TOPICS
Search

Chebyshev Differential Equation

The Chebyshev differential equation of the first kind is

(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+alpha^2y=0
(1)

for |x|<1. The Chebyshev differential equation has regular singular points at -1, 1, and infty. A power series solution about x=0 uses the expansions

y=sum_(n=0)^(infty)a_nx^n
(2)
y^'=sum_(n=1)^(infty)na_nx^(n-1)
(3)
=sum_(n=0)^(infty)(n+1)a_(n+1)x^n
y^('')=sum_(n=1)^(infty)(n+1)na_(n+1)x^(n-1)
(4)
=sum_(n=0)^(infty)(n+2)(n+1)a_(n+2)x^n.

Substituting these expansions into (1) gives

(1-x^2)sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-xsum_(n=0)^infty(n+1)a_(n+1)x^n+alpha^2sum_(n=0)^inftya_nx^n=0
(5)
sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^(n+2) -sum_(n=0)^infty(n+1)a_(n+1)x^(n+1)+alpha^2sum_(n=0)^inftya_nx^n=0
(6)
sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=2)^inftyn(n-1)a_nx^n -sum_(n=1)^inftyna_nx^n+alpha^2sum_(n=0)^inftya_nx^n=0
(7)
2·1a_2+3·2a_3x-1·a_1x+alpha^2a_0+alpha^2a_1x +sum_(n=2)^infty[(n+2)(n+1)a_(n+2)-n(n-1)a_n-na_n+alpha^2a_n]x^n=0
(8)
(2a_2+alpha^2a_0)+[(alpha^2-1)a_1+6a_3]x +sum_(n=2)^infty[(n+2)(n+1)a_(n+2)+(alpha^2-n^2)a_n]x^n=0,
(9)

so

2a_2+alpha^2a_0=0
(10)
(alpha^2-1)a_1+6a_3=0,
(11)

and equating the remaining coefficients gives

a_(n+2)=(n^2-alpha^2)/((n+1)(n+2))a_n
(12)

for n=2, 3, ....

The equations for a_2 and a_3 are special cases, so the recurrence relation

a_(n+2)=(n^2-alpha^2)/((n+1)(n+2))a_n
(13)

holds for n=0, 1, .... It determines all coefficients from a_0=y(0) and a_1=y^'(0). For the even-indexed coefficients,

a_2=(-alpha^2)/2a_0
(14)
a_4=(2^2-alpha^2)/(3·4)a_2=((2^2-alpha^2)(-alpha^2))/(1·2·3·4)a_0
(15)
a_(2n)=(a_0)/((2n)!)product_(j=0)^(n-1)[(2j)^2-alpha^2],
(16)

and for the odd-indexed coefficients,

a_3=(1-alpha^2)/6a_1
(17)
a_5=(3^2-alpha^2)/(4·5)a_3=((3^2-alpha^2)(1^2-alpha^2))/(5!)a_1
(18)
a_(2n+1)=(a_1)/((2n+1)!)product_(j=0)^(n-1)[(2j+1)^2-alpha^2].
(19)

Both product formulas hold for n=0, 1, ..., with the products taken to be 1 when n=0.

For even k>=0, the coefficients can also be expressed using the gamma function as

a_k=(a_0)/(k!)product_(j=1)^(k/2)[(k-2j)^2-alpha^2]
(20)
=(2^(k-1)pialphacsc(1/2pialpha))/(k!Gamma(1-1/2k-1/2alpha)Gamma(1-1/2k+1/2alpha))a_0,

and for odd k>=1 as

a_k=(a_1)/(k!)product_(j=1)^((k-1)/2)[(k-2j)^2-alpha^2]
(21)
=(2^(k-1)pisec(1/2pialpha))/(k!Gamma(1-1/2k-1/2alpha)Gamma(1-1/2k+1/2alpha))a_1.

At parameter values where the gamma function expressions are indeterminate, their limits give the finite products above. The general solution is obtained by combining the even and odd parts of the power series,

y=sum_(n=0)^inftya_(2n)x^(2n)+sum_(n=0)^inftya_(2n+1)x^(2n+1).
(22)

The change of variables x=sint, with Y(t)=y(sint), transforms (1) into

Y^('')(t)+alpha^2Y(t)=0,
(23)

so for alpha!=0 the general solution is

y=a_0cos(alphasin^(-1)x)+(a_1)/alphasin(alphasin^(-1)x).
(24)

For alpha=0, the limiting form is

y=a_0+a_1sin^(-1)x.
(25)

For alpha!=0, an equivalent form of the general solution is

y=b_1cos(alphacos^(-1)x)+b_2sin(alphacos^(-1)x)=b_1T_alpha(x)+b_2sqrt(1-x^2)U_(alpha-1)(x),
(26)

where T_alpha(x) and U_(alpha-1)(x) are Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind, respectively, extended to noninteger orders when necessary. Another equivalent form of the general solution for alpha!=0 is

y=c_1cosh[alphaln(x+sqrt(x^2-1))]+ic_2sinh[alphaln(x+sqrt(x^2-1))].
(27)

The Chebyshev differential equation of the second kind is

(1-x^2)y^('')-3xy^'+n(n+2)y=0.
(28)

For a nonnegative integer n, the Chebyshev polynomial of the second kind U_n(x) is a solution (NIST Digital Library of Mathematical Functions, Table 18.8.1). Differentiating (1) at alpha=n+1 gives the second-kind equation for the derivative of a first-kind solution.

See also

Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind

Explore with Wolfram|Alpha

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 735, 1985.Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 232 and 252, 1986.NIST Digital Library of Mathematical Functions. "Differential Equations." Table 18.8.1. https://dlmf.nist.gov/18.8.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997.

Referenced on Wolfram|Alpha

Chebyshev Differential Equation

Cite this as:

Weisstein, Eric W. "Chebyshev Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChebyshevDifferentialEquation.html

Subject classifications