Al-Salam%E2%80%93Chihara%20polynomials
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1: Bibliography
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Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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Some orthogonal -polynomials.
Math. Nachr. 30, pp. 47–61.
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Characterization theorems for orthogonal polynomials.
In Orthogonal Polynomials (Columbus, OH, 1989),
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.
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A -beta integral on the unit circle and some biorthogonal rational functions.
Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
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Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
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2: 18.3 Definitions
§18.3 Definitions
… ►There are many ways of characterizing the classical OP’s within the general OP’s , see Al-Salam (1990). … ►For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►Bessel polynomials
►Bessel polynomials are often included among the classical OP’s. …3: 18.28 Askey–Wilson Class
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§18.28(iii) Al-Salam–Chihara Polynomials
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18.28.7
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18.28.8
or ; ; .
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§18.28(iv) -Al-Salam–Chihara Polynomials
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18.28.9
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4: 24.14 Sums
§24.14 Sums
►§24.14(i) Quadratic Recurrence Relations
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24.14.5
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►For (24.14.11) and (24.14.12), see Al-Salam and Carlitz (1959).
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►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
5: 18.1 Notation
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►Nor do we consider the shifted Jacobi polynomials:
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Classical OP’s
… ►Hahn Class OP’s
… ►Wilson Class OP’s
… ►Al-Salam–Chihara: .
6: Bibliography W
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Global asymptotics of the Meixner polynomials.
Asymptotic Analysis 75 (3-4), pp. 211–231.
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Asymptotics of orthogonal polynomials via recurrence relations.
Anal. Appl. (Singap.) 10 (2), pp. 215–235.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials.
Ph.D. Thesis, University of Wisconsin, Madison, WI.
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Explicit formulas for the associated Jacobi polynomials and some applications.
Canad. J. Math. 39 (4), pp. 983–1000.
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7: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
… ►Askey
… ►See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle. … ►Then the corresponding orthonormal polynomials are … ►For a polynomial …8: Bibliography D
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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Differential-difference operators associated to reflection groups.
Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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9: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
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Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Nielsen’s generalized polylogarithms.
SIAM J. Math. Anal. 17 (5), pp. 1232–1258.
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10: Bibliography G
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Positive sums of the classical orthogonal polynomials.
SIAM J. Math. Anal. 8 (3), pp. 423–447.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials.
Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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