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Askey Scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic orthogonal polynomials. Askey scheme for hypergeometric orthogonal polynomials give the following version of the Askey scheme: ;_4F_3(4): Wilson , Racah ;_3F_2(3): Continuous dual Hahn , Continuous Hahn , Hahn , dual Hahn ;_2F_1(2): Meixner–Pollaczek , Jacobi , Pseudo Jacobi , Meixner , Krawtchouk ;_2F_0(1)\ \ / \ \ _1F_1(1): Laguerre , Bessel , Charlier ;_2F_0(0): Hermite Here _pF_q(n) indicates a hypergeometric series representation with n parameters Askey scheme for basic hypergeometric orthogonal polynomials give the following scheme for basic hypergeometric orthogonal polynomials: ;4\phi3: Askey–Wilson , q-Racah ;3\phi2: Continuous dual q-Hahn , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal 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. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Racah Polynomials
In mathematics, the ''q''-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ... by :p_n(q^+q^cd;a,b,c,d;q) = _4\phi_3\left begin q^ &abq^&q^&q^cd\\ aq&bdq&cq\\ \end;q;q\right/math> They are sometimes given with changes of variables as :W_n(x;a,b,c,N;q) = _4\phi_3\left begin q^ &abq^&q^&cq^\\ aq&bcq&q^\\ \end;q;q\right/math> Relation to other polynomials q-Racah polynomials→Racah polynomials References * * * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Krawtchouk Polynomials
In mathematics, the ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme . give a detailed list of their properties. showed that the ''q''-Krawtchouk polynomials are spherical functions for 3 different Chevalley groups over finite fields, and showed that they are related to representations of the quantum group SU(2). Definition The polynomials are given in terms of basic hypergeometric functions by :K_n(q^;p,N;q)=_3\phi_2\left begin q^,q^,-pq^n\\ q^,0\end ;q,q\right\quad n=0,1,2,...,N. See also * affine q-Krawtchouk polynomials * dual q-Krawtchouk polynomials In mathematics, the dual ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hyperg ... * quantum q-Krawtchouk polynomials Sources * * * * * {{refend Orthogonal polynomials Q-analogs Special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Q-Krawtchouk Polynomials
In mathematics, the quantum ''q''-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by :K_n^(q^;p,N;q)=_2\phi_1\left begin q^,q^\\ q^\end ;q;pq^\rightqquad n=0,1,2,...,N. References * * * *{{dlmf, id=18, title=Chapter 18 Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Meixner Polynomials
In mathematics, the ''q''-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by :M_n(q^;b,c;q)=_2\phi_1\left begin q^,q^\\ bq\end ;q,-\frac\right References * * * *{{cite thesis , last=Sadjang , first=Patrick Njionou , title=Moments of Classical Orthogonal Polynomials , type=Ph.D. , publisher=Universität Kassel , citeseerx=10.1.1.643.3896 Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Little Q-Jacobi Polynomials
In mathematics, the little ''q''-Jacobi polynomials ''p''''n''(''x'';''a'',''b'';''q'') are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definition The little ''q''-Jacobi polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by :\displaystyle p_n(x;a,b;q) = _2\phi_1(q^,abq^;aq;q,xq) Gallery The following are a set of animation plots for Little ''q''-Jacobi polynomials, with varying q; three density plots of imaginary, real and modulus in complex space; three set of complex 3D plots of imaginary, real and modulus of the said polynomials. References * * * *{{dlmf, id=18, first=Tom H. , last=Koornwinder, first2=Rod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Q-Laguerre Polynomials
In mathematics, the big ''q''-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s and the q-Pochhammer symbol by P_n(x;a,b;q)=\frac_2\phi_1\left(q^,aqx^;aq;q,\frac\right) Relation to other polynomials Big q-Laguerre polynomials→Laguerre polynomials References * * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Q-Jacobi Polynomials
In mathematics, the continuous ''q''-Jacobi polynomials ''P''(''x'', ''q''), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ... by :P_n^(x;q)=\frac_4\phi_3\left begin q^,q^,q^,q^\\ q^,-q^,-q^\end ;q,q\rightqquad x=\cos\,\theta. References * * * * * *{{cite thesis , last=Sadjang , first=Patrick Njionou , title=Moments of Classical Orthogonal Polynomials , type=Ph.D. , publisher=Universität Kassel , citeseerx=10.1.1.643.3896 Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Meixner–Pollaczek Polynomials
In mathematics, the ''q''-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ... by ：Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p 460,Springer : P_(x;a\mid q) = a^ e^ \frac_3\phi_2(q^, ae^, ae^; a^2, 0 \mid q; q),\quad x=\cos(\theta+\phi). References * * * {{DEFAULTSORT:Q-Meixner-Pollaczek polynomials Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Salam–Chihara Polynomials
In mathematics, the Al-Salam–Chihara polynomials ''Q''''n''(''x'';''a'',''b'';''q'') are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of the properties of Al-Salam–Chihara polynomials. Definition The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ... by : Q_n(x;a,b;q) = \frac_3\phi_2(q^, ae^, ae^; ab,0; q,q) where ''x'' = cos(θ). References * * * * Further reading * Bryc, W., Matysiak, W., & Szabłowski, P. (2005). Probabilistic aspects of Al-Salam–Chihara polynomials. Proceedings of the American Mathematical Society, 133(4), 1127-1134. * Floreanini, R., LeTourneux, J., & Vinet, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Q-Hahn Polynomials
In mathematics, the dual ''q''-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s. R_n(q^+\gamma\delta q^,\gamma,\delta,N, q)=_3\phi_2\left begin q^,q^,\gamma\delta q^\\ \gamma q,q^\end ;q,q\right\quad n=0,1,2,...,N References * * * * * Orthogonal polynomials Q-analogs Special hypergeometric functions {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Hahn Polynomials
In mathematics, the ''q''-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions by :Q_n(q^;a,b,N;q)=_3\phi_2\left begin q^,abq^,q^\\ aq,q^\end ;q,q\right Relation to other polynomials q-Hahn polynomials→ Quantum q-Krawtchouk polynomials： \lim_Q_(q^;a;p,N, q)=K_^(q^;p,N;q) q-Hahn polynomials→ Hahn polynomials make the substitution\alpha=q^,\beta=q^ into definition of q-Hahn polynomials, and find the limit q→1, we obtain :_3F_2(-n,\alpha+\beta+n+1,-x,\alpha+1,-N,1)，which is exactly Hahn polynomials In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spec .... References * * * *{{cite journal, la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
