,252,-3240,39990,-378180,1956240},
    {-240,504,-9120,168780,-2691500,31299660,-193818240},
    {-330,924,-22308,573210,-13533950,262134768,-3604679456,25969798400},
    {-440,1584,-49104,1665048,-54028800,1533955752,-34784795304,551021454648,-4524877873152},
    {-572,2574,-99528,4294290,-182338520,7056003570,-232622918920,6027044680440,-107934603537600,994719833856000},
    {-728,4004,-188760,10081500,-540836660,27196564920,-1213669081240,45402131767300,-1320731548020500,26362209822109700,-269367401834854400},
    {-910,6006,-338910,21926190,-1447260750,91405905570,-5269521952170,265270592109600,-11076506267112000,357029036918928000,-7854973969921056000,88120488036962304000},
    {-1120,8736,-581152,44753280,-3559649600,275205604224,-19825712025728,1282796713117920,-71715923149544960,3301368476366175072,-116699890447511246304,2804668390029759121440,-34267109445760293273600},
    {-1360,12376,-958256,86572772,-8159187400,756765661176,-66440724681072,5348371530601974,-382626987630417516,23479203132873399912,-1180109553700743730064,45368332310341259611392,-1182254848944902971766528,15625389962188145791748096},
    {-1632,17136,-1527552,159942120,-17614027080,1928217475800,-202304078444160,19770089799495420,-1752820980572484900,137106641163083684400,-9150469205270049657000,498255795785126793714300,-20689280727893354516039100,580954533672149606803333500,-8258153843323806482092032000},
    {-1938,23256,-2364360,283936380,-36111651940,4603291360920,-568072401325800,66113606126531250,-7091616708136156350,685093642937381277600,-58084005963489507601600,4185232111643968748563200,-245300606116021459523520000,10937978741665549577203200000,-329201103515195643674342400000,5008018989272579747902955520000},
    {-2280, 31008, -3565920, 486748080, -70778344000, 10387428137520, -1488100140390640, 203083381548077800, -25863156057665295600, 3013571052233142231600, -314587396914476897195600, 28707485207091210763643400, -2219712979095004695654352000, 139280212909636939833667975000, -6636253210525016113608507935000, 213097407927242925450657097805000, -3454359206881951401298519654400000},
    {-2660, 40698, -5255856, 809056860, -133342927200, 22312129321188, -3669893137886976, 579814786246525830, -86348985070472364240, 11912853180399067114218, -1495810636771876664855616, 167605806446114219615512080, -16367656617234522958801322880, 1351158196468373411398992679968, -90343714080278210268675472690176, 4580006947756999980100721131530240, -156282112623791407868541623144448000, 2689249605403395016547015123312640000},
    {-3080, 52668, -7589208, 1308328296, -242549291800, 45885645345792, -8583630937867664, 1553160300088906908, -267115992196423617852, 42987425869754619349668, -6375040043352284259869056, 857073372770354425645988028, -102516882515527648341577456052, 10661831289669551213862356652936, -935559064657406376990369478217592, 66392317932445459737857422287168372, -3567745082228659013145697404664186980, 128910715024676759785064905466962718772, -2346816894979813166169796326498705604608},
    {-3542, 67298, -10758066, 2064221940, -427579512340, 90781666964580, -19156534075381220, 3933373867476907490, -773001449364837212650, 143337480955932385601050, -24740379167353794462482250, 3919619596881620663698352400, -561261767174736596981182684800, 71335052610001221107048830848000, -7868282527494059201650532567232000, 731130238635173198053153940388864000, -54874347623661338563124599643795456000, 3115425378725886177371208787427500032000, -118823016992948260463398185243235328000000, 2281667516033630958272834061325819904000000},
    {-4048, 85008, -14997840, 3185310480, -732817955520, 173481958077120, -40999952833277760, 9476856735932804400, -2109071440724185828800, 445975726476730199677200, -88512496883173204230589200, 16287555278375689955554908000, -2742762926014265565760465116000, 416464503552989975128829015232000, -56022670706540265707212091974752000, 6530157038781041616772104173250906000, -640432883214307884870393296312193408000, 50678900925449456482492390925833162170000, -3030888088511779915449136680430563767970000, 121680563022029894895609769265080802797710000, -2457881871574620592645101983798188100812800000},
    {-4600, 106260, -20594200, 4817335050, -1224368027500, 321314420301000, -84511921344096400, 21837002485484744250, -5460560343151268855500, 1305129863913508752619500, -294830327010826293740053000, 62266523988286549276643448150, -12155836300655550173555485100500, 2166552581042493037698711451044000, -347528815311777085029924893736724000, 49310886906769450018463698053718956000, -6055094240863690023007706913404687496000, 624931908053334691607080995207899763360000, -51995246297017813506321403783934541271680000, 3267058354801244958789636823999905461614080000, -137714123890197288080314052933290069397790720000, 2919082688078782495755638635126117397822177280000},
    {-5200, 131560, -27890720, 7153246100, -1998828604500, 578492714221380, -168381395188292160, 48336833335182707700, -13488618129875116559100, 3616198939768951478832120, -921761072167762889584562040, 221195726134157418021734756400, -49475332499912242476184032601200, 10205195620733840007949834678752240, -1918172758706697630190525216812587520, 323985355991420046527323571346102988380, -48344306336853827546608471043597263180580, 6236410605945196309646575950593070996242112, -675574363650914146210117776862820284274422024, 58952707582889583114849809235709300486236865900, -3882542732229399143933348322987701786016750303500, 171441026104474223322555734221353597751602880467500, -3804948188770142619704669041367337098336518799360000},
    {-5850, 161460, -37297260, 10445304870, -3194948279250, 1014976460259720, -325284627793625640, 103172770888592820030, -31935449045956426109370, 9539216762922608385767220, -2723057324780894154211144140, 736160112029528961953639831370, -186796816819349796321290668269630, 44075061969736070597572230035573520, -9572302178958218895730236218841324000, 1891505089131626077283496419557169149440, -335445207934760792427195571281784372615680, 52500216486691388374580499101901862088712192, -7097170407703652020998162722033686958068359168, 805065643420847023775013665323598164107193712640, -73517194182833158880223919841841477149672840232960, 5063910198045757853788580192610834249566744036769792, -233753499049244474320664989973779541903817843793723392, 5421019686584291940290446961296577366550977042379177984},
    {-6552, 196560, -49299120, 15019547400, -5008903972800, 1739242914960600, -610937879343586200, 213044845306016606400, -72754198545840383754000, 24070115922338692488744000, -7644318575328042577024440000, 2310967474707680781916704026400, -659643825387695670545486563900800, 176310960419571195542552313305220000, -43737562948690701530900945112487860000, 9971569996183917023985409450427829525000, -2065816103048706564571284238051613500970000, 383694616243443437568263013812039800791990000, -62837694530317608096687694020291847950887850000, 8882027287256940966231272654559742520461647675000, -1052798325643449998635040396582752781540903534400000, 100403132568689053077039174326824794886423800369525000, -7219010863790387928075409606725932651241502240697925000, 347694912384706890967941398378351924881505314746114375000, -8410211480678023320925498721110295938761779872530432000000},
    {-7308, 237510, -64467000, 21292941150, -7714097921400, 2916404269698150, -1118190692314588200, 426802089352299911250, -160027260019840672173000, 58331250782529140212904250, -20490396263678036008373775000, 6882240483264899117171621464650, -2193778617416596625925140519889000, 658701474719992054755835904812589250, -184849219993358885493062726683645011000, 48071268295949005824672264507034120713000, -11474627010559184001964832102040305176064000, 2486300080568850616517044780683050329279500000, -482557829071841856097474441995986751588159600000, 82519812703910109373696671538744758363463431600000, -12171485983756715937697088833853438913213231139200000, 1504622344753359530620585743898465267509637578086400000, -149578514098722961283327295745791298648255342896128000000, 11206095897313941255176473673738843538864273909178368000000, -562168844769302825642413825011269789161124218168180736000000, 14158685308482113187947385422334786985368814916547903488000000},
    {-8120, 285012, -83467800, 29793593700, -11686535347500, 4793533613422200, -1998556890132738600, 831586576427972253000, -340848228875785588089000, 136238023124447243505363000, -52660395629469710899852464000, 19538961040560845384445253280400, -6910955344258841823610759577910000, 2314356603026298502745751283117512000, -728675543038928155386332670093531348000, 214094057758291962937552370682805572874500, -58219061862163874792551203092332714335562500, 14516218849791911496733341909570194788693657500, -3282604618552285952731079609822870039677157280000, 664411186688887595279287675566612107186757131137500, -118409734996254908439089485033112187166889903398912500, 18191629115197225627312502465880976532595946618885575000, -2341224306602995570500391595079223927042601832325548375000, 242207299534450314667366061440535601337557549950222099062500, -18876082685899484114785412705333370447849929199143588260312500, 984735360414688845599685871865811518568346909249116274589062500, -25783538810629517021266249543138106125941453578155830804480000000},
    {-8990, 339822, -107076294, 41184403650, -17437036139250, 7734593443379670, -3494544596518289790, 1579284525442183941150, -704824443778733315833950, 307603458576783505012150890, -130224815626952571544054365330, 53104886131252741135008778869750, -20725045171508268666798696739515750, 7692149381874530736969377052576265170, -2697951867406866051352410906078675593130, 888312186434005905720786992903999283113280, -272592942573130969721591283451860861263314880, 77338601225244978709744671633189447473744656512, -20101127086406857123404360233486723068548779401984, 4734712312004040060142996428231341452704698192056320, -997560027458845642233086777856612791698591648386293760, 184958250344358621036304852979454062719166092717119602688, -29548178532481410695731393820614988207097594428678936395776, 3952663598502220791832289869795663784663381872742938194739200, -424872593639734370119419756220910340050772936517217787445248000, 34392492457805374774120867589709950335110603672365392763289600000, -1863044156298137915554810903420018148506651555374947876496998400000, 50638772787167674049719466457371229923270837855299620820746240000000}
};

Then the triangle of $m’$ polynomials:

mm = {
1, -6 + 4*x, 41 - 53*x + 18*x^2, -348 + 648*x - 420*x^2 + 96*x^3, 
   3669 - 8734*x + 8067*x^2 - 3482*x^3 + 600*x^4, 
   -47248 + 135328*x - 158672*x^2 + 96800*x^3 - 31248*x^4 + 4320*x^5, 
   727641 - 2423511*x + 3405267*x^2 - 2622141*x^3 + 1188252*x^4 - 305748*x^5 + 
    35280*x^6, -13122720 + 49768960*x - 81172560*x^2 + 74655760*x^3 - 42491760*x^4 + 
    15258160*x^5 - 3258720*x^6 + 322560*x^7, 
   271959293 - 1157733608*x + 2149311530*x^2 - 2291218376*x^3 + 1553100917*x^4 - 
    697389632*x^5 + 206763300*x^6 - 37696464*x^7 + 3265920*x^8, 
   -6373686528 + 30125806848*x - 62800700544*x^2 + 76183473024*x^3 - 
    59781151872*x^4 + 31886134272*x^5 - 11774336256*x^6 + 2965752576*x^7 - 
    471208320*x^8 + 36288000*x^9, 166695335769 - 867080392969*x + 2008379540976*x^2 - 
    2736817207770*x^3 + 2443396898805*x^4 - 1507011320601*x^5 + 659441858394*x^6 - 
    206111203940*x^7 + 45043779816*x^8 - 6336456480*x^9 + 439084800*x^10, 
   -4812534974464 + 27341393304064*x - 69753592234368*x^2 + 105687047754624*x^3 - 
    106030998529344*x^4 + 74392108414848*x^5 - 37603296588032*x^6 + 
    13896748361216*x^7 - 3755526664512*x^8 + 723700309248*x^9 - 91276174080*x^10 + 
    5748019200*x^11, 151999996277925 - 937041564265650*x + 2613327954360525*x^2 - 
    4364620345415250*x^3 + 4871696279607675*x^4 - 3842208959921550*x^5 + 
    2208928948800975*x^6 - 942087609901350*x^7 + 300405312044100*x^8 - 
    71352461709000*x^9 + 12280661164800*x^10 - 1402935292800*x^11 + 80951270400*x^12, 
   -5212950320375808 + 34671923430780928*x - 105013701846865920*x^2 + 
    191879918776057856*x^3 - 236246229569439744*x^4 + 207409229427830784*x^5 - 
    134079476009502720*x^6 + 65028411548069888*x^7 - 23908234488465408*x^8 + 
    6687356722905088*x^9 - 1414323490805760*x^10 + 219693856813056*x^11 - 
    22925711370240*x^12 + 1220496076800*x^13, 
   192908434730267801 - 1377309563570454971*x + 4504240832008823353*x^2 - 
    8944731476737727263*x^3 + 12057054300505697583*x^4 - 11683650400191522273*x^5 + 
    8411790424807054099*x^6 - 4588515109996305829*x^7 + 1918116031712679596*x^8 - 
    618064823563159856*x^9 + 153647212030630848*x^10 - 29242401180913008*x^11 + 
    4135229005230720*x^12 - 396997001452800*x^13 + 19615115520000*x^14, 
   -7661276413060165632 + 58455665081940344832*x - 205362993128521586688*x^2 + 
    440656568659067627520*x^3 - 646000731200138715648*x^4 + 
    685780191107302542336*x^5 - 545270893574020280064*x^6 + 
    331406006965726821120*x^7 - 155847180496082807808*x^8 + 
    57087389250576640512*x^9 - 16335131671115336448*x^10 + 3647848903953212160*x^11 - 
    630293068833472512*x^12 + 81737176968299520*x^13 - 7263281191219200*x^14 + 
    334764638208000*x^15, 325015466875658755821 - 2639668188726987561436*x + 
    9917145842014784419212*x^2 - 22874961160535389523056*x^3 + 
    36258193335964278154110*x^4 - 41887476949175919553816*x^5 + 
    36506794561565447125836*x^6 - 24516370991512685833088*x^7 + 
    12850659286368069730317*x^8 - 5296394247470924322860*x^9 + 
    1722523200312064838592*x^10 - 442409275011858790256*x^11 + 
    89540582470596717552*x^12 - 14150517433376508288*x^13 + 
    1693379120667874560*x^14 - 140015823275059200*x^15 + 6046686277632000*x^16, 
   -14668500851872718848000 + 126360196032621236224000*x - 
    505641446052759169024000*x^2 + 1248039501127422625792000*x^3 - 
    2127811079963561138176000*x^4 + 2659365515136882884608000*x^5 - 
    2523680829463448281088000*x^6 + 1858664349808259201024000*x^7 - 
    1076955696528752952320000*x^8 + 494913795536023414784000*x^9 - 
    181130674396834734080000*x^10 + 52865257295200467968000*x^11 - 
    12296348611822927872000*x^12 + 2272553630125056000000*x^13 - 
    330581514194555904000*x^14 + 36704049561050112000*x^15 - 
    2836877949868032000*x^16 + 115242726703104000*x^17
}

Solutions Collecting From Web of "Found an recursive identity (involving a continued fraction) for which some simplification is needed."