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continuous q-Hermite polynomials

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1: 18.28 Askey–Wilson Class
18.28.16 H n ( cos θ | q ) = = 0 n ( q ; q ) n e i ( n - 2 ) θ ( q ; q ) ( q ; q ) n - = e i n θ ϕ 0 2 ( q - n , 0 - ; q , q n e - 2 i θ ) .
§18.28(vii) Continuous q - 1 -Hermite Polynomials
18.28.18 h n ( sinh t | q ) = = 0 n q 1 2 ( + 1 ) ( q - n ; q ) ( q ; q ) e ( n - 2 ) t = e n t ϕ 1 1 ( q - n 0 ; q , - q e - 2 t ) = i - n H n ( i sinh t | q - 1 ) .
For continuous q - 1 -Hermite polynomials the orthogonality measure is not unique. …
2: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q - 1 -Hermite polynomials see Chen and Ismail (1998).
3: 18.1 Notation
  • Continuous q - 1 -Hermite: h n ( x | q )

  • 4: 28.9 Zeros
    For real q each of the functions ce 2 n ( z , q ) , se 2 n + 1 ( z , q ) , ce 2 n + 1 ( z , q ) , and se 2 n + 2 ( z , q ) has exactly n zeros in 0 < z < 1 2 π . They are continuous in q . For q the zeros of ce 2 n ( z , q ) and se 2 n + 1 ( z , q ) approach asymptotically the zeros of He 2 n ( q 1 / 4 ( π - 2 z ) ) , and the zeros of ce 2 n + 1 ( z , q ) and se 2 n + 2 ( z , q ) approach asymptotically the zeros of He 2 n + 1 ( q 1 / 4 ( π - 2 z ) ) . Here He n ( z ) denotes the Hermite polynomial of degree n 18.3). Furthermore, for q > 0 ce m ( z , q ) and se m ( z , q ) also have purely imaginary zeros that correspond uniquely to the purely imaginary z -zeros of J m ( 2 q cos z ) 10.21(i)), and they are asymptotically equal as q 0 and | z | . …