Chebyshev Equation

Chebyshev's equation is the second order linear

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

(1-x^2)  - x  + p^2 y = 0

where p is a real (or complex) constant. The equation is named after

Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

n mathematician

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...

. The solutions can be obtained by

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...

: :

y = \sum_^\infty a_nx^n

where the coefficients obey the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...

a_ =  a_n.

The series converges for

, x, <1

(note, ''x'' may be complex), as may be seen by applying the

ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...

to the recurrence. The recurrence may be started with arbitrary values of ''a''₀ and ''a''₁, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are: :''a''₀ = 1 ; ''a''₁ = 0, leading to the solution :

F(x) = 1 - \fracx^2 + \fracx^4 - \fracx^6 + \cdots

and :''a''₀ = 0 ; ''a''₁ = 1, leading to the solution :

G(x) = x - \fracx^3 + \fracx^5 - \cdots.

The general solution is any linear combination of these two. When ''p'' is a non-negative integer, one or the other of the two functions has its series terminate after a finite number of terms: ''F'' terminates if ''p'' is even, and ''G'' terminates if ''p'' is odd. In this case, that function is a polynomial of degree ''p'' and it is proportional to the

Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

of the first kind :

T_p(x) = (-1)^\ F(x)\,

if ''p'' is even :

T_p(x) = (-1)^\ p\ G(x)\,

if ''p'' is odd {{PlanetMath attribution, id=3616, title=Chebyshev equation Ordinary differential equations