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Al-Salam–Chihara polynomials

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1: 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 ) .
2: 18.1 Notation
  • Al-SalamChihara: Q n ( x ; a , b | q ) .

  • 3: 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 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.
  • R. Askey (1985) Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.