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Al-Salam%E2%80%93Chihara%20polynomials

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1: Bibliography
  • W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.
  • W. A. Al-Salam and L. Carlitz (1965) Some orthogonal q -polynomials. Math. Nachr. 30, pp. 47–61.
  • W. A. Al-Salam (1990) 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.
  • W. A. Al-Salam and M. E. H. Ismail (1994) A q -beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
  • R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
  • 2: 18.3 Definitions
    §18.3 Definitions
    There are many ways of characterizing the classical OP’s within the general OP’s { p n ( x ) } , see Al-Salam (1990). … For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 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
    §18.28(iii) Al-SalamChihara Polynomials
    18.28.7 Q n ( cos θ ; a , b | q ) = p n ( cos θ ; a , b , 0 , 0 | q ) = a n = 0 n q ( a b q ; q ) n ( q n ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j cos θ + a 2 q 2 j ) = ( a b ; q ) n a n ϕ 2 3 ( q n , a e i θ , a e i θ a b , 0 ; q , q ) = ( b e i θ ; q ) n e i n θ ϕ 1 2 ( q n , a e i θ b 1 q 1 n e i θ ; q , b 1 q e i θ ) .
    18.28.8 1 2 π 0 π Q n ( cos θ ; a , b | q ) Q m ( cos θ ; a , b | q ) | ( e 2 i θ ; q ) ( a e i θ , b e i θ ; q ) | 2 d θ = δ n , m ( q n + 1 , a b q n ; q ) , a , b or a = b ¯ ; a b 1 ; | a | , | b | 1 .
    §18.28(iv) q 1 -Al-SalamChihara Polynomials
    18.28.9 Q n ( 1 2 ( a q y + a 1 q y ) ; a , b | q 1 ) = ( 1 ) n b n q 1 2 n ( n 1 ) ( ( a b ) 1 ; q ) n ϕ 1 3 ( q n , q y , a 2 q y ( a b ) 1 ; q , q n a b 1 ) .
    4: 24.14 Sums
    §24.14 Sums
    §24.14(i) Quadratic Recurrence Relations
    24.14.5 k = 0 n ( n k ) E k ( h ) B n k ( x ) = 2 n B n ( 1 2 ( x + h ) ) ,
    For (24.14.11) and (24.14.12), see Al-Salam and Carlitz (1959). … 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
    Classical OP’s
    Hahn Class OP’s
    Wilson Class OP’s
  • Al-SalamChihara: Q n ( x ; a , b | q ) .

  • Nor do we consider the shifted Jacobi polynomials: …
    6: Bibliography W
  • X.-S. Wang and R. Wong (2011) Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.
  • X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.
  • J. Wimp (1987) Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math. 39 (4), pp. 983–1000.
  • 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
  • K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.
  • B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
  • C. F. Dunkl (1989) Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
  • T. M. Dunster (1997) Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.
  • T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.
  • 9: Bibliography K
  • K. W. J. Kadell (1994) A proof of the q -Macdonald-Morris conjecture for B C n . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
  • E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • K. S. Kölbig (1986) Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.
  • 10: Bibliography G
  • G. Gasper (1977) Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.
  • W. Gautschi (1994) 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.
  • V. X. Genest, L. Vinet, and A. Zhedanov (2016) The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.