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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. (English) Zbl 0982.33003

The paper is devoted to uniform asymptotic approximations in the complex plane of the solutions of the Jacobi’s differential equation:

$\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+\left[\left(q-p\right)-\left(p+q+2\right)x\right]\frac{dy}{dx}+n\left(n+p+q+1\right)y=0·\phantom{\rule{2.em}{0ex}}\left(1\right)$

This equation is characterized by having regular singularities at $x=±1$ and $x=\infty$.

The author obtains asymptotic approximations in the case of nonnegative values of the parameters satisfying the following condition:

$n+\frac{1}{2}\left(p+q+1\right)\to \infty ·$

Here one or more of the parameters $n$, $p$, and $q$ are ”large” (with $n$ not necessarily an integer).

When $n\in ℕ$, the most important solution of (1) is the Jacobi polynomial, given by:

${P}_{n}^{\left(p,q\right)}\left(x\right)=\frac{{\Gamma }\left(n+p+1\right)}{n!{\Gamma }\left(n+p+q+1\right)}\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{\Gamma }\left(n+p+q+k+1\right)}{{\Gamma }\left(p+k+1\right)}{\left(\frac{x-1}{2}\right)}^{k}·\phantom{\rule{2.em}{0ex}}\left(2\right)$

When $n\in ℕ$ and $p=q$, solutions of (1) are the ultraspherical (Gegenbauer) polynomials, given by:

${G}_{n}^{\left(r\right)}\left(x\right)=\frac{{\Gamma }\left(r+\frac{1}{2}\right){\Gamma }\left(2r+n\right)}{{\Gamma }\left(2r\right){\Gamma }\left(r+n+\frac{1}{2}\right)}{P}_{n}^{\left(r-\frac{1}{2},r-\frac{1}{2}\right)}\left(x\right)·$

When $n\notin ℕ$, ${P}_{n}^{\left(p,q\right)}\left(x\right)$ is no longer a polynomial, and (2) generalizes to

${P}_{n}^{\left(p,q\right)}\left(x\right)=\frac{{\Gamma }\left(n+p+1\right)}{{\Gamma }\left(n+1\right){\Gamma }\left(p+1\right)}F\left(-n,n+p+q+1;p+1;\frac{1}{2}\left(1-x\right)\right),$

where $F$ denotes the hypergeometric function.

The problem of uniform asymptotic approximations of the solutions of equation (1) has a long history. One can find a lot of the bibliographical references in the paper. We only mention the monograph of F. W. J. Olver [“Asymptotics and Special Functions”, Academic Press, New York (1974; Zbl 0303.41035; Reprint A K Peters, Wellesley (1997; Zbl 0982.41018)].

The author investigates the asymptotic behavior of the solutions of (1) for the following four cases (the argument $x$ can lie in unbounded complex domains):

1. $q$ is large, $p$ is fixed, and $0\le n\le O\left(q\right)$.

2. $n$ is large, $p$ is fixed, and $0\le q\le O\left(n\right)$.

3. $p$ and $q$ are large and equal, and $0\le n\le O\left(p\right)$.

4. $n$ is large, and $0\le p=q\le O\left(n\right)$ (the ultraspherical polynomials).

The author obtains the results by an application of existing asymptotic theories of a coalescing turning point and simple pole in the complex plane, and of a coalescing turning point and double pole in the complex plane. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either Whittaker confluent hypergeometric functions or Bessel functions. In cases 1 and 2, the roles of $p$ and $q$ can be interchanged using the connection formulas:

${P}_{n}^{\left(p,q\right)}\left(x\right)={e}^{\mp n\pi i}{P}_{n}^{\left(q,p\right)}\left(x{e}^{±\pi i}\right)+\frac{2{e}^{\mp \left(n+p\right)\pi i}sin\left(n\pi \right)}{\pi }{Q}_{n}^{\left(p,q\right)}\left(x\right)$

and

${Q}_{n}^{\left(q,p\right)}\left(x{e}^{±\pi i}\right)=-{e}^{\mp \left(n+p+q\right)\pi i}{Q}_{n}^{\left(p,q\right)}\left(x\right),$

where

${Q}_{n}^{\left(p,q\right)}\left(x\right)={\frac{1}{2}}^{n+1}{\left(x-1\right)}^{p}{\left(x+1\right)}^{q}{\int }_{-1}^{1}\frac{{\left(1-t\right)}^{n+p}{\left(1+t\right)}^{n+q}}{{\left(x-t\right)}^{n+1}}dt=$
$=\frac{1}{2}{\left(x-1\right)}^{p}{\left(x+1\right)}^{q}{\int }_{-1}^{1}\frac{{\left(1-t\right)}^{p}{\left(1+t\right)}^{q}}{x-t}{P}_{n}^{\left(p,q\right)}\left(t\right)dt,\phantom{\rule{1.em}{0ex}}x\notin \left[-1,1\right]·$

The cases 3 and 4 provide asymptotic approximations for the ultraspherical (Gegenbauer) polynomials when $n\in ℕ$. Explicit error bounds are obtained for all the approximations.

To obtain the results, the author widely uses the theory developed in the articles [W. G. C. Boyd and T. M. Dunster, SIAM J. Math. Anal. 17, 422-450 (1986; Zbl 0591.34048) and T. N. Dunster, SIAM J. Math. Anal. 25, No. 322-353 (1994; Zbl 0798.34062)].

MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_{1}{F}_{1}$ 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34E05 Asymptotic expansions (ODE) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)