查看“︁布巴克尔多項式”︁的源代码

==布巴克尔多項式 ==

在數學中，布巴克尔 多項式  <ref>{{citation|first1=OD|last1=Oyodum|first2=OB|last2=Awojoyogbe|first3=MK.|last3=Dada|first4=JN|last4=Magnuson|journal=Eur. Phys. J. Appl. Phys.|volume=46|pages=21201–21202|title=On the earliest definition of the '''布巴克尔 polynomials'''|url=http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246|accessdate=2011-08-04|archive-date=2011-10-05|archive-url=https://web.archive.org/web/20111005015504/http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246|dead-url=yes}}</ref>有两种常见定义。第一种是 :
:<math>
\begin{align}
B_0(x) & {} = 1 \\
B_1(x) & {} = x \\
B_2(x) & {} = x^2+2 \\
B_3(x) & {} = x^3+x \\
B_4(x) & {} = x^4-2 \\
B_5(x) & {} = x^5-x^3-3x \\
B_6(x) & {} = x^6-2x^4-3x^2+2 \\
B_7(x) & {} = x^7-3x^5-2x^3+5x \\
B_8(x) & {} = x^8-4x^6+8x^2-2 \\
B_9(x) & {} = x^9-5x^7+3x^5+10x^3-7x \\
& {}\,\,\, \vdots
\end{align}
</math>
有时也会使用另一种定义,可以通过递归的方式进行定义。首先，规定前三 个布巴克尔多项式为：
:<math>\begin{align}
B_0(x) &= 1, \\
B_1(x) &= x, \\
B_2(x) &= x^2+2, \\
\end{align}</math>

然后运用下面的[[多项式|递推关系]]得到更高阶的多项式。

:<math>\begin{align}
B_m(x) &= xB_{m-1}(x) - B_{m-2}(x) \quad\text{, } m>2.
\end{align}</math>
布巴克尔 多項式也可以用[[母函数]]表示 :                     
: <math>\sum_{n=0}^{\infty}\tilde{B}_n(x) t^n = \frac{1+3t^2}{1-t(t-x)}. \,\!</math>

产生了许多整数序列在[[On-Line Encyclopedia of Integer Sequences]] (''OEIS'')<ref>Sequences [[OEIS:A135929|A135929]] and  [[OEIS:A135936|A135936]] by Neil J. A. Sloane, [[OEIS:A137276|A137276]] , Roger L. Bagula , Gary Adamson,[[OEIS:A138476|A138476]], A. Bannour, [[OEIS:A137289|A137289]], [[OEIS:A136256|A136256]], [[OEIS:A136255|A136255]] , R. L. Bagula 在 [[On-Line Encyclopedia of Integer Sequences]]</ref> e [http://planetmath.org/?op=getobj&from=objects&id=12200 PlanetMath] {{Wayback|url=http://planetmath.org/?op=getobj&from=objects&id=12200 |date=20201203115300 }}. 

===生成解===
布巴克尔 多項式的[[通解]]為 :
:<math>B_n(x)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n-4p}{n-p} \binom{n-p}{p} (-1)^p x^{n-2p} </math>


===微分操作代表===
布巴克尔 多項式亦可記為 :
:<math> (x^2-1)(3nx^2+n-2)y{''}+3x(nx^2+3n-2)y{'}-n(3n^2x^2+n^2-6n+8)y=0 \,</math>
== 用途 ==
布巴克尔 多項式的 用途:
*[[低温]]<ref>{{citation|title= Book:Cryogenics: Theory, Processes and Applications, Chapter 8: Cryogenics Vessels Thermal Profilng Using the 布巴克尔 Polynomials Expansion Scheme Investigation , Editor: Allyson E.Hayes|url= https://www.novapublishers.com/catalog/product_info.php?products_id=17332&osCsid=06f25d4f739dc8ec36c5160f480acaef|accessdate= 2011-08-04|archive-date= 2012-10-11|archive-url= https://web.archive.org/web/20121011101844/https://www.novapublishers.com/catalog/product_info.php?products_id=17332&osCsid=06f25d4f739dc8ec36c5160f480acaef|dead-url= no}}</ref>
*[[生物学]]<ref>{{citation | journal=Journal of Theoretical Biology (Elsevier)|id={{doi:10.1016/j.jtbi.2010.12.002}} | author=B. Dubey, T.G. Zhao, M. Jonsson, H. Rahmanov| title = A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using 布巴克尔 polynomial expansion scheme| url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WMD-4Y88DJ2-3&_user=10&_coverDate=05%2F07%2F2010&_alid=1735682909&_rdoc=1&_fmt=high&_orig=search&_origin=search&_zone=rslt_list_item&_cdi=6932&_sort=r&_st=13&_docanchor=&view=c&_ct=3&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=59c5fe7394a9f231c68071947b70b311&searchtype=a}}</ref>
*[[动态系统]]<ref>{{citation | journal=Journal of Theoretical Biology (Elsevier)|id={{doi:10.1016/j.jtbi.2010.01.026}} | author=A. Milgeam|title = The stability of the 布巴克尔 polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WMD-4Y88DJ2-3&_user=10&_coverDate=05%2F07%2F2010&_alid=1735677182&_rdoc=21&_fmt=high&_orig=search&_origin=search&_zone=rslt_list_item&_cdi=6932&_sort=r&_st=13&_docanchor=&view=c&_ct=68&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8f60d31122c688778a250f496396c43c&searchtype=a }}</ref>
*[[非线性系统]]<ref>{{citation | journal=Mathematical and Computer Modelling(Elsevier)|id={{doi:10.1016/j.mcm.2011.02.031 }} | author=H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din| title = The Comparative 布巴克尔 Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem| url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0V-528GTH4-4&_user=10&_coverDate=07%2F31%2F2011&_alid=1735682480&_rdoc=3&_fmt=high&_orig=search&_origin=search&_zone=rslt_list_item&_cdi=5656&_sort=r&_st=13&_docanchor=&view=c&_ct=51&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b7675a0e4d358b5daf33968de7cf46ec&searchtype=a}}</ref> <ref>{{citation | journal= The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 > | author= A. Yildirim | title= The 布巴克尔 polynomials expansion scheme for solving nonlinear science problems, | url= http://web3.cas.usf.edu/main/depts/mth/7thde/data/Abstracts-7thDEDS-Tampa.pdf | deadurl= yes | archiveurl= https://web.archive.org/web/20110722013350/http://web3.cas.usf.edu/main/depts/mth/7thde/data/Abstracts-7thDEDS-Tampa.pdf | archivedate= 2011年7月22日 }}</ref>
*[[近似论]]<ref>{{citation| journal=Journal of Integer Sequences (JIS)| author=Paul Barry, Aoife Hennessy| title=Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: '''''The 布巴克尔 polynomials '''''| url=http://www.emis.ams.org/journals/JIS/VOL13/Barry5/barry96s.pdf| accessdate=2011-08-04| archive-date=2018-04-28| archive-url=https://web.archive.org/web/20180428175159/http://www.emis.ams.org/journals/JIS/VOL13/Barry5/barry96s.pdf| dead-url=no}}</ref>
*[[热力学]] <ref>{{citation| journal= Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer)| author= H. Koçak, Z. Dahong, A. Yildirim| title= A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the 布巴克尔 polynomials expansion scheme| url= http://www.springerlink.com/content/d78h761823628gl2/}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{citation| journal= Indian Journal of Physics(Springer)| author= H. Koçak, Z. Dahong, A. Yildirim| title= Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the 布巴克尔 Polynomials Expansion Scheme (BPES)| url= http://www.springerlink.com/content/173787083245t267/}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{citation| journal=Jornal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA)| author=A. Belhadj, O. F. Onyango and N. Rozibaeva| title=布巴克尔 Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model| url=http://pdf.aiaa.org/jaPreview/JTHT/2009/PVJA41850.pdf}}{{dead link|date=2017年12月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
*[[力学]] <ref>{{citation | journal= International Journal of Non-Linear Mechanics(Elsevier)| author= D. H. Zhang| title = Study of a non-linear mechanical system using 布巴克尔 polynomials expansion scheme BPES|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ2-51J36F8-1&_user=10&_coverDate=03%2F31%2F2011&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1736224281&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=a7eda39bb611bdf91b105cb6d9f7e420&searchtype=a}}</ref>
*[[水文地理学]] <ref>{{citation| journal=Studies in Nonlinear Sciences (SNS)| author=Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem| title=Modelling Nonlinear Bivariate Dependence Using the 布巴克尔 Polynomials Copula '''The 布巴克尔 polynomials '''| url=http://idosi.org/sns/2(1)11/3.pdf| accessdate=2011-08-04| archive-date=2020-09-22| archive-url=https://web.archive.org/web/20200922040214/http://idosi.org/sns/2(1)11/3.pdf| dead-url=no}}</ref>
*[[分子动态]] <ref>{{citation| journal= Journal of Structural Chemistry (Springer)| author= W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım| title= A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the 布巴克尔 polynomials expansion scheme| url= http://www.springerlink.com/content/57681724u74gvg76/}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
*[[基本数学]] <ref>{{citation| journal= Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press)| author= D. H. Zhang,  L. Naing| title= The 布巴克尔 polynomials expansion scheme BPES for solving a standard boundary value problem| url= http://www.mathem.pub.ro/apps/v12/A12-zh.pdf| accessdate= 2011-08-04| archive-date= 2020-09-28| archive-url= https://web.archive.org/web/20200928161544/http://www.mathem.pub.ro/apps/v12/A12-zh.pdf| dead-url= no}}</ref>
*[[热量测定]] <ref>{{citation| journal=Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.)| id={{doi:10.1007/s10973-009-0094-4  }}| author=A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier| title=Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and 布巴克尔 polynomials expansion| url=http://www.springerlink.com/content/2l03064124057686/?p=15de2fa57ce5478aa8a62c2b3a618213&pi=1}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
*[[生物物理学]] <ref>{{citation| journal=Heat and Mass Transfer(Springer Berlin / Heidelberg)| id=Volume 45, Number 10 / août 2009, pages:1247-1251 {{doi:10.1007/s00231-009-0493-x}}| author=S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses| title=Cut-off cooling velocity profiling inside a keyhole model using the 布巴克尔 polynomials expansion scheme| url=http://www.springerlink.com/content/b125h6166r216313/}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
*[[光电学]] <ref>{{citation | journal=Modern Physics Letters B ([ISSN: 0217-9849, by WS: World Scientific Publishing Co Pte Ltd] )| author=S. Fridjine and M. Amlouk| title = A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING 布巴克尔 POLYNOMIALS EXPANSION SCHEME| url=http://www.worldscinet.com/mplb/23/2317/S0217984909020321.html}}</ref>
*[[复杂分析]] <ref>{{citation | author=T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud | title=Limit and uniqueness of the 布巴克尔-Zhao polynomials single imaginary root sequence | journal=International Journal of Mathematics and Computation | volume=1 | number=08 | ISSN=0974-5718 | url=http://ceser.res.in/ijmc.html | deadurl=yes | archiveurl=https://web.archive.org/web/20110813091845/http://ceser.res.in/ijmc.html | archivedate=2011-08-13 }}</ref>
*[[矩阵分析]]<ref> {{citation | first1=A. | last1=Luzon | first2=M. | last2=Moron |  | title=RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: 布巴克尔 POLYNOMIALS associated  Riordan matrix | url=http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.2672v1.pdf}}</ref> 
*[[ 加密]]<ref>[http://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1523651_code1403499.pdf?abstractid=1523651&mirid=3 B.  Tirimula  Rao,   P.  Srinivsu, C.  Anantha  Rao,  K.  Satya  Vivek  Vardhan,  Jami   Vidyadhari ,Page 8 : 布巴克尔 polynomials]</ref>
*[[代数]]<ref >{{Cite web |url=http://cujse.cankaya.edu.tr/archive/14/02_cujse_10018.pdf |title=Kiliç  Bülent,  Erdal Bas, Page 7, Citation 27: 布巴克尔 polynomials |access-date=2011-08-04 |archive-date=2018-04-20 |archive-url=https://web.archive.org/web/20180420214112/http://cujse.cankaya.edu.tr/archive/14/02_cujse_10018.pdf |dead-url=no }}</ref>


==参考文献==   
{{references |3}}

==外部链接==   
* Encyclopedia of Physics Research: {{citation| title= Encyclopedia of Physics Research, Chapter 21: Cryogenics Vessels Thermal Profilng Using the 布巴克尔多項式, Editors: Nancy B. Devins and Jillian P. Ramos| url= https://www.novapublishers.com/catalog/product_info.php?products_id=26337&osCsid=| accessdate= 2011-08-04| archive-date= 2016-03-04| archive-url= https://web.archive.org/web/20160304050649/https://www.novapublishers.com/catalog/product_info.php?products_id=26337&osCsid=| dead-url= no}}
* [[NASA]]: USA-Physics Abstract Service Database
:{{citation|title=a New Parameter:. AN Abacus for Optimizing Pv-T Hybrid Solar Device Functional Materials Using the 布巴克尔 多項式 Expansion Scheme|url=http://adsabs.harvard.edu/abs/2009MPLB...23.2179F|accessdate=2011-08-04|archive-date=2017-05-24|archive-url=https://web.archive.org/web/20170524161231/http://adsabs.harvard.edu/abs/2009MPLB...23.2179F|dead-url=no}} and [http://adsabs.harvard.edu/cgi-bin/nph-data_query?doi=10.1007/s00231-009-0493-x&db_key=PHY&link_type=ABSTRACT&high=499df58f2b32224]
* [[La Presse]]
:{{citation|title=De Khawarizmi à Euler|journal=La Presse Magazine|date=January 9, 2008|url=http://www.lapresse.tn/index.php?opt=15&categ=4&news=63764|accessdate=2011-08-04|archive-date=2012-03-29|archive-url=https://web.archive.org/web/20120329125429/http://www.lapresse.tn/index.php?opt=15&categ=4&news=63764|dead-url=no}} {{fr}}, vedi anche [https://web.archive.org/web/20110717112823/http://www.tunisie7arts.com/?nomPage=suite&newsid=555 tunisie7arts.com]
:{{citation|title=Le polynôme de 布巴克尔|journal=La Presse Magazine|date=April 22, 2007|issue=1019|page=6|url=http://www.lapresse.tn/pdf/magazine/2007-04-23_weekend22-04-2007.pdf}}{{dead link|date=2017年12月 |bot=InternetArchiveBot |fix-attempted=yes }} {{fr}} 
* [[John Wiley & Sons]]
: {{citation|title=A new polynomial sequence... The 布巴克尔 Polynomials|journal=Numerical Methods for Partial Differential Equations NMPDE|url=http://www3.interscience.wiley.com/journal/120747691/abstract}}{{Dead link|date=2019年4月 |bot=InternetArchiveBot |fix-attempted=yes }} 
* [[AIAA]] [[American Institute of Aeronautics and Astronautics]]''', Inc.
:{{citation|title=Solution to Heat Equation Using 布巴克尔 Polynomials|journal=J. of Thermophysics and Heat Transfer|url=http://pdf.aiaa.org/jaPreview/JTHT/2009/PVJA40216.pdf}}{{dead link|date=2017年12月 |bot=InternetArchiveBot |fix-attempted=yes }}
* [[ASME]] [[American Society of Mechanical Engineers]]''', Inc.
:{{citation|title=布巴克尔 多項式 Weak Solutions to a Robin Boundary Conditioned Dynamic-State Heat Transfer Problem |journal=International Journal of Heat  Transfer |url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JHTRAO000131000011111305000001&idtype=cvips&gifs=yes}}
* [[WS]] World Scientific Publishing Co Pte Ltd
: {{citation|title=AMLOUK–布巴克尔 EXPANSIVITY..USING 布巴克尔 多項式 |journal=Functional Materials Letter|url=http://www.worldscinet.com/fml/02/0201/S1793604709000533.html}}
* Pubblicazioni accademiche
:{{citation|title=A new polynomial sequence ...The 布巴克尔 Polynomials|journal=International Journal of Applied Mathematics|url=http://www.diogenes.bg/ijam/|accessdate=2011-08-04|archive-date=2021-04-05|archive-url=https://web.archive.org/web/20210405002131/http://www.diogenes.bg/ijam/|dead-url=no}}
* 其他文件:
: {{citation|title=.  |journal=Journal of Crystal Growth|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWR-4VH4DK4-1&_user=10&_coverDate=05%2F31%2F2009&_alid=940174383&_rdoc=3&_fmt=high&_orig=search&_cdi=5569&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7c9d97ac48bb3c6519ca159d1a02a9afcitation}} 
:{{citation|title=.  ||journal=Current appied physics |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W7T-4S1C2P8-1&_user=10&_coverDate=01%2F31%2F2009&_alid=940174383&_rdoc=1&_fmt=high&_orig=search&_cdi=6635&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=5826e9cec07723dbee0f6e273f7ee6ca}}
: [http://www.sciencedirect.com Sciendedirect.com] {{Wayback|url=http://www.sciencedirect.com/ |date=20180511125310 }}: {{citation|title=.  |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W7T-4VBDKKB-3&_user=10&_coverDate=09%2F30%2F2009&_alid=940174383&_rdoc=8&_fmt=high&_orig=search&_cdi=6635&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=90e4fd3ad002b2476c4759376ae2f95d}}, {{citation|title=.  |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W7T-4SP3SRJ-2&_user=10&_coverDate=05%2F31%2F2009&_alid=940174383&_rdoc=10&_fmt=high&_orig=search&_cdi=6635&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=87d032acfae22c05bf14cc42cce8857e}},  {{citation|title=.  |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W7T-4WH2M2Y-1&_user=10&_coverDate=06%2F11%2F2009&_alid=940174383&_rdoc=2&_fmt=high&_orig=search&_cdi=6635&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=91419354fbb76cae9ef08046d7639f53}},{{citation|title=.  ||journal=Journal of Alloys and Compounds|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4V1FBRS-1&_user=10&_coverDate=05%2F27%2F2009&_alid=940174383&_rdoc=7&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=79329cf1a9f6436cac799d7c65b3c28b}},{{citation|title=.  ||url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4VB01SB-1&_user=10&_coverDate=06%2F24%2F2009&_alid=940174383&_rdoc=4&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=ffb80ca0d83aa3d648a1bf1277f54988}}{{citation|title=.  |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4W0SJYN-1&_user=10&_coverDate=07%2F29%2F2009&_alid=940174383&_rdoc=6&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=ac0398647b335b92d51b6f9c6594cd4c}},{{citation|title=.  ||url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4TYPJ82-1&_user=10&_coverDate=05%2F12%2F2009&_alid=940174383&_rdoc=9&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=185db3740cc3e081c10c21e1fa1eb9ea}}, {{citation|title=.  ||url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4WKTWVT-B&_user=10&_coverDate=06%2F24%2F2009&_alid=940174383&_rdoc=11&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=e9d6ea619298bd2dde5dcab3edc66878}}, {{citation|title=.  ||journal=Material Letters|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TX9-4VGF3P2-5&_user=10&_coverDate=05%2F15%2F2009&_alid=940174383&_rdoc=12&_fmt=high&_orig=search&_cdi=5585&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=2da24fe79b5884adccf45d7742b10d5f}}, {{citation|title=.  ||journal=Thermochimica Acta|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6THV-4TPX0JH-1&_user=10&_coverDate=01%2F15%2F2009&_alid=940174383&_rdoc=5&_fmt=high&_orig=search&_cdi=5292&_sort=r&_docanchor=&view=c&_ct=20&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=73c5255563fa35550f1528f7bb9140c0}}
* [[ENEA]] Ente Nazionale per le Energie Alternative
: {{citation|title=An attempt... using 布巴克尔 多項式|journal=International Journal of Heat and Technology|url=http://termserv.casaccia.enea.it/eurotherm/indexHT.html|accessdate=2011-08-04|archive-date=2015-06-29|archive-url=https://web.archive.org/web/20150629190757/http://termserv.casaccia.enea.it/eurotherm/indexHT.html|dead-url=no}}
* [[Taylor and Francis]]
: {{citation|title=Numerical Distribution of Temperature  During Welding Using 布巴克尔 Polynomials |journal= Numerical Heat Transfer, Part A, Applications|url=http://www.informaworld.com/smpp/content~db=all~content=a908590869?words=布巴克尔}}
* [[Università di San Pietroburgo]] 
: {{citation|title=Establishment of an Ordinary Generating Function and a Christoffel-Darboux Type First-Order Differential Equation for the Heat Equation Related 布巴克尔-Turki Polynomials|journal= Journal of Differential Equations and C.P.|url=http://www.neva.ru/journal/j/pdf/布巴克尔2.pdf}}
:{{citation|title=Some new properties of the applied-physics related 布巴克尔 polynomials|url=http://www.neva.ru/journal/j/pdf/zhao.pdf|accessdate=2011-08-04|archive-date=2012-02-12|archive-url=https://web.archive.org/web/20120212221032/http://www.neva.ru/journal/j/pdf/zhao.pdf|dead-url=no}}
* [http://ztgz.diytrade.com/sdp/179333/4/main-611341.html China National Publication Corp.] {{Wayback|url=http://ztgz.diytrade.com/sdp/179333/4/main-611341.html |date=20201126122829 }} [https://web.archive.org/web/20120331021327/http://www.zhongtu.com.cn/Article.aspx?Code=17895031&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021336%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D14726201&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021341%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D16998437&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021352%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D18008892&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021411%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D17043888&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021417%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D19317811&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021426%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15703410&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021433%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D20037626&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021455%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D20051040&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021536%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15675269&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021617%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15672825&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021638%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15237167&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021644%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15934518&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021651%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15679889&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021710%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15130669&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021716%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D17988075&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021723%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D19740627&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021737%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D20282958&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021804%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D19721268&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021813%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D17898561&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021823%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D16287761&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021947%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D18076766&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331021953%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D20319383&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331022012%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15322125&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331022049%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D16605334&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331022055%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D17045870&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331022101%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D16252317&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331022111%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D17898404&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331023110%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D16220799&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331023212%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15693933&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331023222%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D15363058&View=baokanarticle%5D%2C%5Bhttps%3A%2F%2Fweb.archive.org%2Fweb%2F20120331023246%2Fhttp%3A%2F%2Fwww.zhongtu.com.cn%2FArticle.aspx%3FCode%3D19995434&View=baokanarticle], and [https://web.archive.org/web/20120331023301/http://www.zhongtu.com.cn/Article.aspx?Code=15537181&View=baokanarticle].

* [http://www.proofwiki.org/wiki/Definition:Boubaker_Polynomials  ProofWiki] {{Wayback|url=http://www.proofwiki.org/wiki/Definition:Boubaker_Polynomials |date=20210127112735 }}
* [[:oeis:wiki/Boubaker polynomials|OEIS ENCYCLOPEDIA]]
* [http://www.proofwiki.org/wiki/Category:Boubaker_Polynomials CATHEGORIES:ProofWiki] {{Wayback|url=http://www.proofwiki.org/wiki/Category:Boubaker_Polynomials |date=20210125131447 }}
* [[:wikiversity:Boubaker Polynomials|WIKIVERSITY]]
* [http://planetmath.org/encyclopedia/12200.html PlanetMath ENCYCLOPEDIA]{{Dead link}}

[[Category:数学公式]]