一些歐拉和

2023-04-22 20:31 作者:艾琳娜的糖果屋 0人讀過 | 我要投稿

去年推導(dǎo)了一些歐拉和的公式，具體如下：

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$%0A%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%7B%5Cleft(%20-1%20%5Cright)%20%5En%5Cfrac%7BH_%7Bmn%7D%7D%7Bn%5E%7B2s%7D%7D%7D%3D%5Cfrac%7Bm%5E%7B2s%7D%7D%7B2%7D%5Czeta%20%5Cleft(%201%2B2s%20%5Cright)%20-%5Cfrac%7B1%2B2s%7D%7B2m%7D%5Ceta%20%5Cleft(%201%2B2s%20%5Cright)%20%2B%5Csum_%7Bk%3D1%7D%5E%7Bs-1%7D%7B%5Ceta%20%5Cleft(%202k%20%5Cright)%20%5Czeta%20%5Cleft(%202s%2B1-2k%20%5Cright)%20m%5E%7B2s-2k%7D%7D-%5Cfrac%7B%5Cpi%7D%7B2%5E%7B2s%2B1%7D%5Cleft(%202s-1%20%5Cright)%20!%7D%5Csum_%7Bk%3D2%7D%5Em%7B%5Cfrac%7B1%7D%7Bm%5Csin%20%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%7D%5Cleft(%20%5Cpsi%20%5E%7B%5Cleft(%202s-1%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%20%5Cright)%20-%5Cpsi%20%5E%7B%5Cleft(%202s-1%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%20%5Cright)%7D%0A%0A%0A$

$%0A%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%7B%5Cfrac%7BH_%7Bmn%7D%5E%7B%5Cleft(%202%20%5Cright)%7D%7D%7Bn%5E%7B2s-1%7D%7D%7D%3D%5Cfrac%7B%5Czeta%20%5Cleft(%202%20%5Cright)%7D%7Bm%5E2%7D%5Czeta%20%5Cleft(%202s-1%20%5Cright)%20-%5Cleft(%20sm%5E%7B2s-1%7D%2B%5Cfrac%7B2s%5E2-s-1%7D%7B2m%5E2%7D%20%5Cright)%20%5Czeta%20%5Cleft(%202s%2B1%20%5Cright)%20%2B%5Csum_%7Bn%3D1%7D%5E%7Bs-1%7D%7B2%5Czeta%20%5Cleft(%202n%20%5Cright)%20%5Cleft(%20s-n%20%5Cright)%20%5Czeta%20%5Cleft(%202s%2B1-2n%20%5Cright)%20m%5E%7B2s-2n-1%7D%7D%0A%5C%5C%0A%2B%5Cfrac%7B%5Cpi%7D%7B2m%5E2%5Cleft(%202s-2%20%5Cright)%20!%7D%5Csum_%7Bk%3D2%7D%5Em%7B%5Ccot%20%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%20%5Cpsi%20%5E%7B%5Cleft(%202s-1%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%20%5Cright)%7D-%5Cfrac%7B%5Cpi%20%5E2%7D%7B2m%5E2%5Cleft(%202s-2%20%5Cright)%20!%7D%5Csum_%7Bk%3D2%7D%5Em%7B%5Ccsc%20%5E2%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%20%5Cpsi%20%5E%7B%5Cleft(%202s-2%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%20%5Cright)%7D%5C%2C%5C%2C%20%5Cleft(%20s%5Cgeqslant%202%20%5Cright)%20%0A%0A%0A$

$%0A%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%7B%5Cfrac%7B%5Cleft(%20-1%20%5Cright)%20%5EnH_%7Bmn%7D%5E%7B%5Cleft(%202%20%5Cright)%7D%7D%7Bn%5E%7B2s-1%7D%7D%7D%3D%5Cfrac%7B1%7D%7B2m%5E2%7D%5Cleft(%202s%5E2-s-1%20%5Cright)%20%5Ceta%20%5Cleft(%202s%2B1%20%5Cright)%20%2B%5Cfrac%7B%5Czeta%20%5Cleft(%202%20%5Cright)%20%5Ceta%20%5Cleft(%202s-1%20%5Cright)%7D%7B2m%5E2%7D-m%5E%7B2s-1%7Ds%5Czeta%20%5Cleft(%202s%2B1%20%5Cright)%20-%5Csum_%7Bn%3D1%7D%5E%7Bs-1%7D%7B2%7D%5Ceta%20%5Cleft(%202n%20%5Cright)%20%5Cleft(%20s-n%20%5Cright)%20%5Czeta%20%5Cleft(%202s%2B1-2n%20%5Cright)%20m%5E%7B2s-2n-1%7D%0A%5C%5C%0A%2B%5Cfrac%7B%5Cpi%7D%7Bm%5E22%5E%7B2s%2B1%7D%5Cleft(%202s-2%20%5Cright)%20!%7D%5Csum_%7Bk%3D2%7D%5Em%7B%5Ccsc%20%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%20%5Cleft(%20%5Cpsi%20%5E%7B%5Cleft(%202s-1%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%20%5Cright)%20-%5Cpsi%20%5E%7B%5Cleft(%202s-1%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%20%5Cright)%7D-%5Cfrac%7B%5Cpi%20%5E2%7D%7Bm%5E22%5E%7B2s%7D%5Cleft(%202s-2%20%5Cright)%20!%7D%5Csum_%7Bk%3D2%7D%5Em%7B%5Ccsc%20%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%20%5Ccot%20%5Cleft(%20%5Cfrac%7Bk-1%7D%7Bm%7D%5Cpi%20%5Cright)%20%5Cleft(%20%5Cpsi%20%5E%7B%5Cleft(%202s-2%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%20%5Cright)%20-%5Cpsi%20%5E%7B%5Cleft(%202s-2%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bk-1%7D%7B2m%7D%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%20%5Cright)%7D%0A%0A$

更高階的就不算了，所有權(quán)為奇數(shù)的情況都可以用ζ函數(shù)η函數(shù)以及polygamma表出，權(quán)為偶數(shù)的情況會涉及polylog而且難度是比這個高多了的，當(dāng)然這些級數(shù)還可以倒推回對數(shù)積分。

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$%0A%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%7B%5Cfrac%7BH_k%7D%7B%5Cleft(%20k%2Ba%20%5Cright)%20%5E%7Bn%2B2%7D%7D%7D%3D%5Cfrac%7B%5Cleft(%20-1%20%5Cright)%20%5En%7D%7B%5Cleft(%20n%2B1%20%5Cright)%20!%7D%5Cleft(%20%5Csum_%7Bm%3D0%7D%5En%7B%5Cleft(%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09n%5C%5C%0A%09m%5C%5C%0A%5Cend%7Barray%7D%20%5Cright)%20%5Cpsi%20%5E%7B%5Cleft(%20m%2B1%20%5Cright)%7D%5Cleft(%20a%20%5Cright)%20%5Cpsi%20%5E%7B%5Cleft(%20n-m%20%5Cright)%7D%5Cleft(%20a%20%5Cright)%7D-%5Cfrac%7B1%7D%7B2%7D%5Cpsi%20%5E%7B%5Cleft(%20n%2B2%20%5Cright)%7D%5Cleft(%20a%20%5Cright)%20%5Cright)%20%5C%2C%5C%2C%2C%5Cpsi%20%5E%7B%5Cleft(%200%20%5Cright)%7D%5Cleft(%20a%20%5Cright)%20%3D%5Cpsi%20%5Cleft(%20a%20%5Cright)%20%2B%5Cgamma%20%5C%2C%5C%2C%0A%0A$

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