![\begin{displaymath}
(1-x^2){d^2y\over dx^2} - x{dy\over dx} + m^2y = 0
\end{displaymath}](c1_1274.gif) |
(1) |
for
. The Chebyshev differential equation has regular Singularities at
, 1, and
. It can be solved by series solution using the expansions
Now, plug (2-4) into the original equation (1) to obtain
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
so
![\begin{displaymath}
2a_2+m^2 a_0=0
\end{displaymath}](c1_1293.gif) |
(10) |
![\begin{displaymath}
(m^2-1)a_1+6a_3=0
\end{displaymath}](c1_1294.gif) |
(11) |
![\begin{displaymath}
a_{n+2} ={n^2-m^2\over(n+1)(n+2)} a_n \qquad \hbox{for }n=2,3,\ldots.
\end{displaymath}](c1_1295.gif) |
(12) |
The first two are special cases of the third, so the general recurrence relation
is
![\begin{displaymath}
a_{n+2} ={n^2-m^2\over(n+1)(n+2)} a_n \qquad \hbox{for }n=0,1,\ldots.
\end{displaymath}](c1_1296.gif) |
(13) |
From this, we obtain for the Even Coefficients
and for the Odd Coefficients
So the general solution is
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(20) |
If
is Even, then
terminates and is a Polynomial solution, whereas if
is Odd, then
terminates and is
a Polynomial solution. The Polynomial solutions defined here are known as Chebyshev Polynomials of the First
Kind. The definition of the Chebyshev Polynomial of the Second Kind gives a similar, but distinct, recurrence relation
![\begin{displaymath}
a_{n+2}' ={(n+1)^2-m^2\over(n+2)(n+3)} a_n' \qquad \hbox{for }n=0,1,\ldots.
\end{displaymath}](c1_1313.gif) |
(21) |
© 1996-9 Eric W. Weisstein
1999-05-26