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Dickson polynomials of the first kind $D_n(x,a)$
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Numbers
$n$ 
0:
2
1:
x
2:
x^2 - 2*a
3:
x^3 - 3*x*a
4:
x^4 - 4*x^2*a + 2*a^2
5:
x^5 - 5*x^3*a + 5*x*a^2
6:
x^6 - 6*x^4*a + 9*x^2*a^2 - 2*a^3
7:
x^7 - 7*x^5*a + 14*x^3*a^2 - 7*x*a^3
8:
x^8 - 8*x^6*a + 20*x^4*a^2 - 16*x^2*a^3 + 2*a^4
9:
x^9 - 9*x^7*a + 27*x^5*a^2 - 30*x^3*a^3 + 9*x*a^4
10:
x^10 - 10*x^8*a + 35*x^6*a^2 - 50*x^4*a^3 + 25*x^2*a^4 - 2*a^5
11:
x^11 - 11*x^9*a + 44*x^7*a^2 - 77*x^5*a^3 + 55*x^3*a^4 - 11*x*a^5
12:
x^12 - 12*x^10*a + 54*x^8*a^2 - 112*x^6*a^3 + 105*x^4*a^4 - 36*x^2*a^5 + 2*a^6
13:
x^13 - 13*x^11*a + 65*x^9*a^2 - 156*x^7*a^3 + 182*x^5*a^4 - 91*x^3*a^5 + 13*x*a^6
14:
x^14 - 14*x^12*a + 77*x^10*a^2 - 210*x^8*a^3 + 294*x^6*a^4 - 196*x^4*a^5 + 49*x^2*a^6 - 2*a^7
15:
x^15 - 15*x^13*a + 90*x^11*a^2 - 275*x^9*a^3 + 450*x^7*a^4 - 378*x^5*a^5 + 140*x^3*a^6 - 15*x*a^7
16:
x^16 - 16*x^14*a + 104*x^12*a^2 - 352*x^10*a^3 + 660*x^8*a^4 - 672*x^6*a^5 + 336*x^4*a^6 - 64*x^2*a^7 + 2*a^8
17:
x^17 - 17*x^15*a + 119*x^13*a^2 - 442*x^11*a^3 + 935*x^9*a^4 - 1122*x^7*a^5 + 714*x^5*a^6 - 204*x^3*a^7 + 17*x*a^8
18:
x^18 - 18*x^16*a + 135*x^14*a^2 - 546*x^12*a^3 + 1287*x^10*a^4 - 1782*x^8*a^5 + 1386*x^6*a^6 - 540*x^4*a^7 + 81*x^2*a^8 - 2*a^9
19:
x^19 - 19*x^17*a + 152*x^15*a^2 - 665*x^13*a^3 + 1729*x^11*a^4 - 2717*x^9*a^5 + 2508*x^7*a^6 - 1254*x^5*a^7 + 285*x^3*a^8 - 19*x*a^9
20:
x^20 - 20*x^18*a + 170*x^16*a^2 - 800*x^14*a^3 + 2275*x^12*a^4 - 4004*x^10*a^5 + 4290*x^8*a^6 - 2640*x^6*a^7 + 825*x^4*a^8 - 100*x^2*a^9 + 2*a^10
21:
x^21 - 21*x^19*a + 189*x^17*a^2 - 952*x^15*a^3 + 2940*x^13*a^4 - 5733*x^11*a^5 + 7007*x^9*a^6 - 5148*x^7*a^7 + 2079*x^5*a^8 - 385*x^3*a^9 + 21*x*a^10
22:
x^22 - 22*x^20*a + 209*x^18*a^2 - 1122*x^16*a^3 + 3740*x^14*a^4 - 8008*x^12*a^5 + 11011*x^10*a^6 - 9438*x^8*a^7 + 4719*x^6*a^8 - 1210*x^4*a^9 + 121*x^2*a^10 - 2*a^11
23:
x^23 - 23*x^21*a + 230*x^19*a^2 - 1311*x^17*a^3 + 4692*x^15*a^4 - 10948*x^13*a^5 + 16744*x^11*a^6 - 16445*x^9*a^7 + 9867*x^7*a^8 - 3289*x^5*a^9 + 506*x^3*a^10 - 23*x*a^11
24:
x^24 - 24*x^22*a + 252*x^20*a^2 - 1520*x^18*a^3 + 5814*x^16*a^4 - 14688*x^14*a^5 + 24752*x^12*a^6 - 27456*x^10*a^7 + 19305*x^8*a^8 - 8008*x^6*a^9 + 1716*x^4*a^10 - 144*x^2*a^11 + 2*a^12
25:
x^25 - 25*x^23*a + 275*x^21*a^2 - 1750*x^19*a^3 + 7125*x^17*a^4 - 19380*x^15*a^5 + 35700*x^13*a^6 - 44200*x^11*a^7 + 35750*x^9*a^8 - 17875*x^7*a^9 + 5005*x^5*a^10 - 650*x^3*a^11 + 25*x*a^12
26:
x^26 - 26*x^24*a + 299*x^22*a^2 - 2002*x^20*a^3 + 8645*x^18*a^4 - 25194*x^16*a^5 + 50388*x^14*a^6 - 68952*x^12*a^7 + 63206*x^10*a^8 - 37180*x^8*a^9 + 13013*x^6*a^10 - 2366*x^4*a^11 + 169*x^2*a^12 - 2*a^13
27:
x^27 - 27*x^25*a + 324*x^23*a^2 - 2277*x^21*a^3 + 10395*x^19*a^4 - 32319*x^17*a^5 + 69768*x^15*a^6 - 104652*x^13*a^7 + 107406*x^11*a^8 - 72930*x^9*a^9 + 30888*x^7*a^10 - 7371*x^5*a^11 + 819*x^3*a^12 - 27*x*a^13
28:
x^28 - 28*x^26*a + 350*x^24*a^2 - 2576*x^22*a^3 + 12397*x^20*a^4 - 40964*x^18*a^5 + 94962*x^16*a^6 - 155040*x^14*a^7 + 176358*x^12*a^8 - 136136*x^10*a^9 + 68068*x^8*a^10 - 20384*x^6*a^11 + 3185*x^4*a^12 - 196*x^2*a^13 + 2*a^14
29:
x^29 - 29*x^27*a + 377*x^25*a^2 - 2900*x^23*a^3 + 14674*x^21*a^4 - 51359*x^19*a^5 + 127281*x^17*a^6 - 224808*x^15*a^7 + 281010*x^13*a^8 - 243542*x^11*a^9 + 140998*x^9*a^10 - 51272*x^7*a^11 + 10556*x^5*a^12 - 1015*x^3*a^13 + 29*x*a^14
30:
x^30 - 30*x^28*a + 405*x^26*a^2 - 3250*x^24*a^3 + 17250*x^22*a^4 - 63756*x^20*a^5 + 168245*x^18*a^6 - 319770*x^16*a^7 + 436050*x^14*a^8 - 419900*x^12*a^9 + 277134*x^10*a^10 - 119340*x^8*a^11 + 30940*x^6*a^12 - 4200*x^4*a^13 + 225*x^2*a^14 - 2*a^15
31:
x^31 - 31*x^29*a + 434*x^27*a^2 - 3627*x^25*a^3 + 20150*x^23*a^4 - 78430*x^21*a^5 + 219604*x^19*a^6 - 447051*x^17*a^7 + 660858*x^15*a^8 - 700910*x^13*a^9 + 520676*x^11*a^10 - 260338*x^9*a^11 + 82212*x^7*a^12 - 14756*x^5*a^13 + 1240*x^3*a^14 - 31*x*a^15
32:
x^32 - 32*x^30*a + 464*x^28*a^2 - 4032*x^26*a^3 + 23400*x^24*a^4 - 95680*x^22*a^5 + 283360*x^20*a^6 - 615296*x^18*a^7 + 980628*x^16*a^8 - 1136960*x^14*a^9 + 940576*x^12*a^10 - 537472*x^10*a^11 + 201552*x^8*a^12 - 45696*x^6*a^13 + 5440*x^4*a^14 - 256*x^2*a^15 + 2*a^16
33:
x^33 - 33*x^31*a + 495*x^29*a^2 - 4466*x^27*a^3 + 27027*x^25*a^4 - 115830*x^23*a^5 + 361790*x^21*a^6 - 834900*x^19*a^7 + 1427679*x^17*a^8 - 1797818*x^15*a^9 + 1641486*x^13*a^10 - 1058148*x^11*a^11 + 461890*x^9*a^12 - 127908*x^7*a^13 + 20196*x^5*a^14 - 1496*x^3*a^15 + 33*x*a^16
34:
x^34 - 34*x^32*a + 527*x^30*a^2 - 4930*x^28*a^3 + 31059*x^26*a^4 - 139230*x^24*a^5 + 457470*x^22*a^6 - 1118260*x^20*a^7 + 2042975*x^18*a^8 - 2778446*x^16*a^9 + 2778446*x^14*a^10 - 1998724*x^12*a^11 + 999362*x^10*a^12 - 329460*x^8*a^13 + 65892*x^6*a^14 - 6936*x^4*a^15 + 289*x^2*a^16 - 2*a^17
35:
x^35 - 35*x^33*a + 560*x^31*a^2 - 5425*x^29*a^3 + 35525*x^27*a^4 - 166257*x^25*a^5 + 573300*x^23*a^6 - 1480050*x^21*a^7 + 2877875*x^19*a^8 - 4206125*x^17*a^9 + 4576264*x^15*a^10 - 3640210*x^13*a^11 + 2057510*x^11*a^12 - 791350*x^9*a^13 + 193800*x^7*a^14 - 27132*x^5*a^15 + 1785*x^3*a^16 - 35*x*a^17
36:
x^36 - 36*x^34*a + 594*x^32*a^2 - 5952*x^30*a^3 + 40455*x^28*a^4 - 197316*x^26*a^5 + 712530*x^24*a^6 - 1937520*x^22*a^7 + 3996135*x^20*a^8 - 6249100*x^18*a^9 + 7354710*x^16*a^10 - 6418656*x^14*a^11 + 4056234*x^12*a^12 - 1790712*x^10*a^13 + 523260*x^8*a^14 - 93024*x^6*a^15 + 8721*x^4*a^16 - 324*x^2*a^17 + 2*a^18
37:
x^37 - 37*x^35*a + 629*x^33*a^2 - 6512*x^31*a^3 + 45880*x^29*a^4 - 232841*x^27*a^5 + 878787*x^25*a^6 - 2510820*x^23*a^7 + 5476185*x^21*a^8 - 9126975*x^19*a^9 + 11560835*x^17*a^10 - 10994920*x^15*a^11 + 7696444*x^13*a^12 - 3848222*x^11*a^13 + 1314610*x^9*a^14 - 286824*x^7*a^15 + 35853*x^5*a^16 - 2109*x^3*a^17 + 37*x*a^18
38:
x^38 - 38*x^36*a + 665*x^34*a^2 - 7106*x^32*a^3 + 51832*x^30*a^4 - 273296*x^28*a^5 + 1076103*x^26*a^6 - 3223350*x^24*a^7 + 7413705*x^22*a^8 - 13123110*x^20*a^9 + 17809935*x^18*a^10 - 18349630*x^16*a^11 + 14115100*x^14*a^12 - 7904456*x^12*a^13 + 3105322*x^10*a^14 - 810084*x^8*a^15 + 128877*x^6*a^16 - 10830*x^4*a^17 + 361*x^2*a^18 - 2*a^19
39:
x^39 - 39*x^37*a + 702*x^35*a^2 - 7735*x^33*a^3 + 58344*x^31*a^4 - 319176*x^29*a^5 + 1308944*x^27*a^6 - 4102137*x^25*a^7 + 9924525*x^23*a^8 - 18599295*x^21*a^9 + 26936910*x^19*a^10 - 29910465*x^17*a^11 + 25110020*x^15*a^12 - 15600900*x^13*a^13 + 6953544*x^11*a^14 - 2124694*x^9*a^15 + 415701*x^7*a^16 - 46683*x^5*a^17 + 2470*x^3*a^18 - 39*x*a^19
40:
x^40 - 40*x^38*a + 740*x^36*a^2 - 8400*x^34*a^3 + 65450*x^32*a^4 - 371008*x^30*a^5 + 1582240*x^28*a^6 - 5178240*x^26*a^7 + 13147875*x^24*a^8 - 26013000*x^22*a^9 + 40060020*x^20*a^10 - 47720400*x^18*a^11 + 43459650*x^16*a^12 - 29716000*x^14*a^13 + 14858000*x^12*a^14 - 5230016*x^10*a^15 + 1225785*x^8*a^16 - 175560*x^6*a^17 + 13300*x^4*a^18 - 400*x^2*a^19 + 2*a^20
Definition
The Dickson polynomials of the first kind are $D_0(x,a) = 2$, $D_1(x,a) = x$, and $D_n(x,a) = x\,D_{n-1}(x,a) - a\,D_{n-2}(x,a)$ for $n \geq 2$. They are polynomials in two variables, $x$ and the parameter $a$, and are characterised by $D_n\!\left(y + \tfrac{a}{y},\, a\right) = y^{n} + \left(\tfrac{a}{y}\right)^{n}$.
Parameters
$n$
—   integer ($0 \leq n \leq 40$)
Formulas
(1)
$D_n(x,a) = x\,D_{n-1}(x,a) - a\,D_{n-2}(x,a)$, with $D_0 = 2$ and $D_1 = x$.
(2)
$D_n\!\left(y + \tfrac{a}{y},\, a\right) = y^{n} + \left(\tfrac{a}{y}\right)^{n}$, which determines them.
(3)
$D_n(2x, 1) = 2\,T_n(x)$, where $T_n$ are the Chebyshev polynomials of the first kind Chebyshev_polynomials_of_the_first_kind.
(4)
$D_n(x,a) = \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{n}{n-k} \binom{n-k}{k} (-a)^{k} x^{n-2k}$ for $n \geq 1$.
(5)
$D_{mn}(x,a) = D_m\!\left(D_n(x,a),\, a^{n}\right)$, so these compose the way powers do.
(6)
$D_n(x, 0) = x^{n}$.
Comments
(7)
Over a finite field $\mathbb{F}_q$, $D_n(x,a)$ with $a \neq 0$ permutes the field exactly when $\gcd(n, q^{2}-1) = 1$; this is what makes them a standard source of permutation polynomials.
(8)
The Dickson polynomials of the second kind $E_n$ satisfy the same recurrence with $E_0 = 1$ and $E_1 = x$, and relate to the Chebyshev polynomials of the second kind Chebyshev_polynomials_of_the_second_kind as these do to the first. They are not in this table.
(9)
Stored as polynomials in $x$ and $a$ together, rather than as a family indexed by $a$: $a$ is a parameter of the polynomial, not of the table.
Programs
(P1)
Sage
R. = ZZ[]
def dickson(n):
    p, q = R(2), x
    for _ in range(n): p, q = q, x*q - a*p
    return p

dickson(41)                      # the next one after this table
Links
Data properties
Entries are of type: integral polynomial
Table is complete: false
How well the digits are known: exact


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