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© 2000-2015   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).

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.

 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) 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

Wikipedia :   Laguerre Polynomials   |   Rook Polynomials of John Riordan (1903-1988)
MathWorld :   Laguerre Polynomials   |   MacTutor :   Edmond Laguerre  (1834-1886; X1853)


(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)


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

En(x)

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

Leonhard Euler  (1707-1783)
MathWorld :   Euler polynomials


 Bernoulli (2013-04-24)   Bernoulli Polynomials

dBn (x) / dx   =   n Bn-1 (x)

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

Introduction to Bernoulli and Euler Polynomials  by  Zhi-Wei Sun,  Nanjing University   (2002-06-06) MathWorld :   Bernoulli polynomials


(2014-12-07)   Bessel Polynomials

The  reverse Bessel polynomial  tabulated below is involved in the transfer function 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

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

MathWorld :   Bessel polynomials

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