Big Q-Legendre Polynomials
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Big Q-Legendre Polynomials
In mathematics, the big q-Legendre polynomials are an orthogonal polynomials, orthogonal family of polynomials defined in terms of basic hypergeometric function, Heine's basic hypergeometric series asRoelof Koekoek, Peter Lesky, Rene Swattouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, p 443, Springer :\displaystyle P_n(x;c;q)=_3\phi_2(q^,q^,x;q,cq;q,q) . They obey the orthogonality relation : \int_^q P_m(x;c;q)P_n(x;c;q) \, dx=q(1-c)\frac\frac(-cq^2)^n q^\delta_ and have the limiting behavior :\displaystyle\lim_ P_n(x;0;q)=P_n(2x-1) where P_n is the nth Legendre polynomial. References Q-analogs Orthogonal polynomials {{polynomial-stub ...
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Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ...
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