home | index | units | counting | geometry | algebra | trigonometry & functions | calculus analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

# Special Polynomials

The pursuit of mathematics is a refuge from
the goading urgency of contingent happenings.

Alfred North Whitehead  (1861-1947)

### Related Links (Outside this Site)

Online Polynomial Calculator  by  Xavier R. Junqué de Fortuny  (Barcelona)

Wikipedia :   Polynomial   |   Calculus with polynomials   |   Stone-Weierstrass theorem

## 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 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 T8 (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.

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

 U0 (x) = 1 Un+2 (x)   =   2 x Un+1 (x)  -  Un (x) U1 (x) = 2 x 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 U8 (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.

(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 2 P2 (x) = -1 + 3 x2 2 P3 (x) = -3 x + 5 x3 8 P4 (x) = 3 - 30 x2 + 35 x4 8 P5 (x) = 15 x - 70 x3 + 63 x5 16 P6 (x) = -5 + 105 x2 - 315 x4 + 231 x6 16 P7 (x) = -35 x + 315 x3 - 693 x5 + 429 x7 128 P8 (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 2 L2(x) = 2 - 4x + x2 6 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 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 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)

(2013-04-24)   Euler Polynomials

En(x)

Leonhard Euler  (1707-1783)
MathWorld :   Euler polynomials

(2013-04-24)   Bernoulli Polynomials

dBn (x) / dx   =   n Bn-1 (x)

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

MathWorld :   Bessel polynomials