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(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
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(10) |
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(11) |
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(12) |
The first two are special cases of the third, so the general recurrence relation
is
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(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
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(21) |
© 1996-9 Eric W. Weisstein
1999-05-26