*(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:

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

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

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

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

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

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

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:

and

where

The cases 3 and 4 provide asymptotic approximations for the ultraspherical (Gegenbauer) polynomials when $n\in \mathbb{N}$. 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) |