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<head><title>组合数学</title><meta http-equiv="Content-Type" content="text/html; charset=gb2312"><link rel="stylesheet" href="../style.css"></head><h1>§2.3 母函数的性质 </h1><p> 一个序列和它的母函数一一对应。给了序列便得知它的母函数；反之，求得母函数序列也随之而定。这种关系颇像数学中的积分变换，特别酷似离散序列的Z变换。如§2的例子所示的那样，为了求满足某种第推关系的序列，可把它转换为求对应的母函数G(x)，G(x)可能满足一代数方程，或代数方程组，甚至于是微分方程。 </p><p> 最后求逆变换，即从已求得的母函数G(x)得到序列{a<sub>n</sub>} 。关键在于要搭起从序列到母函数，从母函数到序列这两座桥。这一节便是以此为目的的。不特别说明下面假设{a<sub>k</sub>}、{a<sub>k</sub>}两个序列对应的母函数分别为A(x)、B(x) </p><p> 性质1：若 </p><p> <img border="0" src="2_3.pic/image008.gif" width="107" height="51"></p><p>则 B(x)=x<sup>l</sup>A(x)</p><p> 证： </p><p> <img border="0" src="2_3.pic/image012.gif" width="253" height="77"></p><p> 例. 已知 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image014.gif" width="201" height="44"></p> </blockquote></blockquote><p>则 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image016.gif" width="276" height="44"></p> </blockquote></blockquote><p> 性质2：若 b<sub>k</sub>=a<sub>k+l</sub>， </p><p>则 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image020.gif" width="175" height="45"></p> </blockquote></blockquote><p> 证： </p><p> <img border="0" src="2_3.pic/image022.gif" width="192" height="25"></p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image024.gif" width="323" height="141"></p> </blockquote></blockquote><p> 例. </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image026.gif" width="196" height="44"></p> <p><img border="0" src="2_3.pic/image028.gif" width="220" height="85"></p> </blockquote></blockquote><p> 性质3：若 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image030.gif" width="68" height="45">，</p> </blockquote></blockquote><p>则 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image032.gif" width="84" height="41"></p> </blockquote></blockquote><p> 证： </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image034.gif" width="205" height="147"></p> <p><img border="0" src="2_3.pic/image036.gif" width="231" height="7"></p> <p><img border="0" src="2_3.pic/image038.gif" width="316" height="51"></p> </blockquote></blockquote><p> 例. 已知 </p><p> <img border="0" src="2_3.pic/image040.gif" width="240" height="41"></p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image042.gif" width="253" height="44"></p> <p><img border="0" src="2_3.pic/image044.gif" width="143" height="45"></p> </blockquote></blockquote><p> 类似可得：</p><p> <img border="0" src="2_3.pic/image046.gif" width="241" height="125"> </p><p> 性质4：若 </p><blockquote> <blockquote> <blockquote> <p><img border="0" src="2_3.pic/image048.gif" width="39" height="45"></p> </blockquote> </blockquote></blockquote><p>收敛，则 </p><p> <img border="0" src="2_3.pic/image050.gif" width="132" height="41"></p><p> 性质5. 若b<sub>k</sub>=ka<sub>k</sub>，则 <img border="0" src="2_3.pic/image061.gif" width="88" height="24">。 </p><p> 例. </p><p> <img border="0" src="2_3.pic/image063.gif" width="180" height="41"></p><p>则 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image065.gif" width="247" height="47"></p> </blockquote></blockquote><p> 性质6. 若 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image067.gif" width="65" height="41">，</p> </blockquote></blockquote><p>则 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image069.gif" width="121" height="41"></p> </blockquote></blockquote><p> 性质5和性质6的结论是显而易见的。 </p><p> 性质7. 若 <img border="0" src="2_3.pic/image071.gif" width="231" height="24"><img border="0" src="2_3.pic/image073.gif" width="71" height="45"></p><p>则 <img border="0" src="2_3.pic/image075.gif" width="240" height="25"></p><p> 证: </p><p> <img border="0" src="2_3.pic/image077.gif" width="271" height="248"></p><p> 例. 已知 </p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image079.gif" width="228" height="132"></p> </blockquote></blockquote><p> 则</p><blockquote> <blockquote> <p><img border="0" src="2_3.pic/image081.gif" width="96" height="45"></p> </blockquote></blockquote>⌨️ 快捷键说明
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