Asymptotic Expansions, Inequalities And Completely Monotonic Functions For Special Functions

In this paper, we prove some asymptotic expansions, inequalities and completely monotonic functions for special functions. The main results are as follows:1. In 2014, C.-P. Chen [1] proposed the following two conjectures:(i) For all m ∈ N0, the functions are completely monotonic on(0, ∞).(ii) For all m ∈ N0, the functions are completely monotonic on(0, ∞).In the second section of the first chapter, we prove these two conjectures(see Theorem 1.2.1 and Theorem 1.2.2). In the third section, we establish asymptotic expansions of the logarithm of the gamma function terms of the polygamma functions(see Theorem 1.3.1 and Theorem 1.3.2). As by products, we deduce the recursive relations of the Bernoulli numbers:The results of the first chapter are published in Math. Inequal. Appl. 18(2015),no. 1, 379–388(SCI Journal).2. The factorial function has the following approximation formulas: where ω =(3-√3)/6. Burnside [2] proved the formula(0.0.8), the formulas(0.0.9) and(0.0.10) due to Mortici [3]. Using Maple, we find the following faster approximations:In the second chapter, our Theorem 2.1.1 unifies the approximation formulas(0.0.7)–(0.0.12), and develops them to complete asymptotic expansions. More precisely, we prove the following result: Let a, b, r ∈ R, r ?= 0. The gamma function has the following asymptotic expansion:: with the coefficients q j≡ q j(a, b, r)(j ∈ N) given by the following formula: summed over all nonnegative integers k j satisfying the equation and Bndenote the Bernoulli numbers.Theorem 2.1.2 gives new asymptotic expansions of the gamma function. Theorem2.1.3 and Theorem 2.1.4 establish the symmetric double inequalities for the gamma function.The results of the second chapter are published in J. Number Theory 149(2015),313–326(SCI Journal).3. In the third chapter, we establish sharp inequalities for the Euler-Mascheroni constant. The results are published in the College Mathematics, 30(2014), No. 6, 26-31.