Abstract not found

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

. The Kneser graph K(n; k) has as vertices all the k-subsets of a fixed n-set and has as edges the pairs fA; Bg of vertices such that A and B are disjoint. It is known that these graphs are Hamiltonian if \Gamma n\Gamma1 k\Gamma1 \Delta \Gamma n\Gammak k \Delta for n 2k + 1. We determine asymptotically for fixed k the minimum value n = e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution of k\Gamma(n)\Gamma(n \Gamma 2k + 1) = \Gamma 2 (n \Gamma k + 1) for n 2k + 1, as k !1, when n and k are not restricted to take integer values. We also show that for all prime powers q and n 2k, k 1, the q-analogues K q (n; k) are Hamiltonian by consideration of the analogous inequality for q-binomial coefficients. 1. Introduction. The Kneser graph K(n; k) is the graph whose vertices are the k-subsets of the set [n] = f1; 2; : : : ; ng and whose edges are the pairs fA; Bg of k-subsets such that A and B are disjoint [15]. If n = 2k + 1, K(n;...

Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and q-gamma functions. 1.

Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality for ratios of gamma

Abstract. The main aim of this paper is to prove that the double inequality