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Error bounds for asymptotic approximations of special functions. (English) Zbl 0879.41016

The aim of this survey paper, which is dedicated to the reviewer on the occasion of his 70th birthday, is two-fold: firstly, it gives a clear account of the reviewer’s work on the construction of numerical bounds associated with asymptotics approximations. Secondly, it illustrates how these results have been used recently by the author to prove certain monotonicity properties of special functions. More precisely, the author illustrates how it has been possible to prove three important conjectures: The first conjecture, due to Szegö, concerns the monotonicity of the Lebesgue constants for Legendre series, defined by

${L}_{n}=\frac{n+1}{2}{\int }_{1}^{1}\left|{P}_{n}^{\left(1,0\right)}\left(x\right)\right|dx,\phantom{\rule{1.em}{0ex}}n=1,2,\cdots ,$

where ${P}_{n}^{\left(1,0\right)}\left(x\right)$ is the Jacobi polynomial with $\alpha =1$ and $\beta =0$. Replacing in the integral the Jacobi polynomial by a uniform asymptotic expansion, C. K. Qu and R. Wong [Pac. J. Math. 135, 157-188 (1988; Zbl 0664.42012)] obtain for ${L}_{n}$ an asymptotic expansion with bound for the error term which allows to prove the conjecture. In a similar way R. Wong and J.-M. Zhang [SIAM J. Math. Anal. 46, No. 6, 1318-1337 (1994; Zbl 0819.33004)] succeed in proving a conjecture of Askey concerning the monotonicity of the relative extrema of the Jacobi polynomial ${P}_{n}^{\left(0,-1\right)}\left(x\right)$. Finally, it is described the approach used by R. Wong and T. Lang [Can. J. Math. 43, 628-651 (1991; Zbl 0731.33001)] to prove a certain monotonicity property of the inflection points of the Bessel function ${J}_{n}\left(x\right)$, conjectured by Lorch and Szegö. This well-written paper, which presents some interesting new results as well as a Bibliography of 46 items, will be very useful to researchers in the area of special functions.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C45 Orthogonal polynomials and functions of hypergeometric type