The paper is devoted to uniform asymptotic approximations in the complex plane of the solutions of the Jacobi’s differential equation: $$ (1-x^2){d^2y\over{dx^2}}+[(q-p)- (p+q+2)x]{dy\over{dx}}+n(n+p+q+1)y=0. \tag 1$$ 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: $$ n+\tfrac 12(p+q+1)\to\infty. $$ Here one or more of the parameters $n$, $p$, and $q$ are ”large” (with $n$ not necessarily an integer).
When $n\in{\Bbb N}$, the most important solution of (1) is the Jacobi polynomial, given by: $$ P_n^{(p,q)}(x)= {\Gamma(n+p+1)\over{n!\Gamma(n+p+q+1)}} \sum_{k=0}^n{n\choose k}{\Gamma(n+p+q+k+1)\over{\Gamma(p+k+1)}} \Biggl({x-1\over 2}\Biggr)^k. \tag 2$$ When $n\in{\Bbb N}$ and $p=q$, solutions of (1) are the ultraspherical (Gegenbauer) polynomials, given by: $$ G_n^{(r)}(x)= {\Gamma(r+{1\over 2})\Gamma(2r+n) \over{\Gamma(2r)\Gamma(r+n+{1\over 2})}} P_n^{(r-{1\over 2},r-{1\over 2})}(x). $$ When $n\not\in{\Bbb N}$, $P_n^{(p,q)}(x)$ is no longer a polynomial, and (2) generalizes to $$ P_n^{(p,q)}(x)= {\Gamma(n+p+1)\over{\Gamma(n+1)\Gamma(p+1)} } F(-n,n+p+q+1;p+1;\tfrac 12(1-x)), $$ 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 {\it 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 $0\le n\le O(q)$.
2. $n$ is large, $p$ is fixed, and $0\le q\le O(n)$.
3. $p$ and $q$ are large and equal, and $0\le n\le O(p)$.
4. $n$ is large, and $0\le p=q\le O(n)$ (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: $$ P_n^{(p,q)}(x)= e^{\mp n\pi i}P_n^{(q,p)}(xe^{\pm\pi i})+ {2e^{\mp(n+p)\pi i}\sin(n\pi)\over\pi} Q_n^{(p,q)}(x) $$ and $$ Q_n^{(q,p)}(xe^{\pm\pi i})= -e^{\mp(n+p+q)\pi i}Q_n^{(p,q)}(x), $$ where $$ Q_n^{(p,q)}(x)= {1\over 2}^{n+1}(x-1)^p(x+1)^q \int_{-1}^1{(1-t)^{n+p}(1+t)^{n+q}\over{(x-t)^{n+1}}} dt = $$ $$ ={1\over 2}(x-1)^p(x+1)^q \int_{-1}^1{(1-t)^p(1+t)^q\over{x-t}} P_n^{(p,q)}(t) dt,\quad x\not\in[-1,1]. $$ The cases 3 and 4 provide asymptotic approximations for the ultraspherical (Gegenbauer) polynomials when $n\in{\Bbb N}$. Explicit error bounds are obtained for all the approximations.
To obtain the results, the author widely uses the theory developed in the articles [{\it W. G. C. Boyd} and {\it T. M. Dunster}, SIAM J. Math. Anal. 17, 422-450 (1986;

Zbl 0591.34048) and {\it T. N. Dunster}, SIAM J. Math. Anal. 25, No. 322-353 (1994;

Zbl 0798.34062)].