[Paper Review] The Askey Scheme for Hypergeometric Orthogonal Polynomials Viewed from Asymptotic Analysis

Nico Μ. Τemme, José L. López|Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands|Sep 24, 2001

Mathematical functions and polynomials3 citations

TL;DR

This paper presents a novel asymptotic analysis method for deriving limit relations in the Askey scheme of hypergeometric orthogonal polynomials by leveraging generating functions and Cauchy-type integrals. It establishes that Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials asymptotically approach Laguerre polynomials under specific parameter scalings, with full asymptotic expansions provided for each limit, offering deeper insight than standard limit relations alone.

ABSTRACT

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.

Motivation & Objective

• To develop a systematic asymptotic method for deriving limit relations between hypergeometric orthogonal polynomials in the Askey scheme.
• To replace traditional limit relations with full asymptotic expansions that reveal finer structure in the limiting behavior.
• To unify and generalize known limits—especially from discrete to continuous orthogonal polynomials—using generating function techniques.
• To demonstrate the method on key families: Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials, showing convergence to Laguerre polynomials.
• To provide quantitative error bounds and parameter scalings that make the asymptotic approximations precise and uniformly valid.

Proposed method

• The method uses generating functions of the form $ F(x,w) = \sum_{n=0}^\infty p_n(x) w^n $, which are analytically continued in $ w $ around 0.
• A Cauchy-type integral representation is derived: $ p_n(x) = \frac{1}{2\pi i} \int_{\mathcal{C}} F(x,w) w^{-n-1} dw $, where $ \mathcal{C} $ is a contour enclosing the origin.
• The generating function is factored as $ F(x,w) = e^{Aw - Bw^2} f(x,w) $, with $ f(x,w) $ analytic and expanded as a power series $ f(x,w) = \sum_{k=0}^\infty c_k w^k $.
• The polynomial $ p_n(x) $ is expressed as a finite sum: $ p_n(x) = \sum_{k=0}^n B^{n-k} c_k \frac{H_{n-k}(\xi)}{(n-k)!} $, where $ H_n $ are Hermite polynomials, $ \xi = A/(2\sqrt{B}) $, and $ B = \frac{1}{2}p_1^2(x) - p_2(x) $.
• To achieve asymptotic equivalence to Laguerre polynomials, the coefficients $ c_k $ are controlled by setting $ c_1 = c_2 = \cdots = 0 $ via parameter scaling, ensuring $ c_k = \mathcal{O}(\mu^{\lfloor k/n \rfloor}) $ as a parameter $ \mu \to \infty $.
• The method is applied to Jacobi, Meixner, and Krawtchouk polynomials by solving $ c_1 = 0 $ under parameter limits, yielding expansions of the form $ p_n(x) = B^n \left[ L_n^{(C)}(\xi) + \mathcal{O}(\cdot) \right] $, with explicit parameter dependencies.

Experimental results

Research questions

• RQ1 How can asymptotic expansions of hypergeometric orthogonal polynomials be derived from their generating functions?
• RQ2 What parameter scalings lead to uniform asymptotic approximations of discrete orthogonal polynomials (e.g., Meixner, Krawtchouk) by Laguerre polynomials?
• RQ3 Can the standard limit relations in the Askey scheme be replaced by full asymptotic expansions that capture higher-order behavior?
• RQ4 What is the role of the generating function's analytic structure in deriving asymptotic representations via contour integration?
• RQ5 How do the coefficients $ c_k $ in the expansion of $ f(x,w) $ control the asymptotic behavior of the polynomial sequence?

Key findings

• The Meixner-Pollaczek polynomials asymptotically approach Laguerre polynomials as $ \lambda \to \infty $, with $ P_n^{(\lambda)}(x;\phi) = B^n \left[ L_n^{(C)}(\xi) + \mathcal{O}(r^{n-3}) \right] $, where $ r = \sqrt{x^2 + \lambda^2} $, and the approximation is uniform in $ \theta $.
• For Jacobi polynomials, under $ \alpha + \beta \to \infty $, $ P_n^{(\alpha,\beta)}(x) = L_n^{(C)}(\xi) + \mathcal{O}(\gamma^{n-1}) $, with $ \gamma = \alpha + \beta $, $ \xi = \frac{1}{2}(\alpha + \beta + 2)(1 - x) $, and the limit $ \lim_{\beta \to \infty} P_n^{(\alpha,\beta)}(1 - 2\xi/(2 + \alpha + \beta)) = L_n^{(\alpha)}(\xi) $ is recovered.
• For Meixner polynomials, under $ \beta = \alpha + 1 $ and $ c \to 1 $, $ M_n(x;\beta,c) = L_n^{(\alpha)}(\xi) + \mathcal{O}(1 - c) $, with $ \xi = \frac{(\alpha - \beta + 1)c + (1 - c)x}{c} $, and the limit $ \lim_{c \to 1} M_n(c\xi/(1 - c); \alpha + 1, c) = \frac{L_n^{(\alpha)}(\xi)}{L_n^{(\alpha)}(0)} $ is obtained.
• For Krawtchouk polynomials, as $ N \to \infty $, $ \binom{N}{n} K_n(x;p,N) = L_n^{(C)}(\xi) + \mathcal{O}(N^{n-1}) $, with $ \xi = \alpha + 1 - N + (1 + q)x $, $ q = (1 - p)/p $, and $ C = \alpha $, showing convergence to Laguerre polynomials.
• The method yields full asymptotic expansions with explicit error terms, such as $ \mathcal{O}(\gamma^{n-1}) $, $ \mathcal{O}(r^{n-3}) $, and $ \mathcal{O}(N^{n-1}) $, which are uniform in the relevant parameters and variables.
• The approach generalizes beyond hypergeometric polynomials, as demonstrated by its successful application to Bernoulli and Euler polynomials in prior work, confirming its robustness.