Secondary polynomials of the Laguerre polynomials $q_n$

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Polynomials

$n$

$q_n(x)$

-1

1/2*x - 3/2

-1/6*x^2 + 4/3*x - 11/6

1/24*x^3 - 5/8*x^2 + 29/12*x - 25/12

-1/120*x^4 + 1/5*x^3 - 59/40*x^2 + 37/10*x - 137/60

1/720*x^5 - 7/144*x^4 + 26/45*x^3 - 14/5*x^2 + 103/20*x - 49/20

-1/5040*x^6 + 1/105*x^5 - 167/1008*x^4 + 82/63*x^3 - 327/70*x^2 + 236/35*x - 363/140

1/40320*x^7 - 1/640*x^6 + 251/6720*x^5 - 193/448*x^4 + 2539/1008*x^3 - 801/112*x^2 + 2369/280*x - 761/280

-1/362880*x^8 + 1/4536*x^7 - 359/51840*x^6 + 95/864*x^5 - 11447/12096*x^4 + 3323/756*x^3 - 10379/1008*x^2 + 2593/252*x - 7129/2520

10:

1/3628800*x^9 - 11/403200*x^8 + 247/226800*x^7 - 737/32400*x^6 + 3887/14400*x^5 - 15983/8640*x^4 + 53783/7560*x^3 - 11891/840*x^2 + 30791/2520*x - 7381/2520

11:

-1/39916800*x^10 + 1/332640*x^9 - 659/4435200*x^8 + 547/138600*x^7 - 21979/356400*x^6 + 5777/9900*x^5 - 21023/6336*x^4 + 64583/5940*x^3 - 173483/9240*x^2 + 32891/2310*x - 83711/27720

12:

1/479001600*x^11 - 13/43545600*x^10 + 857/47900160*x^9 - 3133/5322240*x^8 + 4867/415800*x^7 - 8671/59400*x^6 + 34081/29700*x^5 - 6617/1188*x^4 + 75473/4752*x^3 - 191633/7920*x^2 + 452993/27720*x - 86021/27720

13:

-1/6227020800*x^12 + 1/37065600*x^11 - 1091/566092800*x^10 + 311/4043520*x^9 - 43711/23063040*x^8 + 581/19305*x^7 - 240979/772200*x^6 + 44971/21450*x^5 - 273983/30888*x^4 + 86363/3861*x^3 - 209057/6864*x^2 + 476753/25740*x - 1145993/360360

14:

1/87178291200*x^13 - 1/447068160*x^12 + 31/165110400*x^11 - 59/6604416*x^10 + 105841/396264960*x^9 - 77173/14676480*x^8 + 175223/2522520*x^7 - 665999/1081080*x^6 + 4338001/1201200*x^5 - 216737/16016*x^4 + 826106/27027*x^3 - 112894/3003*x^2 + 499061/24024*x - 1171733/360360

15:

-1/1307674368000*x^14 + 1/5837832000*x^13 - 1679/100590336000*x^12 + 163/174636000*x^11 - 54947/1651104000*x^10 + 16411/20638800*x^9 - 2868071/220147200*x^8 + 253301/1719900*x^7 - 1761427/1544400*x^6 + 2670001/450450*x^5 - 14358011/720720*x^4 + 1834394/45045*x^3 - 4111669/90090*x^2 + 1040164/45045*x - 1195757/360360

16:

1/20922789888000*x^15 - 17/1394852659200*x^14 + 2039/1494484992000*x^13 - 10183/114960384000*x^12 + 27661/7451136000*x^11 - 214999/2032128000*x^10 + 222379/105670656*x^9 - 8664067/293529600*x^8 + 32087467/110073600*x^7 - 4498829/2246400*x^6 + 12827149/1372800*x^5 - 32815117/1153152*x^4 + 76538797/1441440*x^3 - 8748863/160160*x^2 + 18358463/720720*x - 2436559/720720

17:

-1/355687428096000*x^16 + 1/1235025792000*x^15 - 2447/23712495206400*x^14 + 2173/282291609600*x^13 - 732107/1954326528000*x^12 + 2103/167552000*x^11 - 10289453/34546176000*x^10 + 350507/69092352*x^9 - 12411811/199600128*x^8 + 42497867/77968800*x^7 - 128458027/38188800*x^6 + 15098509/1060800*x^5 - 36979277/933504*x^4 + 83265517/1225224*x^3 - 175837997/2722720*x^2 + 18999103/680680*x - 42142223/12252240

18:

1/6402373705728000*x^17 - 19/376610217984000*x^16 + 1453/200074178304000*x^15 - 8227/13338278553600*x^14 + 525769/15243746918400*x^13 - 62947/46903836672*x^12 + 831617/22208256000*x^11 - 2414539/3172608000*x^10 + 14092313/1243662336*x^9 - 5680145/46061568*x^8 + 90887149/93562560*x^7 - 51936329/9547200*x^6 + 133989209/6364800*x^5 - 13729039/254592*x^4 + 29959279/350064*x^3 - 61698415/816816*x^2 + 124184839/4084080*x - 14274301/4084080

19:

-1/121645100408832000*x^18 + 1/337903056691200*x^17 - 263/550430318592000*x^16 + 41/894449267712*x^15 - 246557/84475764172800*x^14 + 1181221/9050974732800*x^13 - 94228597/22279322419200*x^12 + 2234269/22102502400*x^11 - 757950377/421956864000*x^10 + 2863343/120559104*x^9 - 37657973/162068480*x^8 + 22721897/13674528*x^7 - 308913373/36279360*x^6 + 152924489/5038800*x^5 - 348228377/4837248*x^4 + 32123311/302328*x^3 - 64673959/739024*x^2 + 127860511/3879876*x - 275295799/77597520

20:

1/2432902008176640000*x^19 - 1/6097498767360000*x^18 + 3989/135161222676480000*x^17 - 3607/1135808593920000*x^16 + 510773/2236123169280000*x^15 - 2359/202822963200*x^14 + 5227777/12067966310400*x^13 - 33493511/2784915302400*x^12 + 155558441/618870067200*x^11 - 223201501/56260915200*x^10 + 3980464529/84391372800*x^9 - 48508189/115763200*x^8 + 416779967/151939200*x^7 - 361495189/27907200*x^6 + 862274389/20155200*x^5 - 76155805/806208*x^4 + 157532425/1209312*x^3 - 13509215/134368*x^2 + 26274175/739024*x - 55835135/15519504

21:

-1/51090942171709440000*x^20 + 1/116115777662976000*x^19 - 4619/2688996956405760000*x^18 + 1399/6790396354560000*x^17 - 2775313/166963863306240000*x^16 + 113383/118582289280000*x^15 - 338573/8348193165312*x^14 + 876947/677613081600*x^13 - 612558481/19494407116800*x^12 + 21539551/36921225600*x^11 - 1956740711/236295843840*x^10 + 5282490449/59073960960*x^9 - 5305234043/7293081600*x^8 + 500243167/113954400*x^7 - 1251411967/65116800*x^6 + 962430229/16279200*x^5 - 2066648365/16930368*x^4 + 167106145/1058148*x^3 - 107825635/940576*x^2 + 8982005/235144*x - 18858053/5173168

22:

1/1124000727777607680000*x^21 - 23/53523844179886080000*x^20 + 83/878125568576256000*x^19 - 2323/184868540752896000*x^18 + 619007/547758639267840000*x^17 - 7048603/96663289282560000*x^16 + 100275221/28696914005760000*x^15 - 3239389/25508368005120*x^14 + 3321155/937042089984*x^13 - 964392691/12614028134400*x^12 + 760504397/595662439680*x^11 - 13372412497/812266963200*x^10 + 211606656479/1299627141120*x^9 - 6532143083/5348259840*x^8 + 17136142643/2506996800*x^7 - 1426684687/51163200*x^6 + 3194017947/39793600*x^5 - 2229033385/14325696*x^4 + 4412209765/23279256*x^3 - 335821965/2586584*x^2 + 211524139/5173168*x - 19093197/5173168

23:

-1/25852016738884976640000*x^22 + 1/48962152914554880000*x^21 - 467/94696032010567680000*x^20 + 5563/7694052600858624000*x^19 - 188593/2616600884502528000*x^18 + 816433/157480608789504000*x^17 - 206535859/741085217832960000*x^16 + 198764381/17369184792960000*x^15 - 42684419/117338492823552*x^14 + 27001853/2993328898560*x^13 - 7254645707/41446092441600*x^12 + 5350592509/2014740604800*x^11 - 3551722463/113225091840*x^10 + 267277654169/934107007680*x^9 - 244912433813/123009976320*x^8 + 19963875859/1922030880*x^7 - 46675855703/1176753600*x^6 + 3505900287/32687600*x^5 - 21516751385/109830336*x^4 + 4642017805/20593188*x^3 - 8694299577/59491432*x^2 + 648798629/14872858*x - 444316699/118982864

24:

1/620448401733239439360000*x^23 - 1/1079040698666503372800*x^22 + 6899/28202200078783610880000*x^21 - 47/1193743918678671360*x^20 + 13157/3039625718857728000*x^19 - 298147/863893625359564800*x^18 + 312672149/15118138443792384000*x^17 - 169448953/177860452279910400*x^16 + 1188700547/34738369585920000*x^15 - 29845537/30878550743040*x^14 + 645425453/29933288985600*x^13 - 175072753/460512138240*x^12 + 9125530241/1726920518400*x^11 - 5752039789/99904492800*x^10 + 82755537881/169837637760*x^9 - 4860973309/1537624704*x^8 + 237457360361/15376247040*x^7 - 3478358731/62760192*x^6 + 332510951279/2353507200*x^5 - 22963994875/94140288*x^4 + 43804301131/164745504*x^3 - 8983748275/54915168*x^2 + 16562041459/356948592*x - 1347822955/356948592

25:

-1/15511210043330985984000000*x^24 + 1/24857708402774016000000*x^23 - 709/61309130606051328000000*x^22 + 2399/1179021742424064000000*x^21 - 916103/3730449745870848000000*x^20 + 77459/3586970909491200000*x^19 - 464727119/323960109509836800000*x^18 + 50994569/692222456217600000*x^17 - 65997494749/22232556534988800000*x^16 + 158234357/1670113922400000*x^15 - 1032584899/428868760320000*x^14 + 73937513/1514842560000*x^13 - 15129112399/19188005760000*x^12 + 12125271293/1199250360000*x^11 - 36473818793/356801760000*x^10 + 100753984193/124880616000*x^9 - 14277249851/2912168000*x^8 + 270454511933/12012693000*x^7 - 1793421618299/23535072000*x^6 + 360298026287/1961256000*x^5 - 707636463887/2353507200*x^4 + 45789092203/147094200*x^3 - 83379832203/457626400*x^2 + 16891532467/343219800*x - 34052522467/8923714800

26:

1/403291461126605635584000000*x^25 - 1/597468831298675015680000*x^24 + 4387/8401905440137617408000000*x^23 - 4061/40588915169746944000000*x^22 + 5260789/398509348939333632000000*x^21 - 298567/234279452639232000000*x^20 + 67921109/727437700444815360000*x^19 - 339177373/63810324600422400000*x^18 + 7981401653/33424455743078400000*x^17 - 39228254083/4587670396108800000*x^16 + 34652635789/141124626442800000*x^15 - 1185587393/209073520656000*x^14 + 53932267841/512016785280000*x^13 - 61833285947/39385906560000*x^12 + 582552049091/31180509360000*x^11 - 2161981883/12270960000*x^10 + 241683071761/185536915200*x^9 - 66299382529/8907808000*x^8 + 2746563633647/85180914000*x^7 - 658109018183/6374082000*x^6 + 12037584908897/50992656000*x^5 - 249662092629/679902080*x^4 + 1384269783887/3824449200*x^3 - 85831402203/424938800*x^2 + 154883957203/2974571600*x - 34395742267/8923714800

27:

-1/10888869450418352160768000000*x^26 + 1/14957238256069165056000000*x^25 - 9827/435554778016734086430720000*x^24 + 3041/648146991096330485760000*x^23 - 6649651/9863106386248507392000000*x^22 + 325511/4574724668946432000000*x^21 - 977479523/170789720974000128000000*x^20 + 153435109/426974302435000320000*x^19 - 18500193571/1033727258526842880000*x^18 + 102581834071/143573230350950400000*x^17 - 8554504059707/371601302084812800000*x^16 + 109010490703/181445948283600000*x^15 - 861245531231/67739820692544000*x^14 + 1176750019/5400177032250*x^13 - 246584810341/81801498240000*x^12 + 741372925591/22154572440000*x^11 - 4114119009557/13915268640000*x^10 + 17878718321/8697042900*x^9 - 685447285511/61845638400*x^8 + 3073547791147/67643667000*x^7 - 6478854231397/46936422000*x^6 + 12911099729647/43025053500*x^5 - 24489141626497/55072068480*x^4 + 1439511827887/3442004280*x^3 - 2558163218887/11473347600*x^2 + 157646059403/2868336900*x - 312536252003/80313433200

28:

1/304888344611713860501504000000*x^27 - 29/11292160911544957796352000000*x^26 + 10961/11726474792758225403904000000*x^25 - 5191/24687315353175211376640000*x^24 + 694277/21172801709146795868160000*x^23 - 10408883/2761669788149582069760000*x^22 + 1040147893/3138261122897252352000000*x^21 - 66947747/2930215801024512000000*x^20 + 15003913373/11955280468180008960000*x^19 - 34950949321/629225287798947840000*x^18 + 178255310521/89334454440591360000*x^17 - 1256113670587/21497595988377600000*x^16 + 28398172648291/20321946207763200000*x^15 - 2464673235337/90319760923392000*x^14 + 8402130618337/19354234483584000*x^13 - 641606055257/114522097536000*x^12 + 3707673875239/63623387520000*x^11 - 9930891239239/20506711680000*x^10 + 24762892623439/7792550438400*x^9 - 1560078262147/96204326400*x^8 + 853519608793/13528733400*x^7 - 1761601326193/9663381000*x^6 + 3447056028943/9126526500*x^5 - 6436906110343/12047014980*x^4 + 46302843750997/96376119840*x^3 - 2625282302347/10708457760*x^2 + 4649180818987/80313433200*x - 315404588903/80313433200

29:

-1/8841761993739701954543616000000*x^28 + 1/10525907135404407088742400000*x^27 - 641/17235403496568619794432000000*x^26 + 2273/251902051103695212380160000*x^25 - 20706437/13602710759599541468528640000*x^24 + 2242187/11807908645485713080320000*x^23 - 44136461/2426921935040541818880000*x^22 + 2494741033/1820191451280406364160000*x^21 - 1218427787/14740779488827392000000*x^20 + 29145791123/7222981949525422080000*x^19 - 2932433868887/18247533346169487360000*x^18 + 295504333321/56319547364720640000*x^17 - 21457978362257/152394069339832320000*x^16 + 41622509546791/13394010000571200000*x^15 - 147543150800491/2619273066778368000*x^14 + 11172944063737/13363638095808000*x^13 - 1049351048741/103785650892000*x^12 + 4563366262789/46126955952000*x^11 - 11832429878239/15248580480000*x^10 + 28666050882439/5946946387200*x^9 - 65138896391539/2789925465600*x^8 + 470902665659/5449073175*x^7 - 1906068872143/8006801400*x^6 + 3666897946693/7784390250*x^5 - 13496697654311/21173541480*x^4 + 47947261295767/87340858605*x^3 - 83407649889397/310545275040*x^2 + 4724140023307/77636318760*x - 9227046511387/2329089562800

30:

1/265252859812191058636308480000000*x^29 - 31/9146650338351415815045120000000*x^28 + 3371/2368329105465991594967040000000*x^27 - 32581/87715892795036725739520000000*x^26 + 12768281/188926538327771409285120000000*x^25 - 68920967/7557061533110856371404800000*x^24 + 20987581/22139828710285712025600000*x^23 - 149798417/1925202496546583654400000*x^22 + 186763931017/36403829025608127283200000*x^21 - 1389732449/5053981539026534400000*x^20 + 26708935549/2211116923324108800000*x^19 - 2493367227931/5702354170677964800000*x^18 + 22105906013987/1689586420941619200000*x^17 - 32203550663357/99387436525977600000*x^16 + 59275949755741/8929340000380800000*x^15 - 200503471427341/1785868000076160000*x^14 + 1252154952929057/801818285748480000*x^13 - 84292688498699/4744486897920000*x^12 + 56769466664831/345952169640000*x^11 - 3484158125941/2859108840000*x^10 + 32874456133489/4574574144000*x^9 - 72954506143489/2202572736000*x^8 + 76443439303307/653888781000*x^7 - 4105609256711/13344669000*x^6 + 15548006324047/26689338000*x^5 - 28225975660807/37365073200*x^4 + 24779548570466/39700390275*x^3 - 42688835738966/145568097675*x^2 + 148699793966557/2329089562800*x - 9304682830147/2329089562800

Definition

For $n\geq0$ the secondary polynomial of the Laguerre polynomial $L_n$ is $q_n(x)=\int_0^\infty \frac{L_n(t)-L_n(x)}{t-x}e^{-t}\,\mathrm{d}t$ [1], where $L_n(0)=1$. The density $e^{-t}$ on $[0,\infty)$ has total mass $1$, so this is the probability-normalised secondary polynomial.

Parameters

$n$

— degree of the Laguerre polynomial ($n\geq0$)

Formulas

(1)

$q_0(x)=0$, $q_1(x)=-1$, and $(n+1)q_{n+1}(x)=(2n+1-x)q_n(x)-nq_{n-1}(x)$ for $n\geq1$.

(2)

If $L_n(x)=\sum_{j=0}^n c_jx^j$, then $q_n(x)=\sum_{j=1}^n c_j\sum_{i=0}^{j-1} i!\,x^{j-1-i}$.

(3)

For $n\geq1$, $q_n$ has degree $n-1$, leading coefficient $(-1)^n/n!$, and $q_n(0)=-H_n$, where $H_n$ is the harmonic number $1+\frac12+\cdots+\frac1n$.

(4)

At a root $x_k$ of $L_n$, $\dfrac{q_n(x_k)}{L_n'(x_k)}=\dfrac{x_k}{(n+1)^2L_{n+1}(x_k)^2}=\dfrac{1}{x_kL_n'(x_k)^2}=w_k$, the Gauss-Laguerre weight.

(5)

$q_n(z)/L_n(z)$ is the $[n-1/n]$ Pade approximant at $z=\infty$ to $\int_0^\infty e^{-t}(z-t)^{-1}\,\mathrm{d}t$, the Stieltjes transform of the Laguerre weight [2].

Comments

(6)

$L_n$ is the Laguerre polynomial of degree $n$ with $L_n(0)=1$, orthogonal on $[0,\infty)$ for the density $e^{-t}$. The quotient $(L_n(t)-L_n(x))/(t-x)$ is a polynomial in $t$ and $x$, so $q_n$ is a rational polynomial obtained from the moments $\int_0^\infty t^i e^{-t}\,\mathrm{d}t=i!$.

(7)

The first values are $q_0=0$, $q_1=-1$, $q_2=\frac12x-\frac32$, and $q_3=-\frac16x^2+\frac43x-\frac{11}{6}$.

(8)

At the roots $x_1<\cdots<x_n$ of $L_n$, $q_n(x_k)/L_n'(x_k)$ is the weight $w_k$ of the $n$-point Gauss-Laguerre quadrature rule. The nodes and weights are in the table of Gauss-Laguerre nodes and weights.

(9)

The Legendre secondary-polynomial table stores the same construction for the density $1$ on $[-1,1]$, whose total mass is $2$. Thus it is twice the probability-normalised Legendre version, while the Laguerre density used here already has mass $1$.

(10)

These are also the numerator polynomials, or associated polynomials of order one, of the Laguerre family. They are not the generalized Laguerre polynomials $L_n^{(\alpha)}$.

Programs

(P1)

Sage

R.<x> = QQ[]
p = R(laguerre(8, x))
m = lambda i: factorial(i)          # moments of e^-t on [0, infinity)
q = sum(c * sum(m(i) * x^(j - 1 - i) for i in range(j))
        for j, c in enumerate(p.list()))
q          # 1/40320*x^7 - 1/640*x^6 + 251/6720*x^5 - 193/448*x^4 + 2539/1008*x^3 - 801/112*x^2 + 2369/280*x - 761/280

Links

[1]

Wikipedia: Secondary polynomials

[2]

DLMF 18.30.23-25: Stieltjes continued fractions

[3]

Requested in numberdb-data#93

Similar tables

Laguerre polynomials — the $L_n$ these are the secondary polynomials of, and the recurrence they share

Nodes and weights of Gauss-Laguerre quadrature — the weights are $q_n(x_k)/L_n'(x_k)$ at the nodes

Secondary polynomials of the Legendre polynomials — the same construction for the Legendre weight, stored there with density $1$ rather than probability density $\frac12$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every $n\leq30$, the part of the Laguerre-polynomial range $n\leq50$ whose entries stay within the corpus readability bound)

How they were obtained:

Every value is exact, a polynomial with rational coefficients, so there is no precision to choose. The generator computes the recurrence in $\mathbb{Q}[x]$.

Before the entries are written, they are checked against the defining moment formula, the values $q_n(0)=-H_n$, the leading coefficient $(-1)^n/n!$, the Padé property of $q_n/L_n$, a plain Python fractions implementation of the moment formula, and the Gauss-Laguerre weight identity at every stored node for $n\leq30$.