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102 The Tschebychev polynomials (of the 1st kind) \( T_{n}(x) \) are defined through
103 the recurrence relation
117 T_{n+2}(x)=2x\cdot T_{n+1}(x)-T_{n}(x)\]
126 \( T_{n}(x) \) satisfies the differential equation \( (x^{2}-1)\cdot T_{n}^{''}(x)+x\cdot T_{n}^{'}(x)-n^{2}\cdot T_{n}(x)=0 \) for all \( n\geq 0 \).
134 Let \( F:=\sum ^{\infty }_{n=0}T_{n}(x)z^{n} \) be the generating function of the sequence of polynomials. The
135 recurrence is equivalent to the equation
137 (1-2x\cdot z+z^{2})\cdot F=1-x\cdot z\]
146 \( F \) is a rational function in \( z \), \( F=\frac{1-xz}{1-2xz+z^{2}} \). From the theory of recursions with
147 constant coefficients, we know that we have to perform a partial fraction
148 decomposition. So let \( p(z)=z^{2}-2x\cdot z+1 \) be the denominator and \( \alpha =x+\sqrt{x^{2}-1} \) and \( \alpha ^{-1} \) its zeroes.
149 The partial fraction decomposition reads
151 F=\frac{1-xz}{1-2xz+z^{2}}=\frac{1}{2}\left( \frac{1}{1-\alpha z}+\frac{1}{1-\alpha ^{-1}z}\right) \]
152 hence \( T_{n}(x)=\frac{1}{2}(\alpha ^{n}+\alpha ^{-n}) \). Note that the
153 field \( Q(x)(\alpha ) \), being a finite dimensional extension field of \( Q(x) \) in characteristic
154 0, has a unique derivation extending \( \frac{d}{dx} \) on \( Q(x) \). We can therefore try
155 to construct an annihilating differential operator for \( T_{n}(x) \) by combination
156 of annihilating differential operators for \( \alpha ^{n} \) and \( \alpha ^{-n} \). In fact, \( L_{1}:=(\alpha -x)\frac{d}{dx}-n \) satisfies
157 \( L_{1}[\alpha ^{n}]=0 \), and \( L_{2}:=(\alpha -x)\frac{d}{dx}+n \) satisfies \( L_{2}[\alpha ^{-n}]=0 \). A common multiple of \( L_{1} \) and \( L_{2} \) is easily found
158 by solving an appropriate system of linear equations:
160 \( L=(x^{2}-1)\left( \frac{d}{dx}\right) ^{2}+x\frac{d}{dx}-n^{2}=\left( (\alpha -x)\frac{d}{dx}+n\right) \cdot L_{1}=\left( (\alpha -x)\frac{d}{dx}-n\right) \cdot L_{2} \)
162 It follows that both \( L[\alpha ^{n}]=0 \) and \( L[\alpha ^{-n}]=0 \), hence \( L[T_{n}(x)]=0 \).
170 Starting from the above equation, we compute a linear relation for
171 the partial derivatives of \( F \). Write \( \partial _{x}=\frac{d}{dx} \) and \( \Delta _{z}=z\frac{d}{dz} \). One computes
175 \left( 1-2xz+z^{2}\right) \cdot F=1-xz\]
178 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}F=z-z^{3}\]
181 \left( 1-2xz+z^{2}\right) ^{3}\cdot \partial _{x}^{2}F=4z^{2}-4z^{4}\]
184 \left( 1-2xz+z^{2}\right) ^{2}\cdot \Delta _{z}F=xz-2z^{2}+xz^{3}\]
187 \left( 1-2xz+z^{2}\right) ^{3}\cdot \partial _{x}\Delta _{z}F=z+2xz^{2}-6z^{3}+2xz^{4}+z^{5}\]
190 \left( 1-2xz+z^{2}\right) ^{3}\cdot \Delta _{z}^{2}F=xz+(2x^{2}-4)z^{2}-(2x^{2}-4)z^{4}-xz^{5}\]
193 Solve a \( 6\times 6 \) system of linear equations over \( Q(x) \) to get
195 x\cdot \partial _{x}F+(x^{2}-1)\cdot \partial _{x}^{2}F-\Delta _{z}^{2}F=0\]
198 This is equivalent to the claimed equation \( (x^{2}-1)\cdot T_{n}^{''}(x)+x\cdot T_{n}^{'}(x)-n^{2}\cdot T_{n}(x)=0 \).
200 \begin{lyxsectionbibliography}
202 \item [1] Bruno Haible: D-finite power series in several variables. \em Diploma
203 thesis, University of Karlsruhe, June 1989. \em Sections 2.12 and
206 \end{lyxsectionbibliography}