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Uniform asymptotic expansions of the Tricomi-Carlitz polynomials. (English) Zbl 1202.41032

Authors’ abstract: The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation
\[ (n+1)f_{n+1}^{(\alpha)}(x)-(n+\alpha) x f_n^{(\alpha)}(x)+ f_{n-1}^{(\alpha)}(x)=0, \qquad n\geq1, \]
with initial values \( f_0^{(\alpha)}(x)=1\) and \( f_1^{(\alpha)}(x)=\alpha x\), where \( x\) is a real variable and \( \alpha\) is a positive parameter. An asymptotic expansion is derived for these polynomials by using the turning-point theory for three-term recurrence relations developed by Z. Wang and R. Wong [Numer. Math. 91, No. 1, 147–193 (2002; Zbl 0998.65031); ibid. 94, No. 1, 147–194 (2003; Zbl 1030.39016)]. The result holds uniformly in regions containing the critical values \( x=\pm2/\sqrt{\nu}\), where \( \nu=n+2\alpha-1/2\).

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
39A10 Additive difference equations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:

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