Asymptotic approximations of the continuous Hahn polynomials and their zeros. (English) Zbl 1428.41038
Continuous Hahn polynomials have been introduced in [N. M. Atakishiev and S. K. Suslov, J. Phys. A, Math. Gen. 18, 1583–1596 (1985; Zbl 0582.33006)] as a family \(_{3} F_{2} (3) \) in the second level of the Askey tableau of hypergeometric functions (see [R. Koekoek et al., Hypergeometric orthogonal polynomials and their \(q\)-analogues. With a foreword by Tom H. Koornwinder. Berlin: Springer (2010; Zbl 1200.33012)]). They are the images, by the Fourier transform, of Jacobi polynomials as pointed out in [H. T. Koelink, Proc. Am. Math. Soc. 124, No. 3, 887–898 (1996; Zbl 0848.33007)]. The role of continuous Hahn polynomials in differential geometry and algebra and well as in nonrelativistic and relativistic Coulomb problems has been pointed out by many authors.
The aim of the paper under review is to explore the asymptotic behavior of continuous Hahn polynomials. Taking into account a result by M. E. H. Ismail and X. Li [Proc. Am. Math. Soc. 115, No. 1, 131–140 (1992; Zbl 0744.33005)], the zeros of the continuous Hahn polynomials \(\pi_{n}(x)\) lie asymptotically in the interval \((-n/2, n/2)\). Thus, considering the scaling \(x=nt\), then a neighborhood of the interval \((-1/2, 1/2)\) in the complex \(t\)-plane will be referred to as oscillatory region and the complement set is the non-oscillatory region. The end points of the interval are said to be the turning points.
The main results of this contribution are focused on the asymptotic approximations of such a family of orthogonal polynomials, which has not been studied yet in the literature. The authors deduce the nonuniform asymptotics both in the zero free region and the oscillatory region, uniform asymptotic expansions around the turning points and Plancherel-type asymptotics near the turning points, respectively. Notice, that there are several ways to analyze the asymptotics of a sequence of orthogonal polynomials, namely the method of steepest descent for their integral representations, the WKB approximations for differential equations associated with such polynomials, and the nonlinear steepest descent method for Riemann-Hilbert problems, based on the analyticity of the weight function. Unfortunately, for continuous Hahn polynomials the above approaches yield very technical difficulties and, as a consequence, the authors of the paper under review use an alternative method introduced in [X. S. Wang and R. Wong, Anal. Appl., Singap. 10, No. 2, 215–235 (2012; Zbl 1242.41035)], taking into account the fact that for the continuous Hahn polynomials the coefficients of the three-term recurrence relation are rational functions of \(n\) (see formula (1.16)). Thus uniform asymptotic expansions and matching techniques in the complex plane are used as the key techniques.
The asymptotic behavior of zeros follows from the Plancherel type asymptotics and a result by H. W. Hethcote [SIAM J. Math. Anal. 1, 147–152 (1970; Zbl 0199.49902)] dealing with the location of zeros of a continuous function in a symmetric interval assuming it has a decomposition as a sum of a differentiable function and a function with a control of its bound in terms of the evaluation of the first one at the ends of the interval.
The aim of the paper under review is to explore the asymptotic behavior of continuous Hahn polynomials. Taking into account a result by M. E. H. Ismail and X. Li [Proc. Am. Math. Soc. 115, No. 1, 131–140 (1992; Zbl 0744.33005)], the zeros of the continuous Hahn polynomials \(\pi_{n}(x)\) lie asymptotically in the interval \((-n/2, n/2)\). Thus, considering the scaling \(x=nt\), then a neighborhood of the interval \((-1/2, 1/2)\) in the complex \(t\)-plane will be referred to as oscillatory region and the complement set is the non-oscillatory region. The end points of the interval are said to be the turning points.
The main results of this contribution are focused on the asymptotic approximations of such a family of orthogonal polynomials, which has not been studied yet in the literature. The authors deduce the nonuniform asymptotics both in the zero free region and the oscillatory region, uniform asymptotic expansions around the turning points and Plancherel-type asymptotics near the turning points, respectively. Notice, that there are several ways to analyze the asymptotics of a sequence of orthogonal polynomials, namely the method of steepest descent for their integral representations, the WKB approximations for differential equations associated with such polynomials, and the nonlinear steepest descent method for Riemann-Hilbert problems, based on the analyticity of the weight function. Unfortunately, for continuous Hahn polynomials the above approaches yield very technical difficulties and, as a consequence, the authors of the paper under review use an alternative method introduced in [X. S. Wang and R. Wong, Anal. Appl., Singap. 10, No. 2, 215–235 (2012; Zbl 1242.41035)], taking into account the fact that for the continuous Hahn polynomials the coefficients of the three-term recurrence relation are rational functions of \(n\) (see formula (1.16)). Thus uniform asymptotic expansions and matching techniques in the complex plane are used as the key techniques.
The asymptotic behavior of zeros follows from the Plancherel type asymptotics and a result by H. W. Hethcote [SIAM J. Math. Anal. 1, 147–152 (1970; Zbl 0199.49902)] dealing with the location of zeros of a continuous function in a symmetric interval assuming it has a decomposition as a sum of a differentiable function and a function with a control of its bound in terms of the evaluation of the first one at the ends of the interval.
Reviewer: Francisco Marcellán (Leganes)
MSC:
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
continuous Hahn polynomials; three-term recurrence relation; uniform asymptotics; Plancherel-type asymptotics; zerosCitations:
Zbl 0582.33006; Zbl 1200.33012; Zbl 0848.33007; Zbl 0744.33005; Zbl 1242.41035; Zbl 0199.49902Software:
DLMFReferences:
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