M. Foupouagnigni, M.N. Hounkonnou and A. Ronveaux, The fourth-order difference equation satisfied by the associated orthogonal polynomials of the D-Laguerre -- Hahn class, J. Symbolic Comp., 28 (1999), 801 -- 818.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(difference) equation. Continuous Laguerre Hahn orthogonal polynomials were introduced in [7] The corresponding fourth order differential equation has been established in [11] The approach adopted there has been extended as well to the discrete and q Laguerre Hahn orthogonal polynomials in [4, 5] respectively. In [11] as in [4, 5] the equations were written explicitly for the cases of polynomials r associated to the corresponding classical situations, that is Jacobi polynomials and specializations in [11] Hahn, big q Jacobi polynomials and specializations in [4, 5] respectively. ....

....Laguerre Hahn orthogonal polynomials were introduced in [7] The corresponding fourth order differential equation has been established in [11] The approach adopted there has been extended as well to the discrete and q Laguerre Hahn orthogonal polynomials in [4, 5] respectively. In [11] as in [4, 5], the equations were written explicitly for the cases of polynomials r associated to the corresponding classical situations, that is Jacobi polynomials and specializations in [11] Hahn, big q Jacobi polynomials and specializations in [4, 5] respectively. Laguerre Hahn orthogonal on special ....

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M. Foupouagnigni, A. Ronveaux, and M.N. Hounkonnou, "The fourth-order difference equation satisfied by the associated orthogonal polynomials of the D q -Laguerre-Hahn class," Preprint, Konrad-Zuse-Zentrum, Berlin, 1998.

....(difference) equation. Continuous Laguerre Hahn orthogonal polynomials were introduced in [7] The corresponding fourth order differential equation has been established in [11] The approach adopted there has been extended as well to the discrete and q Laguerre Hahn orthogonal polynomials in [4, 5] respectively. In [11] as in [4, 5] the equations were written explicitly for the cases of polynomials r associated to the corresponding classical situations, that is Jacobi polynomials and specializations in [11] Hahn, big q Jacobi polynomials and specializations in [4, 5] respectively. ....

....Laguerre Hahn orthogonal polynomials were introduced in [7] The corresponding fourth order differential equation has been established in [11] The approach adopted there has been extended as well to the discrete and q Laguerre Hahn orthogonal polynomials in [4, 5] respectively. In [11] as in [4, 5], the equations were written explicitly for the cases of polynomials r associated to the corresponding classical situations, that is Jacobi polynomials and specializations in [11] Hahn, big q Jacobi polynomials and specializations in [4, 5] respectively. Laguerre Hahn orthogonal on special ....

[Article contains additional citation context not shown here]

M. Foupouagnigni, A. Ronveaux, and M.N. Hounkonnou, "The fourth-order difference equation satisfied by the associated orthogonal polynomials of the #-Laguerre-Hahn class," J. Symbolic Computation. 28 (1999), 801--818.

....these new polynomials are not semi classical but belong to the discrete Laguerre Hahn class (see Preliminaries and notations section) Many works have been devoted to the derivation of these fourth order difference equations. Their polynomial coefficients have been given explicitly in Refs. [1,7,8,10,19,32,36] for the rth associated classical discrete orthogonal polynomials. In 1999, using symmetry properties inside the three term recurrence relation, hypergeometric representation and symbolic computation, the coefficients of the fourthorder difference equation for the co recursive associated Meixner ....

....in general a simple functional equation living in P , the dual space of P. Appropriate definitions of DU and PU, where P is a polynomial that allows building a simple difference equation for the functional, which generalizes in some way the Pearson type difference equation for the weight r [7,10,13,34]. If the Stieltjes function S(x) satisfies a first order linear difference equation of the form fx Sx 1Cx SxDx; 7 where f, C and D are polynomials, the functional U satisfies in P 0 a first order difference equation with polynomial coefficients. In this case, the functional U and the ....

[Article contains additional citation context not shown here]

M. Foupouagnigni, M.N. Hounkonnou and A. Ronveaux, The fourth-order difference equation satisfied by the associated orthogonal polynomials of the D-Laguerre -- Hahn class, J. Symbolic Comp., 28 (1999), 801 -- 818.

....n = Gamma n( 2 ) oe(j n Gamma1 ) j n Gamma1 ) n 1 (4) jn = Gamma (0) noe (0) Gamma n 2 oe noe it is possible to write the corresponding equations (1) 2) and (3) again in terms of oe and ; for the generic classical discrete polynomials. Proposition 1 [3] The associated polynomials satisfy D r;n = N r 1;n Gamma1 ; 5) D r 1;n Gamma1 N r;n ; 6) where D r;n = a 2 T a 1 T a 0 T ; N r 1;n Gamma1 = a 1 T a 0 T ; 7) D r 1;n Gamma1 = b 2 T b 1 T b 0 T ; N r;n = b 1 T b 0 T ; 8) a 2 = k 9;0 ....

....polynomials k 3 ; k 4 ; k 6 ; k 8 are constant with respect to the variable x and fi r ; fl r are given by (4) If r = 0, from (2) and (16) we have k 4 = K 0 = 0. Then, N 1;n Gamma1 is equal to zero, thus the fourth order difference equation for the first associated P n , factorizes in the form [1,3,10] ( A 1 T B 1 T C 1 T ) A 1 T B 1 T C 1 T ) P n ] 0: For r = 0, if we are inside the Hahn class with ff fi 1 = 0 (discrete Grosjean polynomials) from (2) 16) and [2,9] we have K 0 = 2k 6 = 0. Then N 1;n is equal to zero and the difference equation in this case ....

M.Foupouagnigni, A. Ronveaux, M. N. Hounkonnou, The fourth-order difference equation satisfied by the associated orthogonal polynomials of \DeltaLaguerre -Hahn Class. Konrad-Zuse-Zentrum Berlin, Preprint SC 97-..., 1997.

....(0) noe 0 (0) Gamma n 2 oe 00 2 0 noe 00 ; n 0: 12 M. Foupouagnigni, A. Ronveaux and M. N. Hounkonnou The results in this section allow us to write a fourth order difference equation (3. 27) in terms of oe and which therefore is valid for all classical discrete families (see Foupouagnigni et al. 1997)) in the same spirit as for the classical continuous families (see Zarzo et al. 1993) 4.2. Factorized form of the fourth order difference equation satisfied by the first associated Use of relations (3.19) and (3.22) for r = 0 and taking into account Equations (3.16) 4.5) 4.6) and (4.7) ....

Foupouagnigni, M., Koepf, W., Ronveaux, A. (1997). The fourth-order difference equation satisfied by the associated classical discrete orthogonal polynomials. Konrad-Zuse-Zentrum Berlin, Preprint SC 97-72.

....equal to 2 (1 Gammaa b q) q Gamma1 . Therefore, the first associated of the little q Jacobi polynomials (resp. big q Jacobi polynomials) is still in the little q Jacobi (resp. big q Jacobi) family when a b q = 1. Computations involving the coefficients fi n and fl n (see Equation (8) given in [1,6,11] and uses of Maple V.4 generate the following relations between the monic little q Jacobi (resp. monic big q Jacobi) polynomials and their respective first associated p (1) n (x; a; 1 q a jq) a q) n pn ( x a q ; 1 a ; a q jq) 16) P (1) n (x; a; 1 q a; c; q) a) n Pn ( x ....

.... = a q) n pn ( x a q ; 1 a ; a q jq) 16) P (1) n (x; a; 1 q a; c; q) a) n Pn ( x a ; 1 a ; a q; c q; q) 17) iii) The results given in this paper (see Equations (11) and (13) which agree with the ones obtained using the Stieltjes properties of the associated linear form [6], can be used for connection problems, expanding the first associated P (1) n Gamma1 in terms of Pn , in the same spirit as in [10] We have also computed the coefficients of the fourth order q difference equation satisfied by the first associated of the q classical orthogonal polynomials ....

M.Foupouagnigni, A. Ronveaux, M. N. Hounkonnou, The fourth-order difference equation satisfied by the associated orthogonal polynomials of the D q -Laguerre-Hahn Class (in progress).

....oe 00 2 ) oe(j n Gamma1 ) j n Gamma1 ) n 1 (4) jn = Gamma (0) noe 0 (0) Gamma n 2 oe 00 2 0 noe 00 ; n 0 it is possible to write the corresponding equations (1) 2) and (3) again in terms of oe and ; for the generic classical discrete polynomials. Proposition 1 [3] The associated polynomials satisfy D r;n h P (r) n i = N r 1;n Gamma1 h P (r 1) n Gamma1 i ; 5) D r 1;n Gamma1 h P (r 1) n Gamma1 i = N r;n h P (r) n i ; 6) where D r;n = a 2 T 2 a 1 T a 0 T 0 ; N r 1;n Gamma1 = a 1 T a 0 T 0 ; 7) D r 1;n Gamma1 ....

....k 3 ; k 4 ; k 6 ; k 8 are constant with respect to the variable x and fi r ; fl r are given by (4) If r = 0, from (2) and (16) we have k 4 = K 0 = 0. Then, N 1;n Gamma1 is equal to zero, thus the fourth order difference equation for the first associated P (1) n , factorizes in the form [1,3,10] ( A 1 T 2 B 1 T C 1 T 0 ) A 1 T 2 B 1 T C 1 T 0 ) P (1) n ] 0: For r = 0, if we are inside the Hahn class with ff fi 1 = 0 (discrete Grosjean polynomials) from (2) 16) and [2,9] we have K 0 = 2k 6 = 0. Then N 1;n is equal to zero and the difference equation ....

M.Foupouagnigni, A. Ronveaux, M. N. Hounkonnou, The fourth-order difference equation satisfied by the associated orthogonal polynomials of \DeltaLaguerre -Hahn Class. Konrad-Zuse-Zentrum Berlin, Preprint SC 97-..., 1997.

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