Asymptotic expansions, integral means and applications to the special functions

Mihoković, Lenka (2016) Asymptotic expansions, integral means and applications to the special functions. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Abstract

As an introduction to the main part of this work, the asymptotic expansion of the integral mean of polygamma function was derived. The recursive relation which determines coefficients in this expansion was given. This approach is then generalized to the integral mean of the functions which possess asymptotic power series expansion. To this end, algorithm for solving equations of the form $B(A(x)) = C(x)$ is obtained, where asymptotic expansions of functions B and C are known, and the results are illustrated by the examples of some relevant integral means. Studying asymptotic expansions turned out to be useful when comparing means. Thus, the necessary conditions for the comparison of parametric means were derived. This thesis is divided in several parts. In the introduction, asymptotic expansions were defined and some basic properties about manipulations with asymptotic series were described. Second chapter deals with gamma and related functions. Using known results and expressions through quotient of gamma functions, asymptotic expansion of Wallis sequence $W_n$ was derived, which provided improvements of some existing inequalities related to this sequence. In the same manner, coefficients in asymptotic expansion of the Wallis power function were found and some properties were described. Asymptotic expansion of the Wallis function through polygamma function was derived. Then the coefficients in asymptotic expansion of integral mean of polygamma function were found which motivated the studying of integral means in general. In the third and main chapter the algorithm for calculating coefficients in asymptotic expansion of integral mean of a function possessing asymptotic expansion was described and obtained results were explained on some specific examples. Similar techniques were used in the fourth chapter in the analysis of classical and parameter means. By knowing coefficients in asymptotic expansions, necessary conditions for comparison of those means were derived. Convex combinations of two means which include Seiffert or Neuman-Sándor mean were also observed. Some new inequalities were proved while others were stated in form of conjectures.

Item Type: Thesis (Doctoral thesis)
Supervisor: Elezović, Neven
Date: 2016
Number of Pages: 134
Subjects: NATURAL SCIENCES > Mathematics > Applied Mathematics and Mathematic Modeling
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 01 Jul 2016 10:43
Last Modified: 01 Jul 2016 10:43
URI: http://digre.pmf.unizg.hr/id/eprint/4940

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