1 #This file was created by <bruno> Sun Feb 16 00:38:14 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 Hermite polynomials
25 \begin_inset Formula \( H_{n}(x) \)
31 H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]
42 \begin_inset Formula \( H_{n}(x) \)
45 satisfies the recurrence relation
61 H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
66 \begin_inset Formula \( n\geq 0 \)
69 and the differential equation
70 \begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)
74 \begin_inset Formula \( n\geq 0 \)
85 \begin_inset Formula \( F:=\sum ^{\infty }_{n=0}\frac{H_{n}(x)}{n!}z^{n} \)
88 be the exponential generating function of the sequence of polynomials.
89 Then, because the Taylor series development theorem holds in formal power
90 series rings (see [1], section 2.
94 F & = & e^{x^{2}}\cdot \sum ^{\infty }_{n=0}\frac{1}{n!}\left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \cdot (-z)^{n}\\
95 & = & e^{x^{2}}\cdot e^{-(x-z)^{2}}\\
102 \begin_inset Formula \( \frac{d}{dz}F=(2x-2z)\cdot F \)
106 This is equivalent to the claimed recurrence.
110 Starting from this equation, we compute a linear relation for the partial
112 \begin_inset Formula \( F \)
117 \begin_inset Formula \( \partial _{x}=\frac{d}{dx} \)
121 \begin_inset Formula \( \Delta _{z}=z\frac{d}{dz} \)
135 \partial _{x}F=2z\cdot F\]
142 \partial _{x}^{2}F=4z^{2}\cdot F\]
149 \Delta _{z}F=(2xz-2z^{2})\cdot F\]
156 \partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]
163 \Delta _{z}^{2}F=\left( 2x\cdot z+(4x^{2}-4)\cdot z^{2}-8x\cdot z^{3}+4\cdot z^{4}\right) \cdot F\]
168 \begin_inset Formula \( 5\times 6 \)
171 system of linear equations over
172 \begin_inset Formula \( Q(x) \)
178 (-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
182 This is equivalent to the claimed equation
183 \begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)
190 [1] Bruno Haible: D-finite power series in several variables.
193 Diploma thesis, University of Karlsruhe, June 1989