D.S.Moak. The q-analogue of the Laguerre polynomials. J.Math.Anal.Appl., Vol.81, No.1, pp.20-47, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n L ) H 2n 1 (x) 1) 2n 1 n xL (2.1) where n (z) 1 F 1 1 ( n) k ( 1) k k ; 2.2) and (a) n = a n) a) n = 0; 1; 2; is the shifted factorial. It is also known that a q extension of the Laguerre polynomials n (x; q) de ned [17] [19] as 1 1 = 1 ; x (2.3) satis es two kinds of orthogonality relations, an absolutely continuous one and a discrete one. The former orthogonality relation is given by E q (x) m (x; q) L n (x; q) dx = d n ( mn ; 1 ; 2.4) where E q (x) is the ....

.... does not contradict the general theory of orthogonal polynomials [22, 23] The point is that the Hamburger moment problem associated with f h n (x; q)g is indeterminate, and therefore they are orthogonal with respect to an in nite class of weight functions, both continuous and discrete ones [19], 24] 26] C.Berg studied in [27] some families of discrete solutions to indeterminate moment problems and showed how they can be used to generate absolutely continuous solutions to the same moment problems. In particular, C.Berg derived in [27] the continuous weight function w(x) 1=E q ) ....

D.S.Moak. The q-analogue of the Laguerre polynomials. J.Math.Anal.Appl., Vol.81, No.1, pp.20-47, 1981.

....and beta functions, and furthermore that some of these formulas can be interpreted as giving solutions to an indeterminate moment problem. The corresponding orthogonal polynomials are now called q Laguerre polynomials, and they have been considered by a number of authors, see [3] 14] 17] [21]. Later Askey and Roy, cf. 5] evaluated the following integral c ( Gammaq a c t; Gammaq b 1 Gammac =t; q) 1 Gamma q (a) Gamma q (b) Gamma q (a b) 1.9) where Gamma q is Jackson s q extension of the gamma function Gamma q (x) 1 Gamma q) 1 Gammax : 1.10) Other ....

....consider a 2 ]q; 1] The function (3.5) satisfies L(aq; p) q p : c) The vector integral 1 a;p is equal to the measure (3.3) in the case c = 0. q Laguerre We follow the normalization of the q Laguerre polynomials given in [18] i.e. we remove the factor (1 Gamma q) used by Moak in [21]. They belong to an indeterminate moment problem with moments s n (ff; q) q Gammaffn Gamma ( n 1 2 ) q) n ; 4.1) when 0 q 1, ff Gamma1, as pointed out by Askey in [2] More information about this indeterminate moment problem can be found in [17] In [21] one finds the following ....

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D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20--47.

....determined by being restricted to the countable set f Sigmaq k 1 2 jk 2 Zg. We shall answer this question in the negative. We also treat the same question for the Stieltjes Wigert polynomials [21] which are associated with the log normal distribution, and for the q Laguerre polynomials, cf. [17]. The question by Ruffing was motivated by the study of a q deformation of the harmonic oscillator, see [16] 18] 19] 10] We shall make use of the following elementary result, which is a special case of a theorem of Naimark, cf. 1, p.47] characterizing the extreme points of the convex set V ....

....( Gammacq k ; q) 1 cq k (3.12) for c 0; 0 q 1 and A(c) 1 X k= Gamma1 q k=2 ( Gammacq k ; q) 1 : They all have the moments s n ( q; c) p q; q) n q Gamma 1 2 n 2 : 3. 13) 4 The q Laguerre polynomials The q Laguerre polynomials L (ff) n (x; q) are studied in [17]. We restrict the parameters to 0 q 1; ff Gamma1, and in this case they are associated with an indeterminate Stieltjes moment problem with the moment sequence s n (ff; q) 1 Gamma q) Gamman q Gammaffn Gamma ( n 1 2 ) q ff 1 ; q) n : 4.1) See [11] for a calculation of the ....

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D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20--47.

....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; 0; for which ....

D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20--47.

....( Gammax; Gammaq Gammaff ; q) q; q) n = L (ff) n (x; q) Since the Stieltjes and Hamburger moment problems corresponding to the q Laguerre polynomials are indeterminate there exist many different weight functions. References. 8] 10] 30] 31] 45] 75] 77] 78] 96] 114] [177]. 3.22 Alternative q Charlier Definition. Kn (x; a; q) 2 OE 1 q Gamman ; Gammaaq n 0 fi fi fi fi q; qx (3.22.1) xq 1 Gamman ; q) n Delta 1 OE 1 q Gamman xq 1 Gamman fi fi fi fi q; Gammaaxq n 1 = Gammaaxq n ) n Delta 2 OE 1 q Gamman ; x ....

D.S. Moak : The q-analogue of the Laguerre polynomials. Journal of Mathematical Analysis and Applications 81, 1981, 20-47.

....(q; q) m (fi q Gammam=2 Gamma1=2 ) n Gammam (q; q) n L (n Gammam) m Gammafffi 2 q Gamma1 Gammam 1 Gamma q ; q =q Gammam(m 1) 4 n(n 1) 4 ff 1 Gamma q m Gamman L (m Gamman) n Gammafffi 2 q Gamma1 Gammam 1 Gamma q ; q (7. 4) where [14, 23, 24] (7.5) L (fl) n (x; q) q fl 1 ; q) n (q; q) n 1 OE 1 q Gamman q fl 1 ; q; Gammaxq n fl 1 is a q Laguerre polynomial. Note that L (n Gammam) m (x; q) Gammax) m Gamman (q;q)n (q;q)m L (m Gamman) n (x; q) We can obtain the matrix elements of the operator e q (fiE ....

....there exist many inner products in addition to (7.18) for which the Laguerre polynomials fL (ff) k : k = 0; 1; Delta Delta Delta g are orthogonal. Indeed one can use a technique analogous to the derivation of (3. 10) to obtain a family of orthogonality relations with discrete weight functions, [23, 24], 14, page 194] As a second example of the computation of matrix elements we consider the operator e q (fiE )e q (ffE Gamma ) 7.22) e q (fiE )e q (ffE Gamma )f m = 1 X n=0 Anm (ff; fi)f n : The result is Anm (ff; fi) q (m Gamman) m n 1) 4 (fi ) n Gammam (q; q) n Gammam 2 OE 1 ....

D.S. Moak (1981), The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81, 20--47.

....( q q Gamma x ) of the Euclidean Lie algebra (0; 0) Thus, 0; q Gamma1 ) Omega # ; Gamma q Gamma1 ; 0) Z 1 0 Phi( q q Gamma x ) ae(x) dx: 28) The functions H s n (x; t) are the Clebsch Gordan coefficients for this decomposition. However, as is well known [16, 20], the measure for which the qLaguerre polynomials are orthogonal is not unique. Indeed C has no unique self adjoint extension (the deficiency indices are (1,1) Thus, there is a multiplicity of possible self adjoint extensions for C and each such extension defines a different tensor product. For ....

D.S. Moak, The q-analogue of the Laguerre polynomials , J. Math. Anal. Appl. 81, (1981), 20--47 15

....1 (q Gamman ; 0; qa; q; qx) 1 (q n a Gamma1 ; q) n 2 OE 0 (q Gamman ; x Gamma1 ; Gamma; q; x=a) The second equality is a limit case of (1.39) Wall polynomials are q analogues of Laguerre polynomials on [0; 1) so they might be called little q Laguerre polynomials. 3c) Moak s [29] q Laguerre polynomials (notation as in [15, Exercise 7.43] are given by L ff n (x; q) q ff 1 ; q) n (q; q) n 1 OE 1 (q Gamman ; q ff 1 ; q; Gammaxq n ff 1 ) 1 (q; q) n 2 OE 1 (q Gamman ; Gammax; 0; q; q n ff 1 ) 37 The second equality is a limit case of (1.38) ....

D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20--47.

....[0; 0] 6] Thus, we have derived a direct integral decomposition (3.12) 0; q Gamma1 ] Omega # ; Gamma q Gamma1 ; 0] Z 1 0 Phi( p q Gamma x ) ae(x) dx: The functions H s n (x; t) are the Clebsch Gordan coefficients for this decomposition. However, as is well known [3, 8], the measure (3.7) for which the q Laguerre polynomials are orthogonal is not unique. Indeed the symmetric operator C has no unique self adjoint extension (the deficiency indices are (1,1) Thus, there is a multiplicity of possible selfadjoint extensions for C and each such extension defines a ....

D.S. Moak (1981), The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81, 20--47.

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