. "A Identidade de Vandermonde (tamb\u00E9m conhecida como Teorema de Euler) (desenvolvida em 1772) \u00E9 a seguinte express\u00E3o matem\u00E1tica: Um caso particular interessante ocorre quando m=n=r. Este resultado \u00E9 conhecido como Teorema de Lagrange."@pt . . . "\u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430"@ru . . "7989"^^ . . . . . . . . . . . "De identiteit van Vandermonde, ook convolutie van Vandermonde geheten, is een identiteit uit de combinatoriek, die een betrekking tussen binomiaalco\u00EBffici\u00EBnten geeft: . De identiteit is vernoemd naar de Franse wiskundige Alexandre-Th\u00E9ophile Vandermonde, maar werd al in 1303 vermeld door de Chinese wiskundige Zhu Shijie (Chu Shi-Chieh)."@nl . . . . . . . . . . . "Vandermonde's identity"@en . . . "In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre-Th\u00E9ophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie. There is a q-analog to this theorem called the q-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity"@en . "En math\u00E9matiques combinatoires, l'identit\u00E9 de Vandermonde, ainsi nomm\u00E9e en l'honneur d'Alexandre-Th\u00E9ophile Vandermonde (1772), ou formule de convolution, affirme que , o\u00F9 les nombres sont les coefficients binomiaux, \u00AB ! \u00BB d\u00E9signant la factorielle. Les contributions non nulles \u00E0 cette derni\u00E8re somme proviennent des valeurs de j pour lesquelles les coefficients binomiaux sont non nuls, c'est-\u00E0-dire pour ."@fr . "\u8303\u5FB7\u8499\u6052\u7B49\u5F0F(\u82F1\u6587\uFF1AVandermonde's Identity)\u662F\u4E00\u4E2A\u6709\u5173\u7EC4\u5408\u6570\u7684\u6C42\u548C\u516C\u5F0F\u3002"@zh . . . . . . . "A Identidade de Vandermonde (tamb\u00E9m conhecida como Teorema de Euler) (desenvolvida em 1772) \u00E9 a seguinte express\u00E3o matem\u00E1tica: Um caso particular interessante ocorre quando m=n=r. Este resultado \u00E9 conhecido como Teorema de Lagrange."@pt . "In combinatoria, l'identit\u00E0 di Vandermonde (o convoluzione di Vandermonde) \u00E8 la seguente identit\u00E0 riguardante i coefficienti binomiali: per ogni , , interi non negativi. L'identit\u00E0 deve il suo nome a Alexandre-Th\u00E9ophile Vandermonde (1772), sebbene fosse gi\u00E0 conosciuta nel 1303 dal matematico cinese Zhu Shijie. Si pu\u00F2 generalizzare l'identita di Vandermonde in diversi modi, come ad esempio la seguente versione: ."@it . "\u8303\u5FB7\u8499\u6052\u7B49\u5F0F(\u82F1\u6587\uFF1AVandermonde's Identity)\u662F\u4E00\u4E2A\u6709\u5173\u7EC4\u5408\u6570\u7684\u6C42\u548C\u516C\u5F0F\u3002"@zh . "Identit\u00E0 di Vandermonde"@it . . . . "Identit\u00E9 de Vandermonde"@fr . . . "Vandermondova konvoluce nebo Vandermondova identita je kombinatorick\u00E1 identita pojmenov\u00E1na po francouzsk\u00E9m matematikovi Alexandre-Th\u00E9ophile Vandermonde, kter\u00FD s n\u00ED poprv\u00E9 p\u0159i\u0161el roku 1772.Zn\u011Bn\u00ED identity je: kde je binomick\u00FD koeficient. Navzdory tomu, \u017Ee je konvoluce pojmenovan\u00E1 po Vandermondovi, ve skute\u010Dnosti poch\u00E1z\u00ED ji\u017E z roku 1303, kdy ji objevil \u010D\u00EDnsky matematik ."@cs . "\u0641\u064A \u0627\u0644\u062A\u0648\u0627\u0641\u0642\u064A\u0627\u062A\u060C \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0641\u0627\u0646\u062F\u064A\u0631\u0645\u0648\u0646\u062F (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Vandermonde's identity)\u200F \u0647\u064A \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0627\u0644\u062A\u0627\u0644\u064A\u0629 \u0644\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u0627\u0644\u062B\u0646\u0627\u0626\u064A\u0629: \u062D\u064A\u062B r \u0648 m \u0648 n \u0623\u0639\u062F\u0627\u062F \u0635\u062D\u064A\u062D\u0629. \u0633\u0645\u064A\u062A \u0647\u0630\u0647 \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0647\u0643\u0630\u0627 \u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0623\u0644\u0643\u0633\u0646\u062F\u0631 \u062B\u064A\u0648\u0641\u064A\u0644 \u0641\u0627\u0646\u062F\u064A\u0631\u0645\u0648\u0646\u062F (1772)\u060C \u0631\u063A\u0645 \u0623\u0646\u0647\u0627 \u0643\u0627\u0646\u062A \u0645\u0639\u0631\u0648\u0641\u0629 \u0645\u0646 \u0642\u0628\u0644 \u0645\u0646\u0630 1303 \u0644\u062F\u0649 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u064A \u0627\u0644\u0635\u064A\u0646\u064A \u0632\u0648 \u0634\u064A\u062C\u064A\u0647 (\u0634\u0648 \u0634\u064A-\u0634\u064A\u064A\u0647)."@ar . . "\u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 (\u0438\u043B\u0438 \u0441\u0432\u0451\u0440\u0442\u043A\u0430 \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430) \u2014 \u044D\u0442\u043E \u0441\u043B\u0435\u0434\u0443\u044E\u0449\u0435\u0435 \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0434\u043B\u044F \u0431\u0438\u043D\u043E\u043C\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u043E\u0432: \u0434\u043B\u044F \u043B\u044E\u0431\u044B\u0445 \u043D\u0435\u043E\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043B\u044C\u043D\u044B\u0445 \u0446\u0435\u043B\u044B\u0445 \u0447\u0438\u0441\u0435\u043B r, m, n. \u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u043D\u0430\u0437\u0432\u0430\u043D\u043E \u0438\u043C\u0435\u043D\u0435\u043C \u0410\u043B\u0435\u043A\u0441\u0430\u043D\u0434\u0440\u0430 \u0422\u0435\u043E\u0444\u0438\u043B\u0430 \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 (1772), \u0445\u043E\u0442\u044F \u043E\u043D\u043E \u0431\u044B\u043B\u043E \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u043E \u0435\u0449\u0451 \u0432 1303 \u043A\u0438\u0442\u0430\u0439\u0441\u043A\u043E\u043C\u0443 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0443 \u0427\u0436\u0443 \u0428\u0438\u0446\u0437\u0435. \u0421\u043C. \u0441\u0442\u0430\u0442\u044C\u044E \u0410\u0441\u043A\u0435\u044F \u043F\u043E \u0438\u0441\u0442\u043E\u0440\u0438\u0438 \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u0430. \u0421\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 q-\u0430\u043D\u0430\u043B\u043E\u0433 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B, \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0449\u0438\u0439\u0441\u044F . \u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 \u043C\u043E\u0436\u043D\u043E \u043E\u0431\u043E\u0431\u0449\u0438\u0442\u044C \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E\u043C \u0441\u043F\u043E\u0441\u043E\u0431\u043E\u0432, \u0432\u043A\u043B\u044E\u0447\u0430\u044F \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E ."@ru . "Identiteit van Vandermonde"@nl . "Identidad de Vandermonde"@es . "En math\u00E9matiques combinatoires, l'identit\u00E9 de Vandermonde, ainsi nomm\u00E9e en l'honneur d'Alexandre-Th\u00E9ophile Vandermonde (1772), ou formule de convolution, affirme que , o\u00F9 les nombres sont les coefficients binomiaux, \u00AB ! \u00BB d\u00E9signant la factorielle. Les contributions non nulles \u00E0 cette derni\u00E8re somme proviennent des valeurs de j pour lesquelles les coefficients binomiaux sont non nuls, c'est-\u00E0-dire pour ."@fr . . . "\u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 (\u0438\u043B\u0438 \u0441\u0432\u0451\u0440\u0442\u043A\u0430 \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430) \u2014 \u044D\u0442\u043E \u0441\u043B\u0435\u0434\u0443\u044E\u0449\u0435\u0435 \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0434\u043B\u044F \u0431\u0438\u043D\u043E\u043C\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u043E\u0432: \u0434\u043B\u044F \u043B\u044E\u0431\u044B\u0445 \u043D\u0435\u043E\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043B\u044C\u043D\u044B\u0445 \u0446\u0435\u043B\u044B\u0445 \u0447\u0438\u0441\u0435\u043B r, m, n. \u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u043D\u0430\u0437\u0432\u0430\u043D\u043E \u0438\u043C\u0435\u043D\u0435\u043C \u0410\u043B\u0435\u043A\u0441\u0430\u043D\u0434\u0440\u0430 \u0422\u0435\u043E\u0444\u0438\u043B\u0430 \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 (1772), \u0445\u043E\u0442\u044F \u043E\u043D\u043E \u0431\u044B\u043B\u043E \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u043E \u0435\u0449\u0451 \u0432 1303 \u043A\u0438\u0442\u0430\u0439\u0441\u043A\u043E\u043C\u0443 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0443 \u0427\u0436\u0443 \u0428\u0438\u0446\u0437\u0435. \u0421\u043C. \u0441\u0442\u0430\u0442\u044C\u044E \u0410\u0441\u043A\u0435\u044F \u043F\u043E \u0438\u0441\u0442\u043E\u0440\u0438\u0438 \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u0430. \u0421\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 q-\u0430\u043D\u0430\u043B\u043E\u0433 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B, \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0449\u0438\u0439\u0441\u044F . \u0422\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E \u0412\u0430\u043D\u0434\u0435\u0440\u043C\u043E\u043D\u0434\u0430 \u043C\u043E\u0436\u043D\u043E \u043E\u0431\u043E\u0431\u0449\u0438\u0442\u044C \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E\u043C \u0441\u043F\u043E\u0441\u043E\u0431\u043E\u0432, \u0432\u043A\u043B\u044E\u0447\u0430\u044F \u0442\u043E\u0436\u0434\u0435\u0441\u0442\u0432\u043E ."@ru . . . . "\u8303\u5FB7\u8499\u6052\u7B49\u5F0F"@zh . . . . . . . "1075925768"^^ . . "In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre-Th\u00E9ophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie. There is a q-analog to this theorem called the q-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity"@en . "916157"^^ . . "Identidade de Vandermonde"@pt . . . "Vandermondova konvoluce nebo Vandermondova identita je kombinatorick\u00E1 identita pojmenov\u00E1na po francouzsk\u00E9m matematikovi Alexandre-Th\u00E9ophile Vandermonde, kter\u00FD s n\u00ED poprv\u00E9 p\u0159i\u0161el roku 1772.Zn\u011Bn\u00ED identity je: kde je binomick\u00FD koeficient. Navzdory tomu, \u017Ee je konvoluce pojmenovan\u00E1 po Vandermondovi, ve skute\u010Dnosti poch\u00E1z\u00ED ji\u017E z roku 1303, kdy ji objevil \u010D\u00EDnsky matematik ."@cs . . . . . "En combinatoria, la identidad de Vandermonde o convoluci\u00F3n de Vandermonde, que recibe su nombre del matem\u00E1tico franc\u00E9s Alexandre-Th\u00E9ophile Vandermonde (1772), expresa que: para coeficientes binomiales. Esta identidad ya hab\u00EDa sido descubierta en 1303 por el matem\u00E1tico chino Zhu Shijie (Chu Shi-Chieh).\u200B Existe una de este teorema denominada q-identidad de Vandermonde."@es . . "De identiteit van Vandermonde, ook convolutie van Vandermonde geheten, is een identiteit uit de combinatoriek, die een betrekking tussen binomiaalco\u00EBffici\u00EBnten geeft: . De identiteit is vernoemd naar de Franse wiskundige Alexandre-Th\u00E9ophile Vandermonde, maar werd al in 1303 vermeld door de Chinese wiskundige Zhu Shijie (Chu Shi-Chieh)."@nl . . "Vandermondova konvoluce"@cs . "\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0641\u0627\u0646\u062F\u0631\u0645\u0648\u0646\u062F"@ar . . . . . "In combinatoria, l'identit\u00E0 di Vandermonde (o convoluzione di Vandermonde) \u00E8 la seguente identit\u00E0 riguardante i coefficienti binomiali: per ogni , , interi non negativi. L'identit\u00E0 deve il suo nome a Alexandre-Th\u00E9ophile Vandermonde (1772), sebbene fosse gi\u00E0 conosciuta nel 1303 dal matematico cinese Zhu Shijie. Si pu\u00F2 generalizzare l'identita di Vandermonde in diversi modi, come ad esempio la seguente versione: ."@it . . . "\u0641\u064A \u0627\u0644\u062A\u0648\u0627\u0641\u0642\u064A\u0627\u062A\u060C \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0641\u0627\u0646\u062F\u064A\u0631\u0645\u0648\u0646\u062F (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Vandermonde's identity)\u200F \u0647\u064A \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0627\u0644\u062A\u0627\u0644\u064A\u0629 \u0644\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u0627\u0644\u062B\u0646\u0627\u0626\u064A\u0629: \u062D\u064A\u062B r \u0648 m \u0648 n \u0623\u0639\u062F\u0627\u062F \u0635\u062D\u064A\u062D\u0629. \u0633\u0645\u064A\u062A \u0647\u0630\u0647 \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0647\u0643\u0630\u0627 \u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u0623\u0644\u0643\u0633\u0646\u062F\u0631 \u062B\u064A\u0648\u0641\u064A\u0644 \u0641\u0627\u0646\u062F\u064A\u0631\u0645\u0648\u0646\u062F (1772)\u060C \u0631\u063A\u0645 \u0623\u0646\u0647\u0627 \u0643\u0627\u0646\u062A \u0645\u0639\u0631\u0648\u0641\u0629 \u0645\u0646 \u0642\u0628\u0644 \u0645\u0646\u0630 1303 \u0644\u062F\u0649 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u064A \u0627\u0644\u0635\u064A\u0646\u064A \u0632\u0648 \u0634\u064A\u062C\u064A\u0647 (\u0634\u0648 \u0634\u064A-\u0634\u064A\u064A\u0647)."@ar . "En combinatoria, la identidad de Vandermonde o convoluci\u00F3n de Vandermonde, que recibe su nombre del matem\u00E1tico franc\u00E9s Alexandre-Th\u00E9ophile Vandermonde (1772), expresa que: para coeficientes binomiales. Esta identidad ya hab\u00EDa sido descubierta en 1303 por el matem\u00E1tico chino Zhu Shijie (Chu Shi-Chieh).\u200B Existe una de este teorema denominada q-identidad de Vandermonde."@es . . . .