Hahn, W.: Uber Orthogonalpolynome, die q-Differenzengleichungen genugen. Math. Nachr. 2, 1949, 4--34.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... exactly n, given by (2) is a family of classical q orthogonal polynomials if it is the solution of a q difference equation of the type oe(x) D q D 1 q y(x) x) D q y(x) q;n y(x) 0 ; 6) where D q f(x) f(qx) Gamma f(x) q Gamma 1)x ; q 6= 1; denotes the q difference operator [7], and oe(x) ax 2 bx c and (x) dx e are again polynomials of at most second and of first order, respectively. By equating the coefficients of x n in (6) one gets q;n = Gammaa [n] 1 q [n Gamma 1] q Gamma d [n] q ; 7) where the abbreviation [n] q = 1 Gamma q n 1 Gamma q ....

....of the hypergeometric function hypergeom(upper,lower,z) with summation variable x, see [9] Hahn polynomials are not accessible with Koornwinder Swarttouw s rec2ortho. 17 5 Classical q Orthogonal Polynomials In this section, we consider the same problem for classical q orthogonal polynomials ([7], 12] see e.g. 8] The classical q orthogonal polynomials are given by a q difference equation (6) These polynomials can be classified similarly as in the continuous and discrete cases according to the functions oe(x) and (x) up to linear transformations the classical q orthogonal ....

Hahn, W.: Uber Orthogonalpolynome, die q-Differenzengleichungen genugen. Math. Nachr. 2, 1949, 4--34.

....problem, connection problem AMS Subject classification: Primary 33C25. Secondary 33D45 Numerical Algorithms 23 (2000) 31 50 1. Introduction Let fp k (x; q)g be any system of basic hypergeometric orthogonal polynomials, belonging to the q Hahn class, i.e. q classical orthogonal polynomials [4,6,12]. Corresponding author 2 S. Lewanowicz et al. Recurrence relations for the Fourier coefficients Given a function f(x) satisfying a linear q difference equation of the form P f(x) j t X i=0 w i (x)D q i f(x) g(x) 1.1) where D q is the q derivative operator (see, 2.5) below) and w ....

....f(q k )q k : 2.11) Using (2.8) we obtain the following formula of q integration by parts: Z b a g(x)D q f(x) d q x = f(x)E q Gamma1 g(x) fi fi fi b a Gamma q Gamma1 Z b a f(x)D q Gamma1 g(x) d q x: 2.12) 3. q classical orthogonal polynomials 3.1. Basic properties q Hahn class [4,6] contains all the families of basic hypergeometric orthogonal polynomials fp k (x; q)g such that for any k, p k (x; q) satisfies a second order q difference equation L k p k (x; q) j n oe(x)D q D q Gamma1 (x)D q k I o p k (x; q) 0; 3.1) S. Lewanowicz et al. Recurrence relations ....

W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949) 4--34.

....in x4 (see Theorems 4.2 and 4.5) Some illustrative examples are given in x5. 3. Properties of the little q Jacobi polynomials 3.1. Basic properties. The little q Jacobi polynomials fp k (x; ff; fijq)g (cf. 2. 1) belong to a large q Hahn class of basic hypergeometric orthogonal polynomials [2, 4]. The orthogonality relation for these polynomials reads Z 1 0 (x)p k (x)p l (x) d q x = h k ffi kl ; 3.1) where pm (x) j pm (x; ff; fijq) m = 0; 1; 0 ffq 1, fiq 1, x) x a (qx; q) 1 (fiqx; q) 1 (ff = q a ) 3.2) h k : q k(k a) q; q) k (ffq; q) k (fiq; q) k (fffiq; ....

W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949) 4--34.

....only possible choice and explicitly mentions that it sufficies to have q 1. StieltjesWigert polynomials are also limit cases of q Laguerre polynomials, as was shown by Askey [3] These q Laguerre polynomials are also corresponding to an indeterminate moment problem and they were studied by Hahn [15] and Moak [31] 11 Two other examples are mentioned, the first on p. 695 with Psi 0 (u) 0;1[ u) h 1 sin(u 1=4 ) i e Gammau 1=4 ; Gamma1 1; for which oe n = 4(4n 3) and the second on p. 707 Psi 0 (u) 0;1[ u)u a Gamma1 e Gammabu ; a 0; b 0; ....

W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949), 4--34.

.... Gammabd Gamma1 qx; q) 1 ffi q x: These polynomials are not really different from those defined by (3.5.1) since we have Pn (x; a; b; c; d; q) Pn (ac Gamma1 qx; a; b; Gammaac Gamma1 d; q) and Pn (x; a; b; c; q) Pn (x; a; b; aq; Gammacq; q) References. 8] 10] 25] 114] [121], 127] 140] 142] 160] 163] 176] 180] 181] 212] Special case 3.5.1 Big q Legendre Definition. The big q Legendre polynomials are big q Jacobi polynomials with a = b = 1 : Pn (x; c; q) 3 OE 2 q Gamman ; q n 1 ; x q; cq fi fi fi fi q; q : 3.5.7) Orthogonality. ....

....(3.6.5) can also be written as : 2 OE 1 q x GammaN ; 0 fiq fi fi fi fi q; q Gammax t 1 OE 1 q Gammax ffq fi fi fi fi q; ffqt = N X n=0 (q GammaN ; q) n (fiq; q; q) n Qn (q Gammax ; ff; fi; N jq)t n : References. 10] 25] 43] 45] 88] 111] 114] [121], 142] 145] 158] 160] 180] 197] 215] 217] 218] 3.7 Dual q Hahn Definition. Rn ( x) fl; ffi; N jq) 3 OE 2 q Gamman ; q Gammax ; fl ffi q x 1 flq; q GammaN fi fi fi fi q; q ; n = 0; 1; 2; N; 3.7.1) where (x) q Gammax fl ffi q x 1 ....

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W. Hahn : Uber Orthogonalpolynome, die q-Differenzengleichungen genugen. Mathematische Nachrichten 2, 1949, 4-34.

....(2:11) Here the definition of orthogonal polynomials is relaxed somewhat. We include the possibility that the degree n of the polynomials p n only takes the values n = 0; 1; N and that the orthogonality is with respect to a positive measure having support on a set of N 1 points. Hahn [16] studied the q analogue of this classification (cf. x2.5) and he pointed out how the polynomials satisfying (2.11) come out as limit cases for q 1 of his classification. The generic case for this classification is given by the Hahn polynomials Q n (x; ff; fi; N) 3 F 2 Gamman; n ff fi ....

....transitions between them, form the Askey tableau (or scheme or chart) of hypergeometric orthogonal polynomials. See Askey Wilson [8, Appendix] Labelle [25] or Table 1. See also [20] for group theoretic interpretations. 2.3. Big q Jacobi polynomials. These polynomials were hinted at by Hahn [16] and explicitly introduced by Andrews Askey [4] Here we will show how their basic properties can be derived from a suitable pair of shift operators. We keep the convention of x1 that 0 q 1. First we introduce q integration by parts. This will involve backward and forward q derivatives: D ....

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W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949), 4--34, 379.

.... equation is important for some connection coefficient problems [10] and also in order to represent finite modifications inside the Jacobi matrices of the q classical starting family [15] q classical orthogonal polynomials involved in this work belong to the q Hahn class as introduced by Hahn [8]. They are represented by the basic hypergeometric series appearing 1 E mail: foupouag syfed.bj.refer.org. Fax: 229 21 25 25. Research supportedby: Deutscher Akademischer Austauschdienst (DAAD) Preprint submitted to Elsevier Science 2 June at the level 3 OE 2 and not at the level 4 OE 3 of ....

....3 OE 2 and not at the level 4 OE 3 of the Askey Wilson orthogonal polynomials. The orthogonality weight ae (defined in the interval I) for q classical orthogonal polynomials is defined by a Pearson type q difference equation D q (oe ae) ae; 1) where the q difference operator D q is defined [8] by D q f(x) f(qx) Gamma f(x) q Gamma 1)x ; x 6= 0; 0 q 1; 2) and D q f(0) f 0 (0) by continuity, provided that f 0 (0) exists. oe is a polynomial of degree at most two and is polynomial of degree one. The monic polynomials Pn (x; q) orthogonal with respect to ae satisfy the ....

W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949), 4- 34.

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H. W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949), pp. 4--34.

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