Chebyshev Differential Equation

Chebyshev's equation is the second order linear differential equation :

(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 country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...

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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

: :

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 parameter ...

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 convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero wh ...

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 orthogonal polynomials related to the trigonometric functions, 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 tr ...

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