布巴克尔多項式

在數學中，布巴克尔 多項式 有两种常见定义。第一种是 : 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 − 3 x B 6 ( x ) = x 6 − 2 x 4 − 3 x 2 + 2 B 7 ( x ) = x 7 − 3 x 5 − 2 x 3 + 5 x B 8 ( x ) = x 8 − 4 x 6 + 8 x 2 − 2 B 9 ( x ) = x 9 − 5 x 7 + 3 x 5 + 10 x 3 − 7 x ⋮ {\displaystyle {\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： B 0 ( x ) = 1 , B 1 ( x ) = x , B 2 ( x ) = x 2 + 2 , {\displaystyle {\begin{aligned}B_{0}(x)&=1,\\B_{1}(x)&=x,\\B_{2}(x)&=x^{2}+2,\\\end{aligned}}} 然后运用下面的递推关系得到更高阶的多项式。 B m ( x ) = x B m − 1 ( x ) − B m − 2 ( x ) ,  m > 2. {\displaystyle {\begin{aligned}B_{m}(x)&=xB_{m-1}(x)-B_{m-2}(x)\quad {\text{, }}m>2.\end{aligned}}} 布巴克尔 多項式也可以用母函数表示 : ∑ n = 0 ∞ B ~ n ( x ) t n = 1 + 3 t 2 1 − t ( t − x ) . {\displaystyle \sum _{n=0}^{\infty }{\tilde {B}}_{n}(x)t^{n}={\frac {1+3t^{2}}{1-t(t-x)}}.\,\!} 产生了许多整数序列在On-Line Encyclopedia of Integer Sequences (OEIS) e PlanetMath （页面存档备份，存于互联网档案馆）. 布巴克尔 多項式的通解為 : B n ( x ) = ∑ p = 0 ⌊ n / 2 ⌋ n − 4 p n − p ( n − p p ) ( − 1 ) p x n − 2 p {\displaystyle B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}} 布巴克尔 多項式亦可記為 : ( x 2 − 1 ) ( 3 n x 2 + n − 2 ) y ″ + 3 x ( n x 2 + 3 n − 2 ) y ′ − n ( 3 n 2 x 2 + n 2 − 6 n + 8 ) y = 0 {\displaystyle (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\,} 布巴克尔 多項式的 用途: 低温 生物学 动态系统 非线性系统 近似论 热力学 力学 水文地理学 分子动态 基本数学 热量测定 生物物理学 光电学 复杂分析 矩阵分析 加密 代数 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], （原始内容存档于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], （原始内容存档于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 >, （原始内容 (PDF)存档于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], （原始内容存档 (PDF)于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], （原始内容存档 (PDF)于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], （原始内容存档 (PDF)于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, （原始内容存档于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 : 布巴克尔 polynomials Kiliç Bülent, Erdal Bas, Page 7, Citation 27: 布巴克尔 polynomials (PDF). [2011-08-04]. （原始内容存档 (PDF)于2018-04-20）.  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], （原始内容存档于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], （原始内容存档于2017-05-24）  and [1] La Presse De Khawarizmi à Euler, La Presse Magazine, January 9, 2008 [2011-08-04], （原始内容存档于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], （原始内容存档于2021-04-05）  其他文件: ., Journal of Crystal Growth  ., Current appied physics  Sciendedirect.com （页面存档备份，存于互联网档案馆）: . , . , . ,., 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], （原始内容存档于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], （原始内容存档 (PDF)于2012-02-12）  China National Publication Corp. （页面存档备份，存于互联网档案馆） [2], and [3]. 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