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Sister Celine's Polynomials
In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by . They include Legendre polynomials, Jacobi polynomials, and Bateman polynomials as special cases. References Polynomials {{polynomial-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Religious Sister
A religious sister (abbreviated ''Sr.'' or Sist.) in the Catholic Church is a woman who has taken public vows in a religious institute dedicated to apostolic works, as distinguished from a nun who lives a cloistered monastic life dedicated to prayer. Both nuns and sisters use the term "sister" as a form of address. The ''HarperCollins Encyclopedia of Catholicism'' (1995) defines as "congregations of sisters institutes of women who profess the simple vows of poverty, chastity, and obedience, live a common life, and are engaged in ministering to the needs of society." As William Saunders writes: "When bound by simple vows, a woman is a sister, not a nun, and thereby called 'sister'. Nuns recite the Liturgy of the Hours or Divine Office in common ... ndlive a contemplative, cloistered life in a monastery ... behind the 'papal enclosure'. Nuns are permitted to leave the cloister only under special circumstances and with the proper permission." History Until the 16th century, relig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergeometric Polynomial
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of Classical orthogonal polynomials, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval [-1,1]. The Gegenbauer polynomials, and thus also the Legendre polynomials, Legendre, Zernike polynomials, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer symbol, Pochhammer's symbol (for the rising factorial). In this case, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bateman Polynomials
In mathematics, the Bateman polynomials are a family ''F''''n'' of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by . Bateman polynomials can be defined by the relation :F_n\left(\frac\right)\operatorname(x) = \operatorname(x)P_n(\tanh(x)). where ''P''''n'' is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by :F_n(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end; 1\right). generalized the Bateman polynomials to polynomials ''F'' with :F_n^m\left(\frac\right)\operatorname^(x) = \operatorname^(x)P_n(\tanh(x)) These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely :F_n^m(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end; 1\right). showed that the polynomials ''Q''''n'' studied by , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely : Q_n(x)=(-1)^n2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
