### Citations

900 |
Basic Hypergeometric Series,
- Gasper, Rahman
- 2004
(Show Context)
Citation Context ...ynomials: Pn(x; a, b, c, d|q)(2.9) = 4φ3 [ q−n, abcdqn−1, aeiθ, ae−iθ ab, ac, ad ; q, q ] , n = 0, 1, . . . , where the 4φ3 series is a balanced and terminating basic hypergeometric series defined in =-=[14]-=-, see also [3]. Askey and Wilson [4] had shown in an earlier paper that if one of a, b, c, d is of the form q−n, n = 0, 1, . . ., then φ(a, b) := 4φ3 [ a, b, c, d e, f, g ; q, q ] , efg = abcdq,(2.10)... |

851 |
An Introduction to Orthogonal Polynomials,
- Chihara
- 1978
(Show Context)
Citation Context ...is the ν = −12 special case of the 3- term recurrence relation of the so-called associated Legendre polynomials: (n+ ν + 1)Pn+1(ν, x) = (2n+ 2ν + 1)xPn(ν, x)− (n+ ν)Pn−1(ν, x),(1.10) see also Chihara =-=[9]-=-. In general, the 3-term recurrence relation satisfied by an orthogonal polynomial system {pn(x)} (OPS) is of the form pn+1(x) = (Anx+Bn)pn(x)−CnPn−1(x), n = 0, 1, . . . ,(1.11) with p−1(x) = 0, p0(x)... |

799 |
Asymptotics and Special Functions
- Olver
- 1974
(Show Context)
Citation Context ... in [2] that the asymptotic method of Darboux [45] can be applied to the generating functions of both {Pn(x)} and {P (1)n (x)} to determine their asymptotic behaviour. Darboux’s theorem, as stated in =-=[36]-=-, is as follows. Let f(z) = ∞∑ n=−∞ anz n(2.5) be the Laurent expansion of an analytic function f(z) in an annulus 0 < |z| < r <∞. By Cauchy’s formula an = 1 2πi ∫ C f(z) zn+1 dz,(2.6) where C is a si... |

250 |
Über die partiellen Differenzengleichungen der mathematischen Physik,
- Courant, Friedrichs, et al.
- 1928
(Show Context)
Citation Context ... have axial symmetry the Laplace equation in cylindrical coordinates is 1 r ∂ ∂r ( r ∂V ∂r ) + ∂2V ∂z2 = 0.(1.1) Discrete analogues of Laplace equation were considered by Courant, Friedricks and Lewy =-=[11]-=-. Boyer [7] studied the solutions of the following discretization of eqn. (1.1): ( m+ 1 2 ) k ( V (mh+ h, nk)− V (mh, nk) h ) (1.2) − ( m− 1 2 ) k ( V (mh, nk)− V (mh− h, nk) h ) + mh ( V (mh, nk+ k)−... |

87 | The differential equations of birth-and-death processes, and the Stieltjes moment problem. - Karlin, McGregor - 1957 |

81 |
J.A.: A set of orthogonal polynomials that generalize the Racah coefficients or 6- j symbols
- Askey, Wilson
- 1979
(Show Context)
Citation Context ...4φ3 [ q−n, abcdqn−1, aeiθ, ae−iθ ab, ac, ad ; q, q ] , n = 0, 1, . . . , where the 4φ3 series is a balanced and terminating basic hypergeometric series defined in [14], see also [3]. Askey and Wilson =-=[4]-=- had shown in an earlier paper that if one of a, b, c, d is of the form q−n, n = 0, 1, . . ., then φ(a, b) := 4φ3 [ a, b, c, d e, f, g ; q, q ] , efg = abcdq,(2.10) satisfies the contiguous relation A... |

81 |
Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion”,
- Meixner
- 1934
(Show Context)
Citation Context ...Pλ−1(x) = 0, Pλ0 (x) = 1, 0 < φ < π, 2λ + c > 0, c ≥ 0, or 0 < φ < π, 2λ + c ≥ 1, c > −1. The resulting polynomials, called by Askey and Wimp [5] the associated MeixnerPollaczek polynomials, (Meixner =-=[34]-=- had also found them and their orthogonality relation when c = 0) are orthogonal on the infinite interval −∞ < x < ∞ wrt the same weight function (1.38) with appropriate replacement of the variables. ... |

73 |
Recurrence relations, continued fractions, and orthogonal polynomials,”
- Askey, Ismail
- 1984
(Show Context)
Citation Context ...ystem satisfying (1.11). It is well-known that closely related to (1.11) is the continued fraction F (x) = 1 ∣∣∣∣∣∣A0x+B0 − C1 ∣∣∣∣∣∣A1x+B1 − C2 ∣∣∣∣∣∣A2x+B2 − . . . ,(1.22) see, for example, [9] and =-=[2]-=-. The nth convergent of this continued fraction, say, P (1)n (x)/Pn(x), is a rational function where the denominator polynomial Pn(x) satisfies (1.11) with the same initial conditions, but the numerat... |

61 |
Some basic hypergeometric polynomials that generalize Jacobi polynomials,
- Askey, Wilson
- 1985
(Show Context)
Citation Context ...re γ = (1+α+β−a− b)/2 and η = 0 when µ0 = 0 and η = 1 when µ0 = αβ = 0. Note that Pn(x; a, b, α, β, η) reduces to the continuous dual Hahn polynomials, see Andrews and Askey [1] and Askey and Wilson =-=[3]-=- when α = 0 or β = 0. It may be hazardous to speculate who was the first one to study the associated orthogonal polynomials or who used the adjective “associated” to describe them, but Humbert’s 1918 ... |

43 |
Classical orthogonal polynomials, Polynômes Orthogonaux et Applications (Bar-Le-Duc
- Andrews, Askey
- 1984
(Show Context)
Citation Context ...(a+ γ − i √ x− γ2)j , where γ = (1+α+β−a− b)/2 and η = 0 when µ0 = 0 and η = 1 when µ0 = αβ = 0. Note that Pn(x; a, b, α, β, η) reduces to the continuous dual Hahn polynomials, see Andrews and Askey =-=[1]-=- and Askey and Wilson [3] when α = 0 or β = 0. It may be hazardous to speculate who was the first one to study the associated orthogonal polynomials or who used the adjective “associated” to describe ... |

38 | The associated Askey-Wilson polynomials
- Ismail, Rahman
- 1991
(Show Context)
Citation Context ... + An+α + Cn+α)pαn(x)(3.1) 12 = An+αp α n+1(x) + Cn+αp α n−1(x), n = 0, 1, 2, . . . , with pα−1(x) = 0, p α 0 (x) = 1; An+α, Cn+α being the same as in (2.14) with n replaced by n + α. It was shown in =-=[27]-=- that the two linearly independent solutions of (3.1) are rn+α = (abqn+α, acqn+α, adqn+α, bcdqn+α/z; q)∞ (bcqn+α, bdqn+α, cdqn+α, azqn+α; q)∞ ( a z )n+α (3.2) × 8W7(bcd/qz; b/z, c/z, d/z, abcdqn+α−1, ... |

31 | Dual birth and death processes and orthogonal polynomials,
- Letessier, Valent
- 1986
(Show Context)
Citation Context ...e a characterization theorem of Wall and Wetzel [46] in terms of the so-called chain sequences to find the true interval of orthogonality, or an alternate approach suggested by Chihara [9], [10]. See =-=[21]-=- for a brief summary of Chihara’s ideas. If the support turns out to be (0,∞) or (−∞,∞) then the unboundedness of it poses the additional problem of whether or not one has a determined or an undetermi... |

30 |
Orthogonal Polynomials, 4th edn.
- Szegö
- 1975
(Show Context)
Citation Context ...ther or not one has a determined or an undetermined moment problem. For a comprehensive analysis of this problem see [44]. However, for a finite interval [a, b] the measure is necessarily unique, see =-=[45]-=-, and can be obtained, in principle, in 4 different ways. There is another approach to the associated OPS problem, mainly due to Grunbaum [15], [16], who looks at them from the point of view of the bi... |

29 | Contiguous relations, basic hypergeometric functions, and orthogonal polynomials,
- Ismail, Libis
- 1989
(Show Context)
Citation Context ...lane or from n ≥ 0 to n ≤ 0. An example of how to construct a minimal solution, when it exists, by first deriving a set of solutions of (2.19) (any two of them being linearly independent) is given in =-=[17]-=-. 3 Associated Askey-Wilson polynomials The associated polynomials that generalize the Askey-Wilson polynomials given in (2.9) are solutions of the 3-term recurrence relation (z + z−1 − a− a−1 + An+α ... |

27 |
Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polynomials, in Orthogonal Polynomials and Their Applications,
- Magnus
- 1988
(Show Context)
Citation Context ...olutely continuous measure on [−1, 1]. Unfortunately, however, this simple device does not seem to work for nonclassical OPS, and certainly not for the associated ones, classical or otherwise. Magnus =-=[30]-=- showed that the OPS belonging to what he calls the Laguerre-Hahn class, defined in terms of a Riccati-type difference equation, that includes the classical OPS along with the associated ones, general... |

25 |
Linear birth and death models and associated Laguerre polynomials,
- Ismail, Letessier, et al.
- 1988
(Show Context)
Citation Context ...iated polynomials. The associated orthogonal polynomials also appear in a natural way in the birth-anddeath processes, studied by Karlin and McGregor [28], [29], see also Ismail, Letessier and Valent =-=[22]-=-. Let pmn(t) = Prob{X(t) = n|X(0) = m},(1.27) the transition probabilities of a birth-and-death process, be given by pmn(t) = λmt+ o(t), n = m+ 1 µmt+ o(t), n = m− 1 1− (λm + µm)t+ o(t), n = m,... |

18 |
Hypergeometric series, recurrence relations and some new orthogonal functions
- Wilson
- 1978
(Show Context)
Citation Context ... = 0, P0(x) = 1, x = a2 + t2, where λn = (n+ a + b)(n+ a+ c)(n + a+ d)(n + s− 1) (2n + s− 1)(2n + s) , µn = n(n+ b+ c− 1)(n + b+ d − 1)(n+ c+ d − 1) (2n+ s− 2)(2n + s− 1) ,(4.14) s = a + b+ c+ d, see =-=[47]-=-, [48] and [3]. 19 Wilson [47] found that they are orthogonal on (−∞,∞) wrt the weight function∣∣∣∣∣Γ(a + it)Γ(b+ it)Γ(c+ it)Γ(d+ it)Γ(2it) ∣∣∣∣∣ 2 and that Pn(x; a, b, c, d) = 4F3 [ −n, n+ s− 1, a− i... |

17 |
Associated Laguerre and Hermite polynomials
- Askey, Wimp
- 1984
(Show Context)
Citation Context ...he form Lαn+1(x; c) = 2n+ 2c+ α+ 1− x n + c+ 1 Lαn(x; c)− n + α+ c n + c+ 1 Lαn−1(x; c),(1.33) 5 which is the relation associated to (1.15) with n replaced by n+ c in the coefficients. Askey and Wimp =-=[5]-=- found the measure of orthogonality for these associated Laguerre polynomials Lαn(x; c), as well as their explicit polynomial form: Lαn(x; c) = (α+ 1)n n! n∑ k=0 (−n)k xk (c+ 1)k(α+ 1)k 3F2 [ k − n, k... |

13 | Chain sequences and orthogonal polynomials, - Chihara - 1962 |

13 |
Explicit formulas for the associated Jacobi polynomials and some applications
- Wimp
- 1987
(Show Context)
Citation Context ... c, d|q), which satisfy the 3-term recurrence relation (2.13) with An, Bn, Cn replaced by An+α, Bn+α, Cn+α, but gave a scheme of how to derive the corresponding fourth order difference equation. Wimp =-=[49]-=-, however, was able to find an explicit fourth order differential equation for the associated Jacobi polynomials which are the q → 1 limit cases of pαn(x; q 1/2, qα+1/2,−qβ+1/2,−q1/2|q). Considering t... |

12 |
Two families of associated Wilson polynomials.
- Ismail, Letessier, et al.
- 1990
(Show Context)
Citation Context ... ae−iθ; q)k (q, abqα, acqα, adqα, abcdqα−1; q)k qk(3.16) × 10W9 ( abcdq2α+k−2; qα, bcqα−1, bdqα−1, cdqα−1, qk, abcdq2α+n+k−1, qk−n; q, qa2 ) . Masson’s exceptional Askey-Wilson polynomials, see [32], =-=[25]-=- that correspond to the indeterminate cases abcd = q or q2 (see eqn. (2.14)) turn out to be the limiting case α → 0+ of pαn(x), while qαn(x) approaches the Askey-Wilson polynomials. By using the trans... |

8 |
Wilson polynomials and some continued fractions of Ramanujan.
- Masson
- 1991
(Show Context)
Citation Context ... aeiθ, ae−iθ; q)k (q, abqα, acqα, adqα, abcdqα−1; q)k qk(3.16) × 10W9 ( abcdq2α+k−2; qα, bcqα−1, bdqα−1, cdqα−1, qk, abcdq2α+n+k−1, qk−n; q, qa2 ) . Masson’s exceptional Askey-Wilson polynomials, see =-=[32]-=-, [25] that correspond to the indeterminate cases abcd = q or q2 (see eqn. (2.14)) turn out to be the limiting case α → 0+ of pαn(x), while qαn(x) approaches the Askey-Wilson polynomials. By using the... |

8 |
Sur une généralisation des polynomes de
- Pollaczek
- 1949
(Show Context)
Citation Context ... (1.10) by a clever manipulation of the Legendre functions of both kinds, without realizing that their results follow as special cases of Pollaczek’s formulas given in (1.36)- -(1.39), see also [12], =-=[38]-=-. One may describe the special function methods and even the moment methods pretty adhoc and simple-minded, but they can be quite effective in determining the measure of orthogonality, sometimes even ... |

7 |
On the associated Legendre polynomials, in
- Barrucand, Dickinson
- 1967
(Show Context)
Citation Context ...olynomials it would appear that guessing the measure of orthogonality 11 for the associated OPS or to derive it from special function formulas would be a daunting task indeed. Barrucand and Dickinson =-=[6]-=- discovered the weight function for the associated Legendre polynomials satisfying eqn. (1.10) by a clever manipulation of the Legendre functions of both kinds, without realizing that their results fo... |

7 | Quadratic birth and death processes and associated continuous dual Hahn polynomials - Ismail, Letessier, et al. - 1989 |

6 |
Two families of orthogonal polynomials related to Jacobi polynomials
- Ismail, Masson
- 1991
(Show Context)
Citation Context ... unique, for example, the Vn(x; a) polynomials of Al-Salam and Carlitz, see [9]. IV. Method of minimal solutions. This method, extensively used by Masson [31– 33] and his collaborators [17–18], [21], =-=[26]-=-, relies on the following ideas. A solution X(s)n of the 3-term recurrence relation Xn+1 = AnXn +BnXn−1, n ≥ 0,(2.19) is a minimal (or subdominant) solution if for any other linearly independent solut... |

6 |
une famille de polynomes orthogonaux qui contient les polynomes d’Hermite e de Laguerre comme cas limites. Comptes Rendus de l’Académie Sci
- Pollaczek
- 1950
(Show Context)
Citation Context ...2 (α + 1) and −x/2 sinφ, respectively, and take the limit φ → 0, then (1.41) becomes the 3-term recurrence relation for the associated Laguerre polynomials, Lαn(x; c), as was observed by Pollaczek in =-=[40]-=-. Askey and Wimp [5] took advantage of his property to obtain the weight function for the orthogonality of {Lαn(x; c)} on 0 < x <∞, namely, Wα(x; c) = xαe−x |Ψ(c, 1− α; xe−πi)|2 ,(1.42) where Ψ(a, b; ... |

6 |
Quadratic forms and convergence regions for continued fractions
- Wall, Wetzel
- 1944
(Show Context)
Citation Context ... can be said about this measure without a good deal of additional work. First, one has to determine the bounds of the support of dµ for which one can use a characterization theorem of Wall and Wetzel =-=[46]-=- in terms of the so-called chain sequences to find the true interval of orthogonality, or an alternate approach suggested by Chihara [9], [10]. See [21] for a brief summary of Chihara’s ideas. If the ... |

3 |
On certain polynomials associated with orthogonal polynomials
- Dickinson
- 1958
(Show Context)
Citation Context ...g eqn. (1.10) by a clever manipulation of the Legendre functions of both kinds, without realizing that their results follow as special cases of Pollaczek’s formulas given in (1.36)- -(1.39), see also =-=[12]-=-, [38]. One may describe the special function methods and even the moment methods pretty adhoc and simple-minded, but they can be quite effective in determining the measure of orthogonality, sometimes... |

3 |
Über Orthogonalpolynome mit drei Parametern, Deutche
- Hahn
(Show Context)
Citation Context ...ne to study the associated orthogonal polynomials or who used the adjective “associated” to describe them, but Humbert’s 1918 paper [20] is the earliest work that I could find in the literature. Hahn =-=[19]-=- seems to be the first to study the associated Laguerre polynomials. Although he did not find their orthogonality relation he found the fourth order differential equation that they satisfy by first ex... |

2 |
deux polynomes associés aux polynomes de
- Humbert, Sur
- 1918
(Show Context)
Citation Context ...α = 0 or β = 0. It may be hazardous to speculate who was the first one to study the associated orthogonal polynomials or who used the adjective “associated” to describe them, but Humbert’s 1918 paper =-=[20]-=- is the earliest work that I could find in the literature. Hahn [19] seems to be the first to study the associated Laguerre polynomials. Although he did not find their orthogonality relation he found ... |

2 |
Some results on associated Wilson polynomials
- Ismail, Letessier, et al.
- 1991
(Show Context)
Citation Context ...olynomials are, of course, the solutions of (4.13) with n replaced by n+α in λn and µn. Their weight function and the orthogonality properties were worked out by Ismail et al. [26], see also [33] and =-=[24]-=-. In fact, the authors in [26] found two families of polynomials that correspond to the two cases we have considered in the previous section. Their weight function [26, (3.39)] for the first family ca... |

2 |
Convergence and analytic continuation for a class of regular C-fractions
- Masson
- 1985
(Show Context)
Citation Context ...n−1, n ≥ 0,(2.19) is a minimal (or subdominant) solution if for any other linearly independent solution X(d)n (dominant) one has the property lim n→∞ X(s)n X (d) n = 0.(2.20) Pincherle’s theorem (see =-=[31]-=- and the references therein) states that (2.20) is a necessary and sufficient condition for the convergence of the continued fraction that corresponds to (2.19), namely, B0 A0+ B1 A1+ B2 A2+ · · · .(2... |

2 |
une famille de polynômes orthogonaux à quatre paramitres
- Sur
- 1950
(Show Context)
Citation Context ...significant departure from the classical orthogonal polynomials (belonging to the Szegő class) on one hand, and a generalization to the associated ones on the other. In its full generality Pollaczek =-=[41]-=- gave the following 3-term recurrence relation (n+ c+ 1)Pλn+1(x) = 2[(n+ c+ λ+ a)x+ b]P λ n (x)− (n+ c+ 2λ− 1)Pλn−1(x),(1.37) with Pλ−1(x) = 0, Pλ0 (x) = 1, |x| ≤ 1, and either a > |b|, 2λ + c > 0, c ... |

2 |
Some generating functions for the associated Askey-Wilson polynomials
- Rahman
- 1996
(Show Context)
Citation Context ...to be the limiting case α → 0+ of pαn(x), while qαn(x) approaches the Askey-Wilson polynomials. By using the transformation theory of basic hypergeometric series a simpler form of (3.14) was found in =-=[42]-=- 15 pαn(x) ≡ pαn(x; a, b, c, d|q)(3.17) = (abcdq2α−1, qα+1; q)n (q, abcdqα−1; q)n q−αn n∑ k=0 (q−n, abcdq2α+n−1; q)k (qα+1, abqα; q)k ×(aq αeiθ, aqαe−iθ; q)k (acqα, acqα; q)k qk k∑ j=0 (qα, abqα−1, ac... |

1 |
Discrete Bessel functions
- Boyer
(Show Context)
Citation Context ...symmetry the Laplace equation in cylindrical coordinates is 1 r ∂ ∂r ( r ∂V ∂r ) + ∂2V ∂z2 = 0.(1.1) Discrete analogues of Laplace equation were considered by Courant, Friedricks and Lewy [11]. Boyer =-=[7]-=- studied the solutions of the following discretization of eqn. (1.1): ( m+ 1 2 ) k ( V (mh+ h, nk)− V (mh, nk) h ) (1.2) − ( m− 1 2 ) k ( V (mh, nk)− V (mh− h, nk) h ) + mh ( V (mh, nk+ k)− 2V (mh, nk... |

1 |
The associated ultraspherical polynomials and their
- Bustoz, Ismail
(Show Context)
Citation Context ...n asymptotic behaviour. Therefore, this method is inherently simpler to use than the other methods, and have been used in a number of works on associated OPS. See, for example, [2], [22], [23], [21], =-=[8]-=-. III. Special function methods. These methods are usually quite computational and rely more on inspired guesses than on the elaborate machinery of the theory of general OPS. The first step is to find... |

1 | Polinomi più generali di altri classici e dei loro associati e relazioni tra essi funzioini di seconda specie - Palama - 1953 |

1 | des polynômes biorthogonaux que généralisent les polynômes ultrasphériques - Systèmes - 1949 |

1 |
Poisson kernel for the associated continuous qultraspherical polynomials
- Rahman, Tariq
- 1997
(Show Context)
Citation Context ...ute many formulas for the associated polynomials pαn(x; a, b, c, d|q) from the corresponding formulas for the ordinary Askey-Wilson polynomials pn(x; aq α/2, bqα/2, cqα/2, dqα/2|q), see, for example, =-=[43]-=-. By replacing a, b, c, d and α in (3.17) by q1/2, qα+1/2, −qβ+1/2, −q1/2 and c, respectively, then taking the limit q → 1 and defining P (α,β)n (x; c)(3.21) = (α+ c+ 1)n(α + β + c+ 1)n (c+ 1)n(α + β ... |

1 |
The Problem of Moments, revised edition,Math
- Shohat, Tamarkin
- 1950
(Show Context)
Citation Context ...be (0,∞) or (−∞,∞) then the unboundedness of it poses the additional problem of whether or not one has a determined or an undetermined moment problem. For a comprehensive analysis of this problem see =-=[44]-=-. However, for a finite interval [a, b] the measure is necessarily unique, see [45], and can be obtained, in principle, in 4 different ways. There is another approach to the associated OPS problem, ma... |