Asymptotics of the uniform norm of associated Legendre functions <i>P</i><sup><i>k</i></sup><sub><i>n</i></sub> (the case <i>k</i> ≪ <i>n</i>)
Kholshevnikov K.V., Shaidulin V.Sh. Asymptotics of the uniform norm of associated Legendre functions P <sup>k</sup><sub>n </sub>(the case k << n) // Vestnik St.Petersburg University. Ser. 1. 2009. Issue 2. P. 86-93. The uniform (Chebyshevian) norm of the associated Legendre harmonics P^(x) on the orthogonality segment -1 ≤ x ≤ 1 is investigated. The main goal is to examine an asymptotic behaviour while the subscript tends to infinity. The case of a fixed superscript, or one increasing slower than n<sup>2/3</sup>, is considered. It is obtained that the norm increases as n<sup>k</sup>. A corresponding coefficient is equal to the greatest value of the Bessel function of order k. The asymptotics of the coefficient is also established: decreasing as k<sup>-1//3</sup> with an exact numerical coefficient. Direct calculations show that the asymptotic approximation is satisfactory even for the first small values of the superscript.