布巴克爾多項式

在數學中，布巴克爾多項式 ^[1]有兩種常見定義。第一種是 :

{\begin{aligned}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{aligned}}

有時也會使用另一種定義,可以通過遞歸的方式進行定義。首先，規定前三個布巴克爾多項式為：

{\begin{aligned}B_{0}(x)&=1,\\B_{1}(x)&=x,\\B_{2}(x)&=x^{2}+2,\\\end{aligned}}

然後運用下面的遞推關係得到更高階的多項式。

{\begin{aligned}B_{m}(x)&=xB_{m-1}(x)-B_{m-2}(x)\quad {\text{, }}m>2.\end{aligned}}

布巴克爾多項式也可以用母函數表示 :

\sum _{n=0}^{\infty }{\tilde {B}}_{n}(x)t^{n}={\frac {1+3t^{2}}{1-t(t-x)}}.\,\!

產生了許多整數序列在On-Line Encyclopedia of Integer Sequences (OEIS)^[2] e PlanetMath （頁面存檔備份，存於網際網路檔案館）.

生成解

布巴克爾多項式的通解為 :

B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}

微分操作代表

布巴克爾多項式亦可記為 :

(x^{2}-1)(3nx^{2}+n-2)y{''}+3x(nx^{2}+3n-2)y{'}-n(3n^{2}x^{2}+n^{2}-6n+8)y=0\,

用途

布巴克爾多項式的用途:

參考文獻

^ Oyodum, OD; Awojoyogbe, OB; Dada, MK.; Magnuson, JN, On the earliest definition of the 布巴克尔 polynomials, Eur. Phys. J. Appl. Phys.: 21201–21202, [2011-08-04], （原始內容存檔於2011-10-05）
^ Sequences A135929 and A135936 by Neil J. A. Sloane, A137276 , Roger L. Bagula , Gary Adamson,A138476, A. Bannour, A137289, A136256, A136255 , R. L. Bagula 在 On-Line Encyclopedia of Integer Sequences
^ 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, [2011-08-04], （原始內容存檔於2012-10-11）
^ B. Dubey, T.G. Zhao, M. Jonsson, H. Rahmanov, A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using 布巴克尔 polynomial expansion scheme, Journal of Theoretical Biology (Elsevier), {{doi:10.1016/j.jtbi.2010.12.002}}
^ A. Milgeam, The stability of the 布巴克尔 polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem, Journal of Theoretical Biology (Elsevier), {{doi:10.1016/j.jtbi.2010.01.026}}
^ H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din, 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, Mathematical and Computer Modelling(Elsevier), {{doi:10.1016/j.mcm.2011.02.031 }}
^ A. Yildirim, The 布巴克尔 polynomials expansion scheme for solving nonlinear science problems, (PDF), The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 >, （原始內容 (PDF)存檔於2011年7月22日）
^ Paul Barry, Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The 布巴克尔 polynomials (PDF), Journal of Integer Sequences (JIS), [2011-08-04], （原始內容存檔 (PDF)於2018-04-28）
^ H. Koçak, Z. Dahong, A. Yildirim, A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the 布巴克尔 polynomials expansion scheme, Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer) ^{[永久失效連結]}
^ H. Koçak, Z. Dahong, A. Yildirim, Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the 布巴克尔 Polynomials Expansion Scheme (BPES), Indian Journal of Physics(Springer) ^{[永久失效連結]}
^ A. Belhadj, O. F. Onyango and N. Rozibaeva, 布巴克尔 Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model (PDF), Jornal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA) ^{[永久失效連結]}
^ D. H. Zhang, Study of a non-linear mechanical system using 布巴克尔 polynomials expansion scheme BPES, International Journal of Non-Linear Mechanics(Elsevier)
^ Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem, Modelling Nonlinear Bivariate Dependence Using the 布巴克尔 Polynomials Copula The 布巴克尔 polynomials (PDF), Studies in Nonlinear Sciences (SNS), [2011-08-04], （原始內容存檔 (PDF)於2020-09-22）
^ W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım, A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the 布巴克尔 polynomials expansion scheme, Journal of Structural Chemistry (Springer) ^{[永久失效連結]}
^ D. H. Zhang, L. Naing, The 布巴克尔 polynomials expansion scheme BPES for solving a standard boundary value problem (PDF), Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press), [2011-08-04], （原始內容存檔 (PDF)於2020-09-28）
^ A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and 布巴克尔 polynomials expansion, Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.), {{doi:10.1007/s10973-009-0094-4 }} ^{[永久失效連結]}
^ S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses, Cut-off cooling velocity profiling inside a keyhole model using the 布巴克尔 polynomials expansion scheme, Heat and Mass Transfer(Springer Berlin / Heidelberg), Volume 45, Number 10 / août 2009, pages:1247-1251 {{doi:10.1007/s00231-009-0493-x}} ^{[永久失效連結]}
^ S. Fridjine and M. Amlouk, A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING 布巴克尔 POLYNOMIALS EXPANSION SCHEME, Modern Physics Letters B ([ISSN: 0217-9849, by WS: World Scientific Publishing Co Pte Ltd] )
^ T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud, Limit and uniqueness of the 布巴克尔-Zhao polynomials single imaginary root sequence, International Journal of Mathematics and Computation, ISSN 0974-5718, （原始內容存檔於2011-08-13）
^ Luzon, A.; Moron, M., RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: 布巴克尔 POLYNOMIALS associated Riordan matrix (PDF)
^ B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan, Jami Vidyadhari ,Page 8 : 布巴克爾 polynomials
^ Kiliç Bülent, Erdal Bas, Page 7, Citation 27: 布巴克尔 polynomials (PDF). [2011-08-04]. （原始內容存檔 (PDF)於2018-04-20）.

外部連結

Encyclopedia of Physics Research: 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, [2011-08-04], （原始內容存檔於2016-03-04）
NASA: USA-Physics Abstract Service Database

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, [2011-08-04], （原始內容存檔於2017-05-24） and [1]

La Presse

De Khawarizmi à Euler, La Presse Magazine, January 9, 2008 [2011-08-04], （原始內容存檔於2012-03-29）（法文）, vedi anche tunisie7arts.com

Le polynôme de 布巴克尔 (PDF), La Presse Magazine, April 22, 2007, (1019): 6^{[永久失效連結]} （法文）

John Wiley & Sons

A new polynomial sequence... The 布巴克尔 Polynomials, Numerical Methods for Partial Differential Equations NMPDE^{[永久失效連結]}

AIAA American Institute of Aeronautics and Astronautics, Inc.

Solution to Heat Equation Using 布巴克尔 Polynomials (PDF), J. of Thermophysics and Heat Transfer^{[永久失效連結]}

ASME American Society of Mechanical Engineers, Inc.

布巴克尔多項式 Weak Solutions to a Robin Boundary Conditioned Dynamic-State Heat Transfer Problem, International Journal of Heat Transfer

WS World Scientific Publishing Co Pte Ltd

AMLOUK–布巴克尔 EXPANSIVITY..USING 布巴克尔多項式, Functional Materials Letter

Pubblicazioni accademiche

A new polynomial sequence ...The 布巴克尔 Polynomials, International Journal of Applied Mathematics, [2011-08-04], （原始內容存檔於2021-04-05）

其他文件:

., Journal of Crystal Growth

., Current appied physics

Sciendedirect.com （頁面存檔備份，存於網際網路檔案館）: ., ., .,., Journal of Alloys and Compounds,..,., ., ., Material Letters, ., Thermochimica Acta

ENEA Ente Nazionale per le Energie Alternative

An attempt... using 布巴克尔多項式, International Journal of Heat and Technology, [2011-08-04], （原始內容存檔於2015-06-29）

Taylor and Francis

Numerical Distribution of Temperature During Welding Using 布巴克尔 Polynomials, Numerical Heat Transfer, Part A, Applications

Università di San Pietroburgo

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 (PDF), Journal of Differential Equations and C.P.

Some new properties of the applied-physics related 布巴克尔 polynomials (PDF), [2011-08-04], （原始內容存檔 (PDF)於2012-02-12）

China National Publication Corp. （頁面存檔備份，存於網際網路檔案館） [2], and [3].

ProofWiki （頁面存檔備份，存於網際網路檔案館）
OEIS ENCYCLOPEDIA
CATHEGORIES:ProofWiki （頁面存檔備份，存於網際網路檔案館）
WIKIVERSITY
PlanetMath ENCYCLOPEDIA^{[失效連結]}

[1] Oyodum, OD; Awojoyogbe, OB; Dada, MK.; Magnuson, JN, On the earliest definition of the 布巴克尔 polynomials, Eur. Phys. J. Appl. Phys.: 21201–21202, [2011-08-04], （原始內容存檔於2011-10-05）

[2] Sequences A135929 and A135936 by Neil J. A. Sloane, A137276 , Roger L. Bagula , Gary Adamson,A138476, A. Bannour, A137289, A136256, A136255 , R. L. Bagula 在 On-Line Encyclopedia of Integer Sequences

[3] 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, [2011-08-04], （原始內容存檔於2012-10-11）

[4] B. Dubey, T.G. Zhao, M. Jonsson, H. Rahmanov, A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using 布巴克尔 polynomial expansion scheme, Journal of Theoretical Biology (Elsevier), {{doi:10.1016/j.jtbi.2010.12.002}}

[5] A. Milgeam, The stability of the 布巴克尔 polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem, Journal of Theoretical Biology (Elsevier), {{doi:10.1016/j.jtbi.2010.01.026}}

[6] H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din, 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, Mathematical and Computer Modelling(Elsevier), {{doi:10.1016/j.mcm.2011.02.031 }}

[7] A. Yildirim, The 布巴克尔 polynomials expansion scheme for solving nonlinear science problems, (PDF), The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 >, （原始內容 (PDF)存檔於2011年7月22日）

[8] Paul Barry, Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The 布巴克尔 polynomials (PDF), Journal of Integer Sequences (JIS), [2011-08-04], （原始內容存檔 (PDF)於2018-04-28）

[9] H. Koçak, Z. Dahong, A. Yildirim, A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the 布巴克尔 polynomials expansion scheme, Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer) ^{[永久失效連結]}

[10] H. Koçak, Z. Dahong, A. Yildirim, Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the 布巴克尔 Polynomials Expansion Scheme (BPES), Indian Journal of Physics(Springer) ^{[永久失效連結]}

[11] A. Belhadj, O. F. Onyango and N. Rozibaeva, 布巴克尔 Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model (PDF), Jornal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA) ^{[永久失效連結]}

[12] D. H. Zhang, Study of a non-linear mechanical system using 布巴克尔 polynomials expansion scheme BPES, International Journal of Non-Linear Mechanics(Elsevier)

[13] Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem, Modelling Nonlinear Bivariate Dependence Using the 布巴克尔 Polynomials Copula The 布巴克尔 polynomials (PDF), Studies in Nonlinear Sciences (SNS), [2011-08-04], （原始內容存檔 (PDF)於2020-09-22）

[14] W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım, A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the 布巴克尔 polynomials expansion scheme, Journal of Structural Chemistry (Springer) ^{[永久失效連結]}

[15] D. H. Zhang, L. Naing, The 布巴克尔 polynomials expansion scheme BPES for solving a standard boundary value problem (PDF), Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press), [2011-08-04], （原始內容存檔 (PDF)於2020-09-28）

[16] A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and 布巴克尔 polynomials expansion, Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.), {{doi:10.1007/s10973-009-0094-4 }} ^{[永久失效連結]}

[17] S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses, Cut-off cooling velocity profiling inside a keyhole model using the 布巴克尔 polynomials expansion scheme, Heat and Mass Transfer(Springer Berlin / Heidelberg), Volume 45, Number 10 / août 2009, pages:1247-1251 {{doi:10.1007/s00231-009-0493-x}} ^{[永久失效連結]}

[18] S. Fridjine and M. Amlouk, A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING 布巴克尔 POLYNOMIALS EXPANSION SCHEME, Modern Physics Letters B ([ISSN: 0217-9849, by WS: World Scientific Publishing Co Pte Ltd] )

[19] T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud, Limit and uniqueness of the 布巴克尔-Zhao polynomials single imaginary root sequence, International Journal of Mathematics and Computation, ISSN 0974-5718, （原始內容存檔於2011-08-13）

[20] Luzon, A.; Moron, M., RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: 布巴克尔 POLYNOMIALS associated Riordan matrix (PDF)

[21] B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan, Jami Vidyadhari ,Page 8 : 布巴克爾 polynomials

[22] Kiliç Bülent, Erdal Bas, Page 7, Citation 27: 布巴克尔 polynomials (PDF). [2011-08-04]. （原始內容存檔 (PDF)於2018-04-20）.

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