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The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... 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 asy ..."
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Cited by 15 (1 self)
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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.
NECESSARY AND SUFFICIENT CONDITIONS FOR A FUNCTION INVOLVING DIVIDED DIFFERENCES OF THE DI AND TRIGAMMA FUNCTIONS TO BE COMPLETELY MONOTONIC
, 903
"... Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two g ..."
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Cited by 7 (7 self)
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Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.
Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, submitted
"... Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, clai ..."
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Cited by 5 (5 self)
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Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents
Binomial And QBinomial Coefficient Inequalities Related To The Hamiltonicity Of The Kneser Graphs And Their QAnalogues
 J. Combin. Theory Ser. A
, 1995
"... . The Kneser graph K(n; k) has as vertices all the ksubsets of a fixed nset 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 as ..."
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Cited by 4 (1 self)
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. The Kneser graph K(n; k) has as vertices all the ksubsets of a fixed nset 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 qanalogues K q (n; k) are Hamiltonian by consideration of the analogous inequality for qbinomial coefficients. 1. Introduction. The Kneser graph K(n; k) is the graph whose vertices are the ksubsets of the set [n] = f1; 2; : : : ; ng and whose edges are the pairs fA; Bg of ksubsets such that A and B are disjoint [15]. If n = 2k + 1, K(n;...
Some monotonicity properties of gamma and qgamma functions, Available onlie at http://arxiv.org/abs/0709.1126v2
"... Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and qgamma functions. 1. ..."
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Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and qgamma functions. 1.
Complete monotonicity of some functions involving polygamma functions, submitted
"... Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) ..."
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Cited by 3 (3 self)
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Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) (x) are polygamma functions. 1.
A COMPLETE SOLUTION TO AN OPEN PROBLEM RELATING TO AN INEQUALITY FOR RATIOS OF GAMMA FUNCTIONS
, 902
"... 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 fo ..."
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Cited by 2 (2 self)
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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
Supplements to a class of logarithmically completely monotonic functions associated with the gamma function
 Appl. Math. Comput
"... Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the ps ..."
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Cited by 1 (1 self)
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Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived. This is a continuator of the paper [12]. 1.
(k − 1)! h (k−1)!
, 903
"... Abstract. The main aim of this paper is to prove that the double inequality ..."
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Abstract. The main aim of this paper is to prove that the double inequality