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© 2000-2021   Gérard P. Michon, Ph.D.

Special Polynomials

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Special Polynomials


(2012-02-15)   Chebyshev Polynomials   [ of the first kind ]
A family of  commuting  polynomial functions.   Tn oTp  =  Tp oTn  =  Tnp

cos(nq) is a polynomial function of cos(q).  The following relation defines a polynomial of degree n known as the Chebyshev polynomial of degree n:

cos (nq)   =   Tn (cos q)

The symbol T comes from careful Russian transliterations like  TchebyshevTchebychef (French) or Tschebyschow (German).  Alternate spellings include Tchebychev (French) and "Chebychev".

The trigonometric formula   cos (n+2)q  =  2 cos q cos (n+1)q - cos nq   translates into a simple recurrence relation which makes Chebyshev polynomials very easy to tabulate, namely:

T0 (x)  =  1 Tn+2 (x)   =   2 x Tn+1 (x)  -  Tn (x)
T1 (x)  =   x  Pafnuty Lvovich Chebyshev (1821-1894)
Pafnuty Chebyshev
T2 (x)  =   -1+2 x2
T3 (x)  =   -3 x+4 x3
T4 (x)  =   1-8 x2+8 x4
T5 (x)  =   5 x-20 x3+16 x5
T6 (x)  =   -1+18 x2-48 x4 +32 x6
T7 (x)  =   -7 x+56 x3-112 x5 +64 x7
T(x)  =   1-32 x2+160 x4 -256 x6+128 x8

Knowing only the highest term of  Tn  and its obvious  n  distinct real zeroes,  we obtain immediately  Tn  as a product of  n  factors:

If  n > 0,   then     Tn (x)   =   2 n-1   n-1   ( x - cos (k+½) p/n )
Õ
k=0

The case  Tn (0) = (-1)n  tells something nice about a product of cosines.

Inverse formulas :

x0  =  T0  
x1  =   T1  
2 x2  =   T0+T2
4 x3  =   3 T1+T3
8 x4  =   3 T0+4 T2+T4
16 x5  =   10 T1+5 T3+T5
32 x6  =   10 T0+15 T2+6 T4 +T6
64 x7  =   35 T1+21 T3+7 T5 +T7
128 x8  =   35 T0+56 T2+28 T4 +8 T6+T8

(2014-07-26)  A solution looking for a problem :

Chebyshev polynomials verify   Tm(Tn(x))  =  Tmn(x).  This unique property makes it possible to define pairs of closely related functions from any pair of arithmetic functions u and v  (with subexponential growth)  that are Dirichlet inverses of each other, using the following symmetrical relations:

  ¥
g ( x )   =    å    u(n)  f ( Tn(x) )
  n = 1
  ¥
f ( x )   =    å    v(n)  g ( Tn(x) )
  n = 1
If  f (0) = 0, those series are usually absolutely convergent, because  Tn(x)  decreases exponentially with  n,  for any fixed  x  in  ]-1,+1[.

Proof :   Expand the latter right-hand-side using the definition of  g :

å m  å n  u(n) v(m)  f ( Tmn (x) )   =   å k [ å d|k u(d) v(k/d) ]  f ( T(x) )

u and v being Dirichlet inverses, the bracket is either  1  (if k = 1)  or  0.   QED

This applies, in particular, when  u  is a totally multiplicative arithmetic function  [i.e., such that  u(mn)  =  u(m) u(n)  for  any  m & n ]  in which case its Dirichlet inverse can be expressed using the Möbius function (m) :

v(n)   =   m(n) u(n)

Using Tn(x) = x1/n instead of Chebyshev polynomials, this pattern was used in 1859 by Riemann to link his (normalized) prime-counting function  f = p  with the celebrated  jump function  g = J  he obtained with  u(n) = 1/n.

Bienaymé-Chebyshev inequality   |   Chebyshev economization   |   Pafnuty Chebyshev  (1821-1894)


(2015-12-06)   Chebyshev polynomials  of the second kind.
They are denoted by the symbol  U  (simply because U comes after T ).

They obey exactly the same second-order recurrence relation as the above Chebychev polynomials of the first kind but the starting points are different:

T0 (x)  =  1
T1 (x)  =  x
  U0 (x)  =  1
U1 (x)  =  2x

U0 (x)  =  1 Un+2 (x)   =   2 x Un+1 (x)  -  Un (x)
U1 (x)  =   2 x  Pafnuty Lvovich Chebyshev (1821-1894)
Pafnuty Chebyshev
U2 (x)  =   -1+4 x2
U3 (x)  =   -4 x+8 x3
U4 (x)  =   1-12 x2+16 x4
U5 (x)  =   6 x-32 x3+32 x5
U6 (x)  =   -1+24 x2-80 x4 +64 x6
U7 (x)  =   -8 x+80 x3-192 x5 +128 x7
U(x)  =   1-40 x2+240 x4 -448 x6+256 x8

Inverse formulas :

x0  =  U0  
2 x1  =   U1  
4 x2  =   U0+U2
8 x3  =   2 U1+U3
16 x4  =   2 U0+3 U2+U4
32 x5  =   10 U1+4 U3+U5
64 x6  =   5 U0+9 U2+5 U4 +U6
128 x7  =   14 U1+14 U3+6 U5 +U7
256 x8  =   14 U0+28 U2+20 U4 +7 U6+U8

Generalized Chebychev Polynomials, Planar Trees and Galois Theory  by  Anton Bankevich  (2008-02-28)
Wikipedia :   Chebyshev polynomials  of the first and second kinds.
Dessins d'enfants, trees and Shabat polynomials.


 Adrien-Marie Legendre 
 1752-1833 (2012-02-15)   Legendre Polynomials
Key to Coulombian multipole expansionspherical harmonics.

The Legendre polynomials  (A008316)  are recursively defined by:

P0 (x)= 1; Pn (x)   =   (2-1/n) x Pn-1 (x)  -  (1-1/n) Pn-2 (x)
P1 (x)= x  Signature of Adrien-Marie Legendre  Adrien-Marie Legendre (1752-1833)
2P2 (x)= -1+3 x2
2P3 (x)= -3 x+5 x3
8P4 (x)= 3-30 x2 +35 x4
8P5 (x)= 15 x-70 x3 +63 x5
16P6 (x)= -5+105 x2 -315 x4 +231 x6
16P7 (x)= -35 x+315 x3 -693 x5 +429 x7
128P8 (x)= 35- 1260 x2+ 6930 x4- 12012 x6+ 6435 x8

They are linked to the expressions of  spherical harmonics  in terms of the  colatitude  q Î [0,p[  and the  longitude  f  (modulo 2p).

Electric multipoles   |   Figure of the Earth   |   Dynamic form factors   |   Legendre Polynomials
 
Adrien-Marie Legendre  (1752-1833)   |   MathWorld :   Legendre Polynomials   |   Spherical Harmonics


(2012-02-16)   Laguerre Polynomials
Radial part of the solution of the Schrödinger equation for hydrogenoids.

Laguerre's equation  is a second-order linear differential equation:

x y''  +  (1-x) y'  +  n y   =   0

It has non-singular solutions only when  n  is a non-negative integer.  In that case,  a solution is  Ln(n),  the Laguerre polynomial of order n given by:

 L0(x)  =   1 (n+1) Ln+l (x)  =  (2n+1-x) Ln (x) - n Ln-1 (x)
 L1(x)  =   1- x
L2(x)  =   2- 4x+ x2
L3(x)  =   6- 18x+ 9x2- x3
24 L4(x)  =   24- 96x+ 72x2- 16x3+ x4
120 L5(x)  =   120- 600x+ 600x2- 200x3+ 25x4- x5
720 L6(x)  =   720- 4320x+ 5400x2- 2400x3+ 450x4 - 36x5+ x6
5040 L7(x)  =   5040-35280x+52920x2-29400x3+7350x4-882x5 +49x6-x7
 Edmond Laguerre 
 1834-1886, X1853
Edmond Laguerre
  Edmond Laguerre (1834-1886; X1853) may have devised those polynomials as early as 1860 but the relevant memoir was only published in 1879.  The Laguerre polynomials arose from a remarkable continued fraction expansion of the definite integral from zero to infinity of  exp(x)/x
 
Generalization :

Sorin is credited for the following  generalized Laguerre equation :

x y''  +  (a+1-x) y'  +  n y   =   0

This is satisfied by the Laguerre function,  defined by:

L (a)
n
   =    ¥    ( n+a
n-p
)  (-x)p
 Vinculum
p!
å
n=1

Because of the way  binomial coefficients  vanish,  a polynomial  (a finite sum)  called  associated Laguerre polynomial  is so obtained when  n  is a non-negative integer.  Otherwise,  the above is a  divergent series  which is  Borel-summable.

Ordinary  Laguerre polynomials  correspond to the special case  a = 0.

Wikipedia :   Associated Laguerre Polynomials   |   Nikolay Sonin (1849-1915)
Rook Polynomials of John Riordan (1903-1988)
"On Laguerre's Series"  |  First note  |  second note  |  third note  |  by  Einar Hille  (1926).
MathWorld :   Laguerre Polynomials


(2012-02-18)   Hermite Polynomials  &  Modified Hermite Polynomials
Eigenstates of the quantum harmonic oscillator.

H0 (x)= 1 Hn+1(x)   =   2x Hn(x)  -  2n Hn-1(x)
H1 (x)= 2 x  Charles Hermite 
 1822-1901, X1842
Charles Hermite
H2 (x)= -2+4 x2
H3 (x)= -12 x+8 x3
H4 (x)= 12-48 x2 +16 x4
H5 (x)= 120 x-160 x3 +32 x5
H6 (x)= -120+720 x2 -480 x4 +64 x6
H7 (x)= -1680 x+3360 x3 -1344 x5 +128 x7

The above are more popular than the simpler  modified Hermite polynomials  Hen  which can be defined via:   Hn (x)   =   2n/2 Hen (2½ x)

He0 (x)= 1 Hen+1(x)  =  x Hen(x)  -  n Hen-1(x)
He1 (x)= x  
He2 (x)= -1+x2
He3 (x)= -3 x+x3
He4 (x)= 3-6 x2 +x4
He5 (x)= 15 x-10 x3 +x5
He6 (x)= -15+45 x2 -15 x4 +x6
He7 (x)= -105 x+105 x3 -21 x5 +x7

Hermite Polynomials (Wikipedia)   |   Hermite Polynomials (MathWorld)
Charles Hermite (1822-1901; X1942)

 Friedrich Bessel
(2014-12-07)   Bessel Polynomials

The  reverse Bessel polynomials  tabulated below appear in the transfer functions of Bessel-Thomson filters


q0(s)  =  1
q1(s)  =   1+ s qn   =   (2n-1) qn-1  +  s2 qn-2
q2(s)  =   3+ 3 s+ s2
q3(s)  =   15+ 15 s+ 6 s2+ s3
q4(s)  =   105+ 105 s+ 45 s2+ 10 s3 + s4
q5(s)  =   945+ 945 s+ 420 s2+ 105 s3 + 15 s4+ s5
q6(s)  =   10395+ 10395 s+ 4725 s2+ 1260 s3 + 210 s4+ 21 s5+ s6

MathWorld :   Bessel polynomials


 Bernoulli (2013-04-24)   Bernoulli Polynomials

Like  Hermite polynomials  and  Euler polynomials,  the sequence of  Bernoulli polynomials  start with some nonzero constant polynomial  (namely, 1)  and subsequently verify the  Appell property,  which is to say:

dBn (x) / dx   =   n Bn-1 (x)

This relation becomes a recursive definition if the successive  constants of integration  are given as a prescribed sequence:

Bn   =   Bn (0)

The polynomials can be expressed in terms of that sequence of numbers:

Bn (x)   =    n
å
k = 0
   (  n
k
 )   Bk   xn-k

 Come back later, we're
 still working on this one...

B0 (x)= 1   Change sign of  highlighted 
terms for original definition
of Bernoulli polynomials.
 Jacob Bernoulli 
 1655-1705
Jacob Bernoulli
1655-1705
B1 (x)= x   -  1/2
B2 (x)= x2  -   x +  1/6
B3 (x)= x3  - 3/ x2 + 1/2 x + 0
B4 (x)= x4  -  2 x3 +  x2 -  1/30
B5 (x)= x5  -  5/2 x4 +  5/3 x3 -  1/ x + 0
B6 (x)= x6  -  3 x5 +  5/2 x4 -  1/2 x2 + 1/42
B7 (x)= x7  -  7/2 x6 +  7/2  x5 -  7/6 x3 +  1/6  x + 0
B8 (x)= x8  -  4 x7  + 14/3  x5 -  7/3 x3 +  2/3  x -  1/30

On the two competing definitions of Bernoulli numbers :

With the  convention adopted in Numericana  (i.e.,  B1 = ½)  we have:

Bn = Bn(1)

Authors who posit that  B1 = -½   specify instead that  Bn = Bn(0).  Note that those two conventions only differ in the case  n = 1.

The Bernoulli polynomials are not affected by the choice of convention for Bernoulli numbers.  Neither are relations between those polynomials,  like:

Bn (1 - x)   =   (-1)n Bn (x)
Bn (1 + x)   =   Bn (x)  +  n xn-1

Besides the aforementioned case  n = 1,  Bn  vanishes for odd values of  n.

The even-indexed Bernoulli numbers :   B2n  =  A000367(n) / A002445(n)
B0B2B4B6 B8B10B12B14
11 / 6-1 / 301 / 42 -1 / 305 / 66-691 / 27307 / 6

B16B18B20B22 B24
-3617 / 51043867 / 798-174611 / 330854513 / 138 -236364091 / 2730

B26B28B30
8553103 / 6-23749461029 / 8708615841276005 / 14322

B32B34B36
-7709321041217 / 5102577687858367 / 6 -26315271553053477373 / 1919190

B38B40
2929993913841559 / 6 -261082718496449122051 / 13530

B42B44
1520097643918070802691 / 1806 -27833269579301024235023 / 690

By the  von Staudt-Clausen theorem (1840)  the denominator of  B2n  is the product of all primes  p  for which  p-1  divides  2n.

Introduction to Bernoulli and Euler Polynomials  by  Zhi-Wei Sun,  Nanjing University   (2002-06-06)
Mathworld   }   Wikipedia   }   Encyclopedia of Mathematics   }   Kim Milton
 
von Staudt-Clausen theorem (1840)   }   Karl von Staudt (1798-1867)   }   Thomas Clausen (1801-1885)
 
Darboux's summation formula   }   Umbral calculus
 
Why do Bernoulli numbers arise everywhere?  (MathOverflow, since 2011).

 Johann Faulhaber 
 1580-1635
Johann Faulhaber
1580-1635

(2019-11-24)   Faulhaber Polynomials   (Faulhaber, 1631)

After deriving explicit formulas up to  p = 17,  Johann Faulhaber  observed that,   if  p = 2q+1  is odd,  then the sum of the p-th powers of the integers from 0 to n  is a polynomial of degree  q+1  in the variable  x = n(n+1)/2.  A related expression holds for a nonzero even  p,  namely:

n
å
k = 0
  k 2q+1 = Fq+1(x)
If  q > 0,  then:       n
å
k = 0
  k 2q = n+½
Vinculum
2q+1
    d
Vinculum
dx
 Fq+1(x)

That result was proved in full generality by  Carl Jacobi,  in 1834.

F(x) x
F2 (x) x2
F3 (x) ( 4 x3  -  x2 ) / 3
F4 (x) ( 6 x4  -  4 x3  +  x2 ) / 3
F5 (x) ( 16 x5  -  20 x4  +  12 x3  -  3 x2 ) / 5
F6 (x) ( 16 x6  -  32 x5  +  34 x4  -  20 x3  +  5 x2 ) / 3
F7 (x) ( 960 x7 - 2800 x6 + 4592 x5 - 4720 x4 + 2764 x3 - 691 x2 ) / 105
F8 (x) ( 48 x8 - 192 x7 + 448 x6 - 704 x5 + 718 x4 - 420 x3 + 105 x2 ) / 3
F9 (x)   ( 1280 x9 - 6720 x8 + 21120 x7 - 46880 x6 + 72912 x5
- 74220 x4 + 43404 x3 - 1851 x2 ) / 45  

 Come back later, we're
 still working on this one...

Faulhaber's formula   |   Faulhaber polynomials   |   Johann Faulhaber of Ulm (1580-1635)
 
Johann Faulhaber and Sums of Powers  by  Donald E. Knuth  (Math. Comp, 61, 203, 277-294, July 1993).


 Leonhard Euler  
 1707-1783 (2013-04-24)   Euler Polynomials

En(x)

 Come back later, we're
 still working on this one...

Relation to the Sequence of Euler Numbers :

Euler numbers  can be expressed in terms of the above Euler polynomials:

En   =   2n En (½)

The Euler numbers of odd index vanish.  The signs of even-indexed Euler numbers alternate.

Leonhard Euler  (1707-1783)
MathWorld :   Euler polynomials


(2021-07-15)   Mittag-Leffler Polynomials  Mn (x).
Mn (x)  is the coefficient of  t/n!  in the  expansion  of  (1+t)x / (1-t)x

Mittag-Leffler polynomials were first discussed under that name in 1940,  by  Harry Bateman (1882-1946).

They obey the same  binomial formula  as ordinary powers:

M0 (x)= 1 Mn+1(x)  =  (x/2) [ Mn(x+1) + 2 Mn(x) + Mn(x-1) ]
M1 (x)= 2 x  Mittag-Leffler 
 1846-1927
Gösta Mittag-Leffler
M2 (x)= 4 x2
M3 (x)= 4 x+8 x3
M4 (x)= 32 x2 +16 x4
M5 (x)= 48 x+80 x3 +32 x5
M6 (x)= 736 x2 +640 x4 +64 x6
M7 (x)= 1440 x+6272 x3 +2240 x5 +128 x7

Mittag-Leffler Polynomials (1891)   |   MathWorld   |   Gösta Mittag-Leffler (1846-1927)
 
Umbral calculus   |   Sheffer sequence (poweroids)   |   Isador M. Sheffer (1901-1992, PhD 1927)
 
The polynomial of Mittag-Leffler  by  Harry Bateman (1882-1946)   Proc. N.A.S., 26, 491-496 (1940-07-13).
 
Generalization of power polynomials  by  John D. Cook (2020-01-28).


(2021-07-20)   Binomial Polynomials and Umbral Operator
The binomial polynomials form a group under umbral composition.

 Come back later, we're
 still working on this one...

Umbral calculus   |   Sheffer sequence (poweroids)   |   Isador M. Sheffer (1901-1992, PhD 1927)
 
Polynomials of binomial type   |   Cumulants   |   Moments


 Carl Friedrich Gauss (2020-06-02)   Cyclotomic Polynomials
Irreducible divisors of   x n - 1   over the rationals.

The  nth  cyclotomic polynomial  Fn is the  unique  monic polynomial dividing  x k - 1   for  k = n  but not for any lesser value of  k.

When n > 1,  Fn  is  palindromic.  If  n  has at most two distinct odd prime factors,  then the coefficients of  Fn  stay within  {-1,0,1}.  That holds for n < 105;  the first product of three distinct odd primes  (Adolph Migotti, 1883).  Those coefficients can be arbitrarily large  (Issai Schur, 1931).  Furthermore,  any  given integer occurs as a coefficient of  some  cyclotomic polynomial  (Jiro Suzuki, 1987).

Fn  is an  irreducible polynomial  over the rationals, whose degree is equal to the  Euler totient  f (n).  That nontrivial fact is due to  Carl F. Gauss.

The following definition also holds for  n = 0  (as an  empty product  is 1).

 F(x)   =   Õ   (  x - exp( i 2kp / n)  )
1 ≤ k ≤ n
GCD(k,n) = 1

For n > 0,  the cyclotomic polynomial  Fn  can thus be defined as the unique monic polynomial whose roots are the  primitive  nth  roots of unity.

As with any  multiplicative function,  the  (rarely used)  value of  f  at zero is  f (0) = 0  (as its  one-line definition  implies)  which confirms  F0  = 1.

Unfortunately,  the  OEIS  (A013595)  is still following a dubious ad-hoc definition for  F0  from  Maple®  (albeit with due apologies).

The First Cyclotomic Polynomials
   1  x  x2x3x4 x5x6x7x8x9 10111213 1415161718 19202122
F(x) 1
F(x) -11
F(x) 11
F(x) 111
F(x) 101
F(x) 11111
F(x) 1-11
F(x) 1111111
F(x) 10001
F(x) 1001001
F10 (x) 1-11-11
F11 (x) 1111111111
F12 (x) 10-101
F13 (x) 111111111111
F14 (x) 1-11-11-11
F15 (x) 1-101-110-11
F16 (x) 100000001
F17 (x) 11111111 111111111
F18 (x) 100-1001
F19 (x) 111111111 1111111111
F20 (x) 10-1010-10-1
F21 (x) 1-101 -1010-1 10-11
F22 (x) 1-11-11-1 1-11-11
F23 (x) 111111111 1111111111 1111
F24 (x) 1000-10001
F25 (x) 10000 10000 10000 100001
F26 (x) 1-11-1 1-11-1 1-11-11
F27 (x) 100000000 1000000001
F28 (x) 10-10 10-10 10-10 1
   1  x  x2x3x4 x5x6x7x8x9 10111213 1415161718 19202122

The following factorization yields as many factors as there are divisors of n:

 xn - 1   =   Õ   Fk (x)
k | n

Equating polynomial degrees retrieves a  Dirichlet convolution:   N = u*f

The following interesting equation involves the  Möbius function  m :

 Fn (x)   =   Õ   ( xn/k - 1 ) m(k)
k | n

In this,  all the negative exponents do cancel "miraculously".

Encyclopedia of Mathematics   |   MathWorld   |   Wikipedia
 
"Some properties of coefficients of cyclotomic polynomialsMarcin Mazur, Bogdan V. Petrenko  (2019-02-12).
 
"Cyclotomic polynomials"  by  Brett Porter  (student project at Whitman College, 2015-05-20).
 
Bungers-Lehmer Theorem on Cyclotomic Coefficients  by  Robin Whitty  (Theorem of the Day #175).
 
Conditional proof in the 1934 thesis of Rolf Bungers, possibly the future seismologist (1909-1942)
[former student of Gustav Angenheister (1878-1945) who died in a plane crash in Norway, on 1942-12-24]
 
On the magnitude of the coefficients of the cyclotomic polynomial (June 1936) Emma Lehmer (1906-2007)
 
Cyclotomic polynomials (19:42)  by  MathDoctorBob  (2013-01-11).
 
Tips and tricks for computing cyclotomic polynumbers (27:09)  by  Norman J. Wildberger  (2020-09-12).


 Gerard Michon (2020-06-03)   Lucas coefficients   (Edouard Lucas, 1878)
Nontrivial factors of   (p x2) p ± 1   when  p  is an odd prime.

A polynomial  Pp  can be defined for which the following identity holds,  which provides a  nontrivial factorization  of some special integers:

( p x2 ) p - (-1)m   =   ( p x2 - (-1)m )  Pp (-x)  Pp (x)

Here,  p = 2m+1  is an odd prime  (see  Sophie Germain identity  for p=2).
P(x)   =   Ap ( p x2 )  +  (p x) Bp ( p x2 )   where  Ap   and  Bp   are both  palindromic  monic  polynomials.  Ap  has degree  m.   Bp  has degree m-1.

Polynomials   Pp (x) = Ap (y) + (p x) Bp (y)   with   y = p x 2
± Prime p  1  y y2 y3y4y5y6y7 y8y9y10y11y12 y13y14y15
+p=3 A11
B1
-p=5 A131
B11
+p=7 A1331
B111
+p=11 A15-1-151
B11-111
-p=13 A1715191571
B135531
-p=17 A1911-5-15-51191
B131-3-3131
+p=19 A1917273131271791
B135777531
+p=23 A1119-19-152525-15-199111
B13-1-5171-5-131
-p=29 A1153313155745 194557151333151
B15517115 51171551
+p=31 A1154383125151169173 1731691511258343151
B151119252931 31312925191151

For p=31 (and x=9) this factors a nice 102-digit  semiprime:   (251131+1) / 2512   =  889923919072997985238634558820908333948499157179463
× 1111413273683146858652465162019244587926917356315577

That factorization would take a long time with a  general-purpose  program.

For compactness, we'll give palindromic polynomials as lists of coefficients with underlined central ones  (so the mirror endings can be freely truncated).

A37- = (1, 19, 79, 183, 285, 349, 397, 477, 579, 627, 579, 477, 397, 349, 285, 183, 79, 19, 1)
B37- = (1, 7, 21, 39, 53, 61, 71, 87, 101, 101, 87, 71, 61, 53, 39, 21, 7, 1)
 
A41- = (1, 21, 67, 49, 7, 35, 15, 11, -23, -65, -31, -65, -23, 11, 15, 35, 7, 49, 67, 21, 1)
B41- = (1, 7, 11, 3, 3, 5, 1, 1, -9, -7, -7, -9, 1, 1, 5, 3, 3, 11, 7, 1)
 
A43+ = (1, 21, 81, 169, 223, 225, 213, 223, 229, 197, 159, 159, 197, 229, 223, 213, 225, 223, 169...
B43+ = (1, 7, 19, 31, 35, 33, 33, 35, 33, 27, 23, 27, 33, 35, 33, 33, 35, 31, 19, 7, 1)
 
A47+ = (1, 23, 65, -15, -169, -97, 179, 287, -37, -375, -149, 311, 311, -149, -375, -37, 287, 179...
B47+ = (1, 7, 7, -15, -25, 5, 41, 25, -37, -49, 15, 57, 15, -49, -37, 25, 41, 5, -25, -15, 7, 7, 1)
 
A53- = (1, 27, 113, 103, -155, -219, 263, 513, -59, -465, 75, 551, 93, -357, 93, 551, 75, -465, -59...
B53- = (1, 9, 19, -1, -35, -3, 67, 41, -51, -39, 57, 57, -31, -31, 57, 57, -39, -51, 41, 67, -3, -35, -1...
 
A59+ = (1, 29, 111, 55, -85, 47, 11, 53, 131, -245, 41, 103, -111, 227, -103, -103, 227, -111, 103...
B59+ = (1, 9, 15, -5, -5, 9, -3, 21, -9, -25, 25, -11, 9, 19, -31, 19, 9, -11, 25, -25, -9, 21, -3, 9, -5, -5...
 
A61- = (1,31,191,637,1541,2979,4881,7029,9125,10953,12397,13511,14379, 15053,15511,15667...
B61- = (1, 11, 47, 131, 281, 497, 761, 1037, 1291, 1501, 1663, 1789, 1887, 1961, 2001, 2001...
 
A67+ = (1,33,193,565,1055,1429,1599,1803,2225,2637,2617,2195,1869,1875,1865,1469,991,991...
B67+ = (1, 43, 99, 155, 187, 205, 243, 301, 329, 297, 243, 225, 233, 209, 147, 111, 147, 209, 233...
 
A71+ = (1, 35, 169, 155, -109, 233, 597, 39, 101, 445, 163, 293, 89, -203, 249, -49, -505, 37, 37...
B71+ = (1, 11, 25, 1, -5, 63, 43, -9, 43, 37, 21, 35, -19, 1, 29, -47, -35, 23, -35, -47, 29, 1, -19, 35...

For p=61 (with x=2) this gives the  factorization  of the 144-digit semiprime  (24461-1) / 35  =
254180335737792836487420059360430288526895310810588085366845580859576779 × 691880648894768106905652479597579967344338476040716833288367161850591919

Those are linked to  cyclotomic polynomials  via the following equality:

(p x2)p ± 1   =   ( p x2 ± 1)  [ Ap± (p x2)2  -  p2 x2  Bp± (p x2)2 ]

As a polynomial identity,  that's equivalent  (using  y = p x2 )  to a simpler relation  (albeit not directly applicable to integer factorization):

yp ± 1   =   (y ± 1)  [ Ap± (y)2 - (p y) Bp± (y)2 ]    (with  ±1 = (-1)(p+1)/2 )

"Formules de Cauchy et de Lejeune-Dirichlet"  by  Edouard Lucas  (Compte Rendu, pp. 164-173. 1878-08-29)
 
Prime Numbers and Computer Methods for Factorization (Table 24, p. 444)  by  Hans Riesel (1929-2014).
 
The Cunningham project   |   Numericana :   Aurifeuillian Factorizations  and  Beyond


(2020-10-20)   On the Art of Polynomial Factorizations
Manufacturing remarkable identities.

Clearly,   x5 + x4 + x3 + x2 + x + 1      =    x3 (x2 + x + 1) + x2 + x + 1
= (x3 + 1) (x2 + x + 1)

This can be used to factor   x5 + x4 + 1   :

x5 + x4 + 1      =    x5 + x4 + x3 + x2 + x + 1 - x (x2 + x + 1)
= (x3 - x + 1) (x2 + x + 1)

My first quintic equation (10:28)  by  Steve Chow  (blackpenredpen, 2020-06-03).

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