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

(2012-02-15)   Chebyshev Polynomials   [ of the first kind ]
A family of  commuting  polynomial functions.   T_n oT_p  =  T_p oT_n  =  T_np

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)   =   T_n (cos q)

The symbol T comes from careful Russian transliterations like  Tchebyshev,  Tchebychef (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  T_n  and its obvious  n  distinct real zeroes,  we obtain immediately  T_n  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  T_n (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   T_m(T_n(x))  =  T_mn(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  T_n(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 ( T_mn (x) )   =   å _k [ å _d|k u(d) v(k/d) ]  f ( T_k (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 T_n(x) = x^1/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 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 expansion &  spherical 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.

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  He_n  which can be defined via:   H_n (x)   =   2^n/2 He_n (2^½ x)

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

(2013-04-24)   Euler Polynomials

E_n(x)

Leonhard Euler  (1707-1783)
MathWorld :   Euler polynomials

(2013-04-24)   Bernoulli Polynomials

dB_n (x) / dx   =   n B_n-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

MathWorld :   Bessel polynomials