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 and .
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 , , and are ”large” (with not necessarily an integer).
When , the most important solution of (1) is the Jacobi polynomial, given by:
When and , solutions of (1) are the ultraspherical (Gegenbauer) polynomials, given by:
When , is no longer a polynomial, and (2) generalizes to
where 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 can lie in unbounded complex domains):
1. is large, is fixed, and .
2. is large, is fixed, and .
3. and are large and equal, and .
4. is large, and (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 and can be interchanged using the connection formulas:
The cases 3 and 4 provide asymptotic approximations for the ultraspherical (Gegenbauer) polynomials when . 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)].