[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/107740#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/107740","headline":"\u6bcd\u95a2\u6570 – Wikipedia","name":"\u6bcd\u95a2\u6570 – Wikipedia","description":"\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u6bcd\u95a2\u6570\uff08\u307c\u304b\u3093\u3059\u3046\u3001\u82f1: generating function; \u751f\u6210\u95a2\u6570\uff09\u306f\u3001\uff08\u81ea\u7136\u6570\u3067\u6dfb\u5b57\u4ed8\u3051\u3089\u308c\u305f\uff09\u6570\u5217 {an} \u306b\u95a2\u3059\u308b\u60c5\u5831\u3092\u5185\u5305\u3057\u305f\u4fc2\u6570\u3092\u6301\u3064\u3001\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u3067\u3042\u308b\u3002\u6bcd\u95a2\u6570\u306f\u3001\u4e00\u822c\u7dda\u578b\u56de\u5e30\u554f\u984c\u306e\u89e3\u6c7a\u306e\u305f\u3081\u306b\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306b\u3088\u3063\u30661730\u5e74\u306b\u521d\u3081\u3066\u7528\u3044\u3089\u308c\u305f[1]\u3002\u8907\u6570\u306e\u81ea\u7136\u6570\u3067\u6dfb\u5b57\u4ed8\u3051\u3089\u308c\u308b\u6570\u306e\u914d\u5217\uff08\u591a\u91cd\u6570\u5217\uff09\u306e\u60c5\u5831\u3092\u53d6\u308a\u8fbc\u3093\u3060\u591a\u5909\u6570\u51aa\u7d1a\u6570\u3092\u540c\u69d8\u306b\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002 \u6bcd\u95a2\u6570\u306b\u306f\u3001\u901a\u5e38\u578b\u6bcd\u95a2\u6570 (ordinary generating function)\u3001\u6307\u6570\u578b\u6bcd\u95a2\u6570 (exponential generating function)\u3001\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570 (Lambert series)\u3001\u30d9\u30eb\u7d1a\u6570 (Bell series)\u3001\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570 (Dirichlet series) \u306a\u3069\u69d8\u3005\u306a\u3082\u306e\u304c\u3042\u308b\u3002\u3053\u308c\u3089\u306b\u3064\u3044\u3066\u306f\u5b9a\u7fa9\u3068\u4f8b\u3092\u5f8c\u8ff0\u3059\u308b\u3002\u539f\u7406\u7684\u306b\u306f\u3042\u3089\u3086\u308b\u5217\u306b\u3064\u3044\u3066\u305d\u308c\u305e\u308c\u306e\u7a2e\u985e\u306e\u6bcd\u95a2\u6570\u304c\u5b58\u5728\u3059\u308b\uff08\u305f\u3060\u3057\u3001\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570\u3068\u30c7\u30a3\u30ea\u30af\u30ec\u578b\u306f\u6dfb\u5b57\u3092","datePublished":"2022-03-23","dateModified":"2022-03-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a89087cb1dd32673dce62eb5b442b06479e9f7ff","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a89087cb1dd32673dce62eb5b442b06479e9f7ff","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/107740","about":["Wiki"],"wordCount":7740,"articleBody":"\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u6bcd\u95a2\u6570\uff08\u307c\u304b\u3093\u3059\u3046\u3001\u82f1: generating function; \u751f\u6210\u95a2\u6570\uff09\u306f\u3001\uff08\u81ea\u7136\u6570\u3067\u6dfb\u5b57\u4ed8\u3051\u3089\u308c\u305f\uff09\u6570\u5217 {an} \u306b\u95a2\u3059\u308b\u60c5\u5831\u3092\u5185\u5305\u3057\u305f\u4fc2\u6570\u3092\u6301\u3064\u3001\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u3067\u3042\u308b\u3002\u6bcd\u95a2\u6570\u306f\u3001\u4e00\u822c\u7dda\u578b\u56de\u5e30\u554f\u984c\u306e\u89e3\u6c7a\u306e\u305f\u3081\u306b\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306b\u3088\u3063\u30661730\u5e74\u306b\u521d\u3081\u3066\u7528\u3044\u3089\u308c\u305f[1]\u3002\u8907\u6570\u306e\u81ea\u7136\u6570\u3067\u6dfb\u5b57\u4ed8\u3051\u3089\u308c\u308b\u6570\u306e\u914d\u5217\uff08\u591a\u91cd\u6570\u5217\uff09\u306e\u60c5\u5831\u3092\u53d6\u308a\u8fbc\u3093\u3060\u591a\u5909\u6570\u51aa\u7d1a\u6570\u3092\u540c\u69d8\u306b\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u6bcd\u95a2\u6570\u306b\u306f\u3001\u901a\u5e38\u578b\u6bcd\u95a2\u6570 (ordinary generating function)\u3001\u6307\u6570\u578b\u6bcd\u95a2\u6570 (exponential generating function)\u3001\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570 (Lambert series)\u3001\u30d9\u30eb\u7d1a\u6570 (Bell series)\u3001\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570 (Dirichlet series) \u306a\u3069\u69d8\u3005\u306a\u3082\u306e\u304c\u3042\u308b\u3002\u3053\u308c\u3089\u306b\u3064\u3044\u3066\u306f\u5b9a\u7fa9\u3068\u4f8b\u3092\u5f8c\u8ff0\u3059\u308b\u3002\u539f\u7406\u7684\u306b\u306f\u3042\u3089\u3086\u308b\u5217\u306b\u3064\u3044\u3066\u305d\u308c\u305e\u308c\u306e\u7a2e\u985e\u306e\u6bcd\u95a2\u6570\u304c\u5b58\u5728\u3059\u308b\uff08\u305f\u3060\u3057\u3001\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570\u3068\u30c7\u30a3\u30ea\u30af\u30ec\u578b\u306f\u6dfb\u5b57\u3092 1 \u304b\u3089\u59cb\u3081\u308b\u3053\u3068\u304c\u5fc5\u8981\uff09\u304c\u3001\u6271\u3044\u6613\u3055\u306b\u3064\u3044\u3066\u306f\u305d\u308c\u305e\u308c\u306e\u7a2e\u985e\u3067\u76f8\u5f53\u7570\u306a\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u3069\u306e\u6bcd\u95a2\u6570\u304c\u6700\u3082\u6709\u52b9\u304b\u306f\u3001\u305d\u306e\u5217\u306e\u6027\u8cea\u3068\u89e3\u304f\u3079\u304d\u554f\u984c\u306e\u8a73\u7d30\u306b\u4f9d\u5b58\u3059\u308b\u3002\u6bcd\u95a2\u6570\u3092\u3001\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u306b\u5bfe\u3059\u308b\u6f14\u7b97\u30fb\u64cd\u4f5c\u3092\u7528\u3044\u308b\u306a\u3069\u3057\u3066\uff08\u7d1a\u6570\u306e\u5f62\u3067\u306f\u306a\u304f\uff09\u9589\u3058\u305f\u5f62\uff08\u82f1\u8a9e\u7248\uff09\u306e\u5f0f\u3067\u8868\u3059\u3053\u3068\u3082\u3088\u304f\u884c\u308f\u308c\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u6bcd\u95a2\u6570\u306e\u8868\u793a\u306f\u3001\u6bcd\u95a2\u6570\u306e\u4e0d\u5b9a\u5143\u3092 x \u3068\u3059\u308c\u3070\u3001\u56db\u5247\u6f14\u7b97\u3001\u6bcd\u95a2\u6570\u306ex \u306b\u95a2\u3059\u308b\u5fae\u5206\u3001\u4ed6\u306e\u6bcd\u95a2\u6570\u3078\u4ee3\u5165\u3059\u308b\u3053\u3068\u3001\u306a\u3069\u3092\u884c\u3063\u305f\u7d50\u679c\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u3002\u3053\u308c\u3089\u306e\u64cd\u4f5c\u306f\u95a2\u6570\u306b\u5bfe\u3057\u3066\u3082\u5b9a\u7fa9\u3055\u308c\u308b\u3082\u306e\u3067\u3042\u308b\u3057\u3001\u7d50\u679c\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u5f0f\u3082\u3084\u306f\u308a x \u306e\u95a2\u6570\u3067\u3042\u308b\u304b\u306e\u3088\u3046\u306b\u898b\u3048\u308b\u3002\u5b9f\u969b\u3001\u6bcd\u95a2\u6570\u3092 x \u306e\uff08\u5341\u5206\u5c0f\u3055\u3044\uff09\u5177\u4f53\u7684\u306a\u5024\u3067\u8a55\u4fa1\u3059\u308b\u3053\u3068\u306e\u3067\u304d\u308b\u95a2\u6570\u3068\u3057\u3066\u89e3\u91c8\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u5834\u5408\u3082\u5c11\u306a\u304f\u306a\u3044\uff08\u3053\u306e\u3068\u304d\u3001\u6bcd\u95a2\u6570\u306e\u51aa\u7d1a\u6570\u8868\u793a\u306f\u3001\u6bcd\u95a2\u6570\u306e\u9589\u3058\u305f\u5f62\u306e\u5f0f\u306e\u30c6\u30a4\u30e9\u30fc\u7d1a\u6570\u3068\u89e3\u91c8\u3055\u308c\u308b\uff09\u306e\u3067\u3042\u308a\u3001\u305d\u308c\u304c\u3053\u306e\u5f0f\u304c\u300c\u6bcd\u95a2\u6570\u300d\u3068\u547c\u3070\u308c\u308b\u6240\u4ee5\u3067\u3082\u3042\u308b\u3002\u3057\u304b\u3057\u3001\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u306f x \u306b\u4f55\u3089\u304b\u306e\u6570\u5024\u3092\u4ee3\u5165\u3057\u305f\u3068\u304d\u306b\u53ce\u675f\u3059\u308b\u304b\u3069\u3046\u304b\u306f\u554f\u984c\u306b\u3057\u306a\u3044\u306e\u3067\u3042\u3063\u3066\u3001\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u305d\u306e\u3088\u3046\u306a\u95a2\u6570\u3068\u3057\u3066\u306e\u89e3\u91c8\u304c\u53ef\u80fd\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u5fc5\u305a\u3057\u3082\u8981\u6c42\u3055\u308c\u308b\u3082\u306e\u3067\u306f\u306a\u3044\u3057\u3001\u540c\u69d8\u306b x \u306e\u95a2\u6570\u3068\u3057\u3066\u610f\u5473\u3092\u6301\u3064\u5f0f\u304c\u3044\u305a\u308c\u3082\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u306b\u5bfe\u3057\u3066\u610f\u5473\u3092\u6301\u3064\u308f\u3051\u3067\u306f\u306a\u3044\u3002\u6163\u4f8b\u7684\u306b\u6bcd\u300c\u95a2\u6570\u300d\u3068\u547c\u3070\u308c\u3066\u306f\u3044\u308b\u304c\u3001\u59cb\u57df\u304b\u3089\u7d42\u57df\u3078\u306e\u5199\u50cf\u3068\u3044\u3046\u95a2\u6570\u306e\u53b3\u5bc6\u306a\u610f\u5473\u306b\u7167\u3089\u3057\u3066\u8a00\u3048\u3070\u6bcd\u95a2\u6570\u306f\u95a2\u6570\u3067\u306f\u306a\u304f\u3001\u4eca\u65e5\u7684\u306b\u306f\u751f\u6210\u7d1a\u6570\uff08\u6bcd\u7d1a\u6570\uff09\u3068\u547c\u3076\u3053\u3068\u3082\u3057\u3070\u3057\u3070\u3067\u3042\u308b\u3002Table of Contents\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6307\u6570\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u30dd\u30a2\u30bd\u30f3\u6bcd\u95a2\u6570[\u7de8\u96c6]\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570[\u7de8\u96c6]\u30d9\u30eb\u7d1a\u6570[\u7de8\u96c6]\u6bcd\u95a2\u6570\u3068\u3057\u3066\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570[\u7de8\u96c6]\u591a\u9805\u5f0f\u5217\u306e\u6bcd\u95a2\u6570[\u7de8\u96c6]\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6709\u7406\u95a2\u6570[\u7de8\u96c6]\u7573\u307f\u8fbc\u307f\u7a4d[\u7de8\u96c6]\u591a\u5909\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6307\u6570\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u30d9\u30eb\u7d1a\u6570[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]\u591a\u5909\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]\u305d\u306e\u4ed6\u306e\u6bcd\u95a2\u6570[\u7de8\u96c6]\u985e\u4f3c\u306e\u6982\u5ff5[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6570\u5217 {an} \u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u3068\u306f\u3001\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570G(an;x)=\u2211n=0\u221eanxn{displaystyle G(a_{n};x)=sum _{n=0}^{infty }a_{n}x^{n}}  \u306e\u3053\u3068\u3067\u3042\u308b\u3002\u5358\u306b\u300c\u6bcd\u95a2\u6570\u300d\u3068\u8a00\u3063\u305f\u5834\u5408\u3001\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u3092\u610f\u5473\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3002an \u304c\u96e2\u6563\u78ba\u7387\u5909\u6570\u306e\u78ba\u7387\u8cea\u91cf\u95a2\u6570\u306a\u3089\u3001\u305d\u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u3092\u78ba\u7387\u6bcd\u95a2\u6570\uff08\u82f1\u8a9e\u7248\uff09\u3068\u547c\u3076\u3002\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u306f\u591a\u91cd\u6dfb\u5b57\u3092\u6301\u3064\u5217\u306b\u5bfe\u3059\u308b\u3082\u306e\u306b\u4e00\u822c\u5316\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070\u3001\u4e8c\u91cd\u6570\u5217 {am,n}\uff08n \u3068 m \u306f\u81ea\u7136\u6570\uff09\u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u306f  G(am,n;x,y)=\u2211m,n=0\u221eam,nxmyn{displaystyle G(a_{m,n};x,y)=sum _{m,n=0}^{infty }a_{m,n}x^{m}y^{n}}\u3067\u3042\u308b\u3002\u6307\u6570\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6570\u5217 {an} \u306e\u6307\u6570\u578b\u6bcd\u95a2\u6570\u3068\u306f\u3001  EG(an;x)=\u2211n=0\u221eanxnn!{displaystyle EG(a_{n};x)=sum _{n=0}^{infty }a_{n}{frac {x^{n}}{n!}}}\u3068\u3044\u3046\u7d1a\u6570\u3067\u3042\u308b\u3002\u30dd\u30a2\u30bd\u30f3\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6570\u5217 {an} \u306e\u30dd\u30a2\u30bd\u30f3\u6bcd\u95a2\u6570 (Poisson generating function) \u3068\u306fPG(an;x)=\u2211n=0\u221eane\u2212xxnn!{displaystyle PG(a_{n};x)=sum _{n=0}^{infty }a_{n}e^{-x}{frac {x^{n}}{n!}}}\u306e\u3053\u3068\u3067\u3042\u308b\u3002\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570[\u7de8\u96c6]\u6570\u5217 {an} \u306e\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570\u306fLG(an;x)=\u2211n=1\u221eanxn1\u2212xn{displaystyle LG(a_{n};x)=sum _{n=1}^{infty }a_{n}{frac {x^{n}}{1-x^{n}}}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570\u3067\u306f\u3001\u6dfb\u5b57 n \u306f 0 \u304b\u3089\u3067\u306f\u306a\u304f 1 \u304b\u3089\u59cb\u307e\u308b\u70b9\u306b\u6ce8\u610f\u3002\u30d9\u30eb\u7d1a\u6570[\u7de8\u96c6]\u6570\u8ad6\u7684\u95a2\u6570 f(n) \u3068\u7d20\u6570 p \u306b\u5bfe\u3059\u308b\u30d9\u30eb\u7d1a\u6570\u306f\u3001fp(x)=\u2211n=0\u221ef(pn)xn{displaystyle f_{p}(x)=sum _{n=0}^{infty }f(p^{n})x^{n}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u6bcd\u95a2\u6570\u3068\u3057\u3066\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u306f\u53b3\u5bc6\u306a\u610f\u5473\u3067\u306f\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u3067\u306a\u3044\u306b\u3082\u304b\u304b\u308f\u3089\u305a\u3001\u6bcd\u95a2\u6570\u306e\u4e00\u7a2e\u306b\u3057\u3070\u3057\u3070\u5206\u985e\u3055\u308c\u308b\u3002\u6570\u5217 {an} \u306e\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u578b\u306e\u6bcd\u95a2\u6570\u3068\u306fDG(an;s)=\u2211n=1\u221eanns{displaystyle DG(a_{n};s)=sum _{n=1}^{infty }{frac {a_{n}}{n^{s}}}}\u3067\u3042\u308b\u3002\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u578b\u306e\u6bcd\u95a2\u6570\u306f an \u304c\u4e57\u6cd5\u7684\u95a2\u6570\u3067\u305d\u306e\u95a2\u6570\u306e\u30d9\u30eb\u7d1a\u6570\u3092\u4f7f\u3063\u305f\u30aa\u30a4\u30e9\u30fc\u7a4d\u8868\u793a\u304c\u3042\u308c\u3070\u3001\u7279\u306b\u4fbf\u5229\u3067\u3042\u308b\u3002DG(an;s)=\u220fpfp(p\u2212s){displaystyle DG(a_{n};s)=prod _{p}f_{p}(p^{-s}),}an \u304c\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19\u306a\u3089\u3001\u305d\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u6bcd\u95a2\u6570\u3092\u30c7\u30a3\u30ea\u30af\u30ec\u306eL\u95a2\u6570\u3068\u547c\u3076\u3002\u591a\u9805\u5f0f\u5217\u306e\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6bcd\u95a2\u6570\u306e\u6982\u5ff5\u3092\u4ed6\u306e\u6570\u5b66\u7684\u5bfe\u8c61\u306e\u5217\u306b\u5bfe\u3057\u3066\u3082\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070\u3001\u4e8c\u9805\u578b\u306e\u591a\u9805\u5f0f\u5217\u306e\u6bcd\u95a2\u6570\u306fexf(t)=\u2211n=0\u221epn(x)n!tn{displaystyle e^{xf(t)}=sum _{n=0}^{infty }{p_{n}(x) over n!}t^{n}}\u306e\u3088\u3046\u306b\u306a\u308b\u3002\u3053\u3053\u3067\u3001pn(x) \u306f\u591a\u9805\u5f0f\u5217\u3001f(t) \u306f\u3042\u308b\u5f62\u5f0f\u306e\u95a2\u6570\u3067\u3042\u308b\u3002\u30b7\u30a7\u30d5\u30a1\u30fc\u5217\u3082\u540c\u69d8\u306b\u3057\u3066\u751f\u6210\u3055\u308c\u308b\u3002\u8a73\u7d30\u306f\u4e00\u822c\u5316\u30a2\u30da\u30eb\u591a\u9805\u5f0f\u3092\u53c2\u7167\u3002\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]\u6709\u9650\u5217\uff08\u3042\u308b\u3044\u306f\u540c\u3058\u3053\u3068\u3060\u304c\u3001\u3042\u308b\u756a\u53f7\u4ee5\u964d\u306e\u9805\u304c\u5168\u3066 0 \u3068\u306a\u308b\u5b9f\u8cea\u6709\u9650\u5217\uff09\u306b\u5bfe\u5fdc\u3059\u308b\u7279\u5225\u306e\u5834\u5408\u306b\u306f\u3001\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u306f\u591a\u9805\u5f0f\u306b\u306a\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u591a\u304f\u306e\u6709\u9650\u5217\u3092\u3001\u30dd\u30ef\u30f3\u30ab\u30ec\u591a\u9805\u5f0f\u306a\u3069\u306e\u6bcd\u95a2\u6570\u306b\u3088\u3063\u3066\u6709\u52b9\u306b\u89e3\u91c8\u3067\u304d\u308b\u3068\u3044\u3046\u70b9\u3067\u91cd\u8981\u3067\u3042\u308b\u3002\u91cd\u8981\u306a\u6bcd\u95a2\u6570\u3068\u3057\u3066\u3001\u5b9a\u6570\u5217 1, 1, 1, 1, … \u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u2211n=0\u221exn=11\u2212x{displaystyle sum _{n=0}^{infty }x^{n}={1 over 1-x}}\u304c\u3042\u308b\u3002\u53f3\u8fba\u306e\u5f0f\u306f\u3001\u5de6\u8fba\u306e\u51aa\u7d1a\u6570\u306b 1 \u2212 x \u3092\u639b\u3051\u308b\u3068\u305d\u306e\u7d50\u679c\u304c\u5b9a\u51aa\u7d1a\u6570\uff08\u3064\u307e\u308a x0 \u306e\u9805\u3092\u9664\u304f\u5168\u3066\u306e\u4fc2\u6570\u304c 0 \u306e\u7d1a\u6570\uff091 \u306b\u4e00\u81f4\u3059\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3067\u6b63\u5f53\u5316\u3067\u304d\u308b\u3002\u3082\u3063\u3068\u3044\u3048\u3070\u3001\u3053\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3064\u51aa\u7d1a\u6570\u306f\u4ed6\u306b\u5b58\u5728\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u305a\u3001\u3057\u305f\u304c\u3063\u3066\u5de6\u8fba\u306e\u51aa\u7d1a\u6570\u306f\u5f62\u5f0f\u7684\u51aa\u7d1a\u6570\u74b0\u306b\u65bc\u3051\u308b 1 \u2212 x \u306e\u4e57\u6cd5\u7684\u9006\u5143\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u3053\u308c\u3092\u4f7f\u3048\u3070\u3001\u4ed6\u306e\u3044\u304f\u3064\u304b\u306e\u5217\u306b\u3064\u3044\u3066\u306f\u3001\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u306e\u9589\u3058\u305f\u5f0f\u3092\u5bb9\u6613\u306b\u5c0e\u51fa\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070\u3001a \u3092\u4efb\u610f\u306e\u5b9a\u6570\u3068\u3059\u308b\u7b49\u6bd4\u6570\u5217 1, a, a2, a3, … \u306e\u6bcd\u95a2\u6570\u306f\u2211n=0\u221eanxn=11\u2212ax{displaystyle sum _{n=0}^{infty }a^{n}x^{n}={1 over 1-ax}}\u3067\u3042\u308a\u3001\u7279\u306b a \u304c \u22121 \u3068\u3057\u3066\u2211n=0\u221e(\u22121)nxn=11+x{displaystyle sum _{n=0}^{infty }(-1)^{n}x^{n}={1 over 1+x}}\u304c\u5f97\u3089\u308c\u308b\u3002x \u3092 x\u306e\u3042\u308b\u51aa\u4e57\u3067\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001\u5217\u306b\u898f\u5247\u7684\u306a\u30ae\u30e3\u30c3\u30d7\u3092\u5c0e\u5165\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070\u30011, 0, 1, 0, 1, 0, …. \u3068\u3044\u3046\u5217\u306e\u6bcd\u95a2\u6570\u306f\u2211n=0\u221ex2n=11\u2212x2{displaystyle sum _{n=0}^{infty }x^{2n}={1 over 1-x^{2}}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u6700\u521d\u306e\u6bcd\u95a2\u6570\u306e\u5e73\u65b9\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001\u4fc2\u6570\u5217\u304c1, 2, 3, 4, 5, … \u3068\u3044\u3046\u6570\u5217\u3092\u6210\u3059\u3053\u3068\u306f\u5bb9\u6613\u306b\u78ba\u8a8d\u3067\u304d\u308b\u3002\u3064\u307e\u308a\u3001\u6bcd\u95a2\u6570\u306b\u3064\u3044\u3066\u8a00\u3048\u3070\u2211n=0\u221e(n+1)xn=1(1\u2212x)2{displaystyle sum _{n=0}^{infty }(n+1)x^{n}={1 over (1-x)^{2}}}\u304c\u6210\u7acb\u3059\u308b\u3002\u307e\u305f\u7acb\u65b9\u306f\u4fc2\u6570\u5217\u3068\u3057\u3066\u4e09\u89d2\u6570 1, 3, 6, 10, 15, 21, … \u3092\u6301\u3061\u3001n \u756a\u76ee\u306e\u4e09\u89d2\u6570\u306f\u4e8c\u9805\u4fc2\u6570 (n+22){displaystyle {tbinom {n+2}{2}}} \u3067\u3042\u308b\u304b\u3089\u3001\u2211n=0\u221e(n+22)xn=1(1\u2212x)3{displaystyle sum _{n=0}^{infty }{tbinom {n+2}{2}}x^{n}={1 over (1-x)^{3}}}\u304c\u5f97\u3089\u308c\u308b\u3002\u307e\u305f\u3001(n+22)=12(n+1)(n+2)=12(n2+3n+2){displaystyle {binom {n+2}{2}}={frac {1}{2}}{(n+1)(n+2)}={frac {1}{2}}(n^{2}+3n+2)}\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070\u3001\u4e0a\u8ff0\u306e\u6570\u5217\u306e\u6bcd\u95a2\u6570\u306e\u7dda\u578b\u7d50\u5408\u3092\u3068\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u5e73\u65b9\u6570\u306e\u5217 0, 1, 4, 9, 16, … \u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u3092G(n2;x)=\u2211n=0\u221en2xn=2(1\u2212x)3\u22123(1\u2212x)2+11\u2212x=x(x+1)(1\u2212x)3{displaystyle G(n^{2};x)=sum _{n=0}^{infty }n^{2}x^{n}={2 over (1-x)^{3}}-{3 over (1-x)^{2}}+{1 over 1-x}={frac {x(x+1)}{(1-x)^{3}}}}\u3068\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u6709\u7406\u95a2\u6570[\u7de8\u96c6]\u6570\u5217\u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u304c\u6709\u7406\u5f0f\uff082\u3064\u306e\u591a\u9805\u5f0f\u306e\u6bd4\uff09\u3067\u8868\u3055\u308c\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f\u3001\u305d\u306e\u5217\u304c\u7dda\u578b\u6f38\u5316\u5f0f\u3092\u6301\u3064\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u308c\u306f\u3001\u4e0a\u8ff0\u306e\u4f8b\u3092\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002\u7573\u307f\u8fbc\u307f\u7a4d[\u7de8\u96c6]\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u306e\u9593\u306e\u4e57\u6cd5\u306f\u3001\u7d1a\u6570\u306e\u96e2\u6563\u7573\u307f\u8fbc\u307f\uff08\u30b3\u30fc\u30b7\u30fc\u7a4d\uff09\u3092\u751f\u3058\u308b\u3002\u591a\u5909\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]\u591a\u91cd\u6dfb\u5b57\u3092\u3082\u3064\u7d1a\u6570\u306b\u5bfe\u3057\u3066\u3001\u591a\u5909\u6570\u306e\u6bcd\u95a2\u6570\u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u306f\u3057\u3070\u3057\u3070\u8d85\u6bcd\u95a2\u6570 super generating function) \u3068\u547c\u3070\u308c\u308b\u3002\u7279\u306b2\u5909\u6570\u306e\u5834\u5408\u30922\u5909\u6570\u6bcd\u95a2\u6570 (bivariate generating function) \u3068\u547c\u3076\u3002\u4f8b\u3048\u3070\u3001(1 + x)n \u304c\u56fa\u5b9a\u3055\u308c\u305f n \u306b\u5bfe\u3059\u308b\u4e8c\u9805\u4fc2\u6570\u306e\u901a\u5e38\u578b\u6bcd\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\u3001\uff08n \u3092\u3082\u52d5\u304b\u3057\u3066\uff09\u4efb\u610f\u306e k \u3068 n \u306b\u5bfe\u3057\u3066\u4e8c\u9805\u4fc2\u6570 (nk){displaystyle {tbinom {n}{k}}} \u3092\u751f\u6210\u3059\u308b\u4e8c\u5909\u6570\u6bcd\u95a2\u6570\u304c\u3069\u3046\u306a\u308b\u306e\u304b\u3068\u8003\u3048\u308b\u306e\u306f\u81ea\u7136\u306a\u767a\u60f3\u3067\u3042\u308b\u3002\u3053\u308c\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u306f\u3001(1 + x)n \u81ea\u8eab\u3092 n \u3092\u6dfb\u5b57\u3068\u3059\u308b\u6570\u5217\u3068\u8003\u3048\u3001\u305d\u308c\u3092\u4fc2\u6570\u306b\u6301\u3061\u3001y \u3092\u4e0d\u5b9a\u5143\u3068\u3059\u308b\u6bcd\u95a2\u6570\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u3002an \u306e\u6bcd\u95a2\u6570\u306f\u3061\u3087\u3046\u3069 1\/(1 \u2212 ay) \u306b\u7b49\u3057\u3044\u304b\u3089\u3001\u6c42\u3081\u308b\u4e8c\u9805\u4fc2\u6570\u306e\u6bcd\u95a2\u6570\u306f11\u2212(1+x)y=1+(1+x)y+(1+x)2y2+\u2026{displaystyle {frac {1}{1-(1+x)y}}=1+(1+x)y+(1+x)^{2}y^{2}+dots }\u3067\u3042\u308a\u3001xkyn \u306e\u4fc2\u6570\u304c\u4e8c\u9805\u4fc2\u6570 (nk){displaystyle {tbinom {n}{k}}} \u3068\u306a\u308b\u3002\u5e73\u65b9\u6570\u306e\u5217 an = n2 \u306e\u5404\u7a2e\u6bcd\u95a2\u6570\u3092\u4ee5\u4e0b\u306b\u793a\u3059\u3002\u901a\u5e38\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]G(n2;x)=\u2211n=0\u221en2xn=x(x+1)(1\u2212x)3{displaystyle G(n^{2};x)=sum _{n=0}^{infty }n^{2}x^{n}={frac {x(x+1)}{(1-x)^{3}}}}\u6307\u6570\u578b\u6bcd\u95a2\u6570[\u7de8\u96c6]EG(n2;x)=\u2211n=0\u221en2xnn!=x(x+1)ex{displaystyle EG(n^{2};x)=sum _{n=0}^{infty }{frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}\u30d9\u30eb\u7d1a\u6570[\u7de8\u96c6]fp(x)=\u2211n=0\u221ep2nxn=11\u2212p2x{displaystyle f_{p}(x)=sum _{n=0}^{infty }p^{2n}x^{n}={frac {1}{1-p^{2}x}}}\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]DG(n2;s)=\u2211n=1\u221en2ns=\u03b6(s\u22122){displaystyle DG(n^{2};s)=sum _{n=1}^{infty }{frac {n^{2}}{n^{s}}}=zeta (s-2)}\u591a\u5909\u6570\u6bcd\u95a2\u6570[\u7de8\u96c6]\u591a\u5909\u6570\u6bcd\u95a2\u6570\uff08\u591a\u5909\u91cf\u751f\u6210\u95a2\u6570\uff09\u306f\u3001\u884c\u3068\u5217\u306e\u5408\u8a08\u3092\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u975e\u8ca0\u6574\u6570\u306e\u5206\u5272\u8868\u306e\u6570\u3092\u5b9f\u969b\u306b\u8a08\u7b97\u3059\u308b\u969b\u306b\u751f\u3058\u308b\u3002\u8868\u306b r \u500b\u306e\u884c\u3068 c \u500b\u306e\u5217\u304c\u3042\u308a\u3001\u884c\u306e\u5408\u8a08\u304c t1,\u2026tr{displaystyle t_{1},ldots t_{r}}\u3001\u5217\u306e\u5408\u8a08\u304c s1,\u2026sc{displaystyle s_{1},ldots s_{c}} \u3068\u3059\u308b\u3002\u30a2\u30fc\u30d3\u30f3\u30fb\u30b8\u30e7\u30f3\u30fb\u30b0\u30c3\u30c9\uff08\u82f1\u8a9e\u7248\uff09\u306b\u3088\u308c\u3070[2]\u3001\u6b21\u306e\u5f0f\u306b\u304a\u3051\u308b x1t1\u2026xrtry1s1\u2026ycsc{displaystyle x_{1}^{t_{1}}ldots x_{r}^{t_{r}}y_{1}^{s_{1}}ldots y_{c}^{s_{c}}} \u306e\u4fc2\u6570\u304c\u305d\u306e\u8868\u306e\u6570\u3067\u3042\u308b\u3002\u220fi=1r\u220fj=1c11\u2212xiyj{displaystyle prod _{i=1}^{r}prod _{j=1}^{c}{frac {1}{1-x_{i}y_{j}}}}\u6bcd\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306a\u7528\u9014\u306b\u4f7f\u308f\u308c\u308b\u3002\u6f38\u5316\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306b\u5bfe\u3057\u3066\u3001\u305d\u306e\u4e00\u822c\u9805\u306e\u9589\u3058\u305f\u5f62\u306e\u5f0f\u3092\u6c42\u3081\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u30d5\u30a3\u30dc\u30ca\u30c3\u30c1\u6570\u5217\u306a\u3069\u306b\u3064\u3044\u3066\u3053\u308c\u3092\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u6570\u5217\u306b\u5bfe\u3057\u3066\u3001\u305d\u308c\u304c\u6e80\u305f\u3059\u6f38\u5316\u5f0f\u3092\u6c42\u3081\u308b\u3002\u6bcd\u95a2\u6570\u306e\u5f62\u304b\u3089\u6f38\u5316\u5f0f\u3092\u3042\u308b\u7a0b\u5ea6\u4e88\u60f3\u3067\u304d\u308b[3]\u3002\u6570\u5217\u306e\u9593\u306b\u6210\u7acb\u3059\u308b\u95a2\u4fc2\u3092\u6c42\u3081\u308b\u3002\u4e8c\u3064\u306e\u6570\u5217\u306e\u6bcd\u95a2\u6570\u304c\u4f3c\u305f\u5f62\u3067\u3042\u308c\u3070\u3001\u5217\u81ea\u4f53\u306b\u3082\u306a\u3093\u3089\u304b\u306e\u95a2\u4fc2\u304c\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u6570\u5217\u306e\u6f38\u8fd1\u7684\u306a\u6319\u52d5\u3092\u8abf\u3079\u308b\u3002\u3053\u308c\u306b\u306f\u8907\u7d20\u95a2\u6570\u8ad6\u306e\u77e5\u8b58\u304c\u7528\u3044\u3089\u308c\u308b\u3002\u6570\u5217\u306e\u9593\u3067\u6e80\u305f\u3055\u308c\u308b\u95a2\u4fc2\u5f0f\uff08\u6052\u7b49\u5f0f\uff09\u3092\u6c42\u3081\u308b\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u5206\u5272\u6052\u7b49\u5f0f\u306f\u305d\u306e\u4e00\u4f8b\u3067\u3042\u308b\u3002\u7d44\u5408\u305b\u8ad6\u306b\u304a\u3051\u308b\u6570\u3048\u4e0a\u3052\u554f\u984c\u3092\u89e3\u3044\u3066\u3001\u305d\u308c\u3089\u306e\u89e3\u3092\u7d50\u3073\u3064\u3051\u308b\u3002\u30eb\u30fc\u30af\u591a\u9805\u5f0f\uff08\u82f1\u8a9e\u7248\uff09\u306f\u7d44\u5408\u305b\u8ad6\u306b\u304a\u3051\u308b\u5fdc\u7528\u4f8b\u3067\u3042\u308b\u3002\u7121\u9650\u548c\u3092\u8a55\u4fa1\u3059\u308b\u3002\u305d\u306e\u4ed6\u306e\u6bcd\u95a2\u6570[\u7de8\u96c6]\u3055\u3089\u306b\u8907\u96d1\u306a\u6bcd\u95a2\u6570\u3067\u751f\u6210\u3059\u308b\u591a\u9805\u5f0f\u5217\u3068\u3057\u3066\u3001\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u3042\u308b\u3002\u985e\u4f3c\u306e\u6982\u5ff5[\u7de8\u96c6]\u591a\u9805\u5f0f\u88dc\u9593\u306f\u3001\uff08\u4fc2\u6570\u3067\u306f\u306a\u304f\uff09\u5024\u3092\u6570\u5217\u3067\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u305d\u306e\u591a\u9805\u5f0f\u3092\u6c42\u3081\u308b\u554f\u984c\u3067\u3042\u308b\u3002\u307e\u305f\u3001\u3053\u308c\u3092\u53ef\u63db\u74b0\u8ad6\u306b\u304a\u3044\u3066\u62bd\u8c61\u5316\u3057\u305f\u3082\u306e\u304c\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f\u3067\u3042\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Wilf, Herbert S. (1994), Generatingfunctionology (Second ed.), Academic Press, ISBN\u00a00-12-751956-4, http:\/\/www.math.upenn.edu\/%7Ewilf\/DownldGF.html\u00a0.Knuth, Donald E., \u201cSection 1.2.9: Generating Functions\u201d, The Art of Computer Programming, 1,  Fundamental Algorithms (Third ed.), Addison-Wesley, pp.\u00a087\u201396, ISBN\u00a00-201-89683-4\u00a0Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren, \u201cChapter 7: Generating Functions\u201d, Concrete Mathematics. A foundation for computer science (Second Edition ed.), Addison-Wesley, pp.\u00a0320\u2013380, ISBN\u00a00-201-55802-5\u00a0\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki29\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki29\/archives\/107740#breadcrumbitem","name":"\u6bcd\u95a2\u6570 – Wikipedia"}}]}]