ក្នុងគណិតវិទ្យា ពហុធាប៊ែរនូយី (Bernoulli polynomials) លេចឡើងក្នុងការសិក្សាផ្នែកជាច្រើននៃអនុគមន៍ និង ជាពិសេសអនុគមន៍ហ្សេតារីម៉ាន (Riemann zeta function) ។
ពហុធាប៊ែរនូយីគឺជាស្វ៊ីតពហុធាតែមួយគត់
ដែល
![{\displaystyle B_{0}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7161c7471f54b29d020a7b1bee6cbd914c2486b0)
![{\displaystyle \forall n\in \mathbb {N} ,B'_{n+1}=(n+1)B_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daedfd5617475794cc642155b8373f38e2d08cb2)
![{\displaystyle \forall n\in \mathbb {N^{*}} ,\int _{0}^{1}B_{n}(x)dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee377d70c5d8424a8f4ecb5326e61a633b2d9d59)
អនុគមន៍តំនពូជ[កែប្រែ]
អនុគមន៍តំនពូជ (Generating function) ចំពោះពហុធាប៊ែរនូយីគឺ
.
អនុគមន៍តំនពូជ (Generating function) ចំពោះពហុធាអយល័រ (Euler polynomials) គឺ
.
ផលបូកនៃស្វ័យគុណទី p[កែប្រែ]
យើងមាន
![{\displaystyle \sum _{k=0}^{x}k^{p}={\frac {B_{p+1}(x+1)-B_{p+1}(0)}{p+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66dece7acc4ef5fabc034403f3671e602ea2e576)
ចំពោះសេចក្តីលំអិតអំពីរូបមន្តនេះ សូមមើលរូបមន្តហ្វូលហាប័រ (Faulhaber's formula)
ចំនួនអយល័រ និង ចំនួនប៊ែរនូយី[កែប្រែ]
- ចំនួនប៊ែរនូយីអោយដោយ
។
- ចំនួនអយល័រអោតយដោយ
។
កន្សោមអិចភ្លីស៊ីតចំពោះលំដាប់ទាប[កែប្រែ]
ពហុធាប៊ែរនូយីដំបូងមួយចំនួន
![{\displaystyle B_{0}(x)=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9872db89470749143fb817f16f5d83ffd34da7cb)
![{\displaystyle B_{1}(x)=x-1/2\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb7f24e15f19d0cafcf4bfd32544f0684863ce4)
![{\displaystyle B_{2}(x)=x^{2}-x+1/6\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc56150e3fdf89b62cbab3c1e7aba3e4b7f2ad7)
![{\displaystyle B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2417f589f3ea143002c97faa5529955bfc9b57)
![{\displaystyle B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed179b6ff941e63864d41e19fd28f5e24b3a9862)
![{\displaystyle B_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6993438423c5e9661c046d185d4be4d9c34c956)
![{\displaystyle B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd81a18b955481627502e6d7efeb4abddc599fb9)
ពហុធាអយល័រដំបូងមួយចំនួន
![{\displaystyle E_{0}(x)=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/147a3acf96ca5fb9f5627aebad0ae6b52fdfbe99)
![{\displaystyle E_{1}(x)=x-1/2\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c6f2532807441e8115ef072dc56ac4938ff4b3)
![{\displaystyle E_{2}(x)=x^{2}-x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fae5da7509daebda7b71f2d703fbebeb14fa318)
![{\displaystyle E_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{4}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc0ba4eda1779cc6b31ce68fab9c36b3419e87a)
![{\displaystyle E_{4}(x)=x^{4}-2x^{3}+x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0806d7c6339411c56457599349e64878123a0256)
![{\displaystyle E_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{2}}x^{2}-{\frac {1}{2}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/765848dfdf795d0b98cd9885ebd8d74d8688d057)
![{\displaystyle E_{6}(x)=x^{6}-3x^{5}+5x^{3}-3x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d2ce45bb828470cb170ad328eae0ce88956e38)
ពហុធាប៊ែរនូយី និង ពហុធាអយល័រគោរពតាមទំនាក់ទំនងជាច្រើនពីការការគណនានិមិត្តរូប (umbral calculus ឬ symbolic calculus)
![{\displaystyle B_{n}(x+1)-B_{n}(x)=nx^{n-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510de3a6cccd2abc21ed0cfbb0656122336fb7b)
![{\displaystyle E_{n}(x+1)+E_{n}(x)=2x^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2ec9d77f988ad8abc5ec9c2a9a7220e2ed508e)
![{\displaystyle B_{n}'(x)=nB_{n-1}(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4b374acf5361879f73f4d28971fb81a5da2435)
![{\displaystyle E_{n}'(x)=nE_{n-1}(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45b0ab83f0678b84f657d37c46cb386291e2c0bb)
![{\displaystyle B_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e438a90bf791124a6dbe694b8f6dab7461aebd)
![{\displaystyle E_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e02c745283a6da8b1fabf94e13e34904e192c73)
![{\displaystyle B_{n}(1-x)=(-1)^{n}B_{n}(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e15fad8a10a2990df21d0a35a80f66db21185a9)
![{\displaystyle E_{n}(1-x)=(-1)^{n}E_{n}(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab02300945973c072ad4f81ac30733ac2c535db)
![{\displaystyle (-1)^{n}B_{n}(-x)=B_{n}(x)+nx^{n-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df75c7e9c443adcaa0d9136fdb8968e316c4ab5c)
![{\displaystyle (-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0e1b7a29c0bad4641545c2203a7cd2a2b14d11)
លក្ខណៈផ្សេងទៀតនៃពហុធាប៊ែរនូយី[កែប្រែ]
![{\displaystyle \forall n\in \mathbb {N} ,B_{n}(x)=2^{n-1}\left(B_{n}\left({\frac {x}{2}}\right)+B_{n}\left({\frac {x+1}{2}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49a58711d8cd97e04ffe99c4bd65c6233669d19)
![{\displaystyle \forall n>1,B_{n}(0)=B_{n}(1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea3ff840da495cc7d6f39a7a991d1004d358e61)
![{\displaystyle \forall p\in \mathbb {N} ^{*},B_{2p+1}(0)=B_{2p+1}(1)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/936c8de1a81c5b6abb3b51b88466e666a2a4e18d)
![{\displaystyle \forall p\in \mathbb {N} ^{*},B_{2p}\left({\frac {1}{2}}\right)=\left({\frac {1}{2^{2p-1}}}-1\right)B_{2p}(0),B_{2p+1}\left({\frac {1}{2}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df379e1cd6e296f7b1e964580b0fc5501ca1725)
ស៊េរីហ្វួរា[កែប្រែ]
ស៊េរីហ្វួរា (Fourier series) នៃពហុធាប៊ែរនូយីក៏ជាស៊េរីឌីរិចឡេអោយដោយការពន្លាត
![{\displaystyle B_{n}(x)=-{\frac {n!}{(2\pi i)^{n}}}\sum _{k\not =0}{\frac {e^{2\pi ikx}}{k^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a026ecaa16d7e24c176f95e13d108f7a9ed452d3)
នេះជាករណីពិសេសនៃទំរង់អាណាឡូក (analogous form) ចំពោះអនុគមន៍ហ្សេតាហឺវីត (Hurwitz zeta function)
![{\displaystyle B_{n}(x)=-\Gamma (n+1)\sum _{k=1}^{\infty }{\frac {e^{(2\pi ikx)}+e^{(2\pi ik(1-x))}}{(2\pi ik)^{n}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7046e53813e54e80c371ee91fa8c09f567de1f7)
ការពន្លាតនេះគឺត្រឹមត្រូវតែចំពោះ 0 ≤ x ≤ 1 ដែល n ≥ 2 និងត្រឹមត្រូវចំពោះ 0 < x < 1ដែល n = 1 ។
ស៊េរីហ្វួរានៃពហុធាអយល័រអាចគណនាបានផងដែរ ។ កំនត់អនុគមន៍
![{\displaystyle C_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23729ee2bfcbe50ac7b401557397ec08668c4aeb)
និង
![{\displaystyle S_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00bfdd4f6072f35fce566fbb1f008d232ab46c66)
ចំពោះ
ពហុធាអយល័រមានស៊េរីហ្វួរា
![{\displaystyle C_{2n}(x)={\frac {(-1)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281216204f06bf8fdad699814a8bbb9010de2a6d)
និង
![{\displaystyle S_{2n+1}(x)={\frac {(-1)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be3363c06416f33139dba9ab73278f7e29e4cb9d)
សំគាល់ថា
និង
គឺអនុគមន៍សេសនិងគូរៀងគ្នា
![{\displaystyle \ C_{\nu }(x)=-C_{\nu }(1-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd60789ea94e66266fac11d24c978fc9f8d3cbc)
និង
![{\displaystyle \ S_{\nu }(x)=S_{\nu }(1-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/651cefd72b5ae76fbe5d15165d7395308a33409f)
អនុគមន៍ទាំងនេះមានទំនាក់ទំនងនឹងអនុគមន៍ឈីឡឺហ្សង់ (Legendre chi function)
ជា
![{\displaystyle \ C_{\nu }(x)={\mbox{Re}}\chi _{\nu }(e^{ix})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cce6e1cfb73be075da59ce0ee1c3dc31a4ccf42c)
និង
![{\displaystyle \ S_{\nu }(x)={\mbox{Im}}\chi _{\nu }(e^{ix}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed0f3894da0444605642e3764203283f84728db)
ទ្រឹស្តីបទផលគុណ[កែប្រែ]
ទ្រឹស្តីបទផលគុណ (Multiplication theorems) ត្រូវបានផ្តល់អោយដោយ Joseph Ludwig Raabe ក្នុងឆ្នាំ ១៨៥១
![{\displaystyle B_{n}(mx)=m^{n-1}\sum _{k=0}^{m-1}B_{n}\left(x+{\frac {k}{m}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58d8c0197fcacc58546991e1dba593001f303e90)
ចំពោះ ![{\displaystyle m=1,3,\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/add5223f86d923c777e79934a81ccdb2b569ee20)
ចំពោះ ![{\displaystyle m=2,4,\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ee8eaf229e526b03807888e1717586674b74e1)
អាំងតេក្រាលមិនកំនត់
![{\displaystyle \int _{a}^{x}B_{n}(t)\,dt={\frac {B_{n+1}(x)-B_{n+1}(a)}{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c04c282d8224633ebb723c781589f0a8057399b3)
![{\displaystyle \int _{a}^{x}E_{n}(t)\,dt={\frac {E_{n+1}(x)-E_{n+1}(a)}{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/898f49aa4e320fd100c7aaa8fef06cc368ab93b9)
អាំងតេក្រាលកំនត់
![{\displaystyle \int _{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}{\frac {m!n!}{(m+n)!}}B_{n+m}\quad {\mbox{ for }}m,n\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/381a848f7a185b375e622d5f5c45de55aca71f79)
![{\displaystyle \int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!n!}{(m+n+2)!}}B_{n+m+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e04364c1e61a099e1bb53797b180788761e8a7)