1 #This file was created by <bruno> Sun Feb 16 00:32:21 1997
2 #LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
14 \paragraph_separation indent
15 \quotes_language english
17 \paperorientation portrait
24 The Tschebychev polynomials (of the 1st kind)
25 \begin_inset Formula \( T_{n}(x) \)
28 are defined through the recurrence relation
54 T_{n+2}(x)=2x\cdot T_{n+1}(x)-T_{n}(x)\]
59 \begin_inset Formula \( n\geq 0 \)
70 \begin_inset Formula \( T_{n}(x) \)
73 satisfies the differential equation
74 \begin_inset Formula \( (x^{2}-1)\cdot T_{n}^{''}(x)+x\cdot T_{n}^{'}(x)-n^{2}\cdot T_{n}(x)=0 \)
78 \begin_inset Formula \( n\geq 0 \)
89 \begin_inset Formula \( F:=\sum ^{\infty }_{n=0}T_{n}(x)z^{n} \)
92 be the generating function of the sequence of polynomials.
93 The recurrence is equivalent to the equation
96 (1-2x\cdot z+z^{2})\cdot F=1-x\cdot z\]
109 \begin_inset Formula \( F \)
112 is a rational function in
113 \begin_inset Formula \( z \)
117 \begin_inset Formula \( F=\frac{1-xz}{1-2xz+z^{2}} \)
121 From the theory of recursions with constant coefficients, we know that
122 we have to perform a partial fraction decomposition.
124 \begin_inset Formula \( p(z)=z^{2}-2x\cdot z+1 \)
127 be the denominator and
128 \begin_inset Formula \( \alpha =x+\sqrt{x^{2}-1} \)
132 \begin_inset Formula \( \alpha ^{-1} \)
136 The partial fraction decomposition reads
139 F=\frac{1-xz}{1-2xz+z^{2}}=\frac{1}{2}\left( \frac{1}{1-\alpha z}+\frac{1}{1-\alpha ^{-1}z}\right) \]
144 \begin_inset Formula \( T_{n}(x)=\frac{1}{2}(\alpha ^{n}+\alpha ^{-n}) \)
149 \begin_inset Formula \( Q(x)(\alpha ) \)
152 , being a finite dimensional extension field of
153 \begin_inset Formula \( Q(x) \)
156 in characteristic 0, has a unique derivation extending
157 \begin_inset Formula \( \frac{d}{dx} \)
161 \begin_inset Formula \( Q(x) \)
165 We can therefore try to construct an annihilating differential operator
167 \begin_inset Formula \( T_{n}(x) \)
170 by combination of annihilating differential operators for
171 \begin_inset Formula \( \alpha ^{n} \)
175 \begin_inset Formula \( \alpha ^{-n} \)
180 \begin_inset Formula \( L_{1}:=(\alpha -x)\frac{d}{dx}-n \)
184 \begin_inset Formula \( L_{1}[\alpha ^{n}]=0 \)
188 \begin_inset Formula \( L_{2}:=(\alpha -x)\frac{d}{dx}+n \)
192 \begin_inset Formula \( L_{2}[\alpha ^{-n}]=0 \)
197 \begin_inset Formula \( L_{1} \)
201 \begin_inset Formula \( L_{2} \)
204 is easily found by solving an appropriate system of linear equations:
208 \begin_inset Formula \( L=(x^{2}-1)\left( \frac{d}{dx}\right) ^{2}+x\frac{d}{dx}-n^{2}=\left( (\alpha -x)\frac{d}{dx}+n\right) \cdot L_{1}=\left( (\alpha -x)\frac{d}{dx}-n\right) \cdot L_{2} \)
215 \begin_inset Formula \( L[\alpha ^{n}]=0 \)
219 \begin_inset Formula \( L[\alpha ^{-n}]=0 \)
223 \begin_inset Formula \( L[T_{n}(x)]=0 \)
235 Starting from the above equation, we compute a linear relation for the partial
237 \begin_inset Formula \( F \)
242 \begin_inset Formula \( \partial _{x}=\frac{d}{dx} \)
246 \begin_inset Formula \( \Delta _{z}=z\frac{d}{dz} \)
256 \left( 1-2xz+z^{2}\right) \cdot F=1-xz\]
263 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}F=z-z^{3}\]
270 \left( 1-2xz+z^{2}\right) ^{3}\cdot \partial _{x}^{2}F=4z^{2}-4z^{4}\]
277 \left( 1-2xz+z^{2}\right) ^{2}\cdot \Delta _{z}F=xz-2z^{2}+xz^{3}\]
284 \left( 1-2xz+z^{2}\right) ^{3}\cdot \partial _{x}\Delta _{z}F=z+2xz^{2}-6z^{3}+2xz^{4}+z^{5}\]
291 \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}\]
299 \begin_inset Formula \( 6\times 6 \)
302 system of linear equations over
303 \begin_inset Formula \( Q(x) \)
309 x\cdot \partial _{x}F+(x^{2}-1)\cdot \partial _{x}^{2}F-\Delta _{z}^{2}F=0\]
316 This is equivalent to the claimed equation
317 \begin_inset Formula \( (x^{2}-1)\cdot T_{n}^{''}(x)+x\cdot T_{n}^{'}(x)-n^{2}\cdot T_{n}(x)=0 \)
324 [1] Bruno Haible: D-finite power series in several variables.
327 Diploma thesis, University of Karlsruhe, June 1989.