Integral Representation of Some Functions Related to the Gamma Function

By Christian Berg. 2004

164 citations recorded by the source corpus.

Abstract.We prove that the functions $$\Phi (x) = [\Gamma (x + 1)]^{1/x} (1 + 1/x)^x /x$$ and $$\log \Phi (x)$$ are Stieltjes transforms.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/b9a69c3d706b853b6fa59204ebea0a0961f640a3

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In Applied Mathematics

Citations in the corpus

A PROPERTY OF LOGARITHMICALLY ABSOLUTELY MONOTONIC FUNCTIONS AND THE LOGARITHMICALLY COMPLETE MONOTONICITY OF A POWER-EXPONENTIAL FUNCTION (2009, background): “It is said in [7] that “In various papers complete monotonicity for special functions has been established by proving the stronger statement that the function is a Stieltjes transform”.”
Bounds for the Ratio of Two Gamma Functions (2009, background): “For more information, please refer to [10, 59, 60] and related references therein.”
Logarithmically completely monotonic functions relating to the gamma function (2006, background): “Motivated by the papers [25,26], among other things, it is proved in [9] that S \ {0} ⊂ L⊂ C.”
Mills' ratio: Monotonicity patterns and functional inequalities (2008, result): “Berg [7] pointed out that the logarithmically complete monotonic functions in fact are the same as those studied by R.”
Refinements of lower bounds for polygamma functions (2009, background): “that all logarithmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [15]. For more information, please refer to [7, 20]. Lemma2. Theinequality ψ′(t) <e1/t−1 (16) holdson(0,∞). Moregenerally, thefunction e1/t−ψ”
TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION (2009, background): “Recall also [4, 5, 8, 16, 18] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (3) (−1)n[ln f(x)] ≥ 0 for all x ∈ I and n ∈ N.”
Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity (2007, unknown): “See Example (c) in Section 4.)”
The Best Bounds in Gautschi-Kershaw Inequalities (2006, unknown): “([ 2, 14, 15, 16 ]) A logarithmically completely monotonic function is also completely monotonic.”
Completely monotonic functions involving the gamma and q-gamma functions (2005, background): “This holds in a wider sense without the standard normalization f (0) = 1, and also for f (0) = ∞ (cf. [8]).”
Three classes of logarithmically completely monotonic functions involving gamma and psi functions (2007, background): “[10] that S \ {0} ⊂ L[(0, ∞)] ⊂ C[(0, ∞)], where S denotes the set of Stieltjes transforms.”
A Completely Monotonic Function Involving Divided Differences of Psi and Polygamma Functions and an Application (2006, unknown): “It was proved explicitly in [10, 42, 45, 46, 53] by different approaches that any logarithmically completely monotonic function must be completely monotonic, but not conversely.”
On potentially negative space time covariances obtained as sum of products of marginal ones (2008, background): “De Iaco, Myers and Posa (2001), with some algebraic manipulation, showed that model (4) can be rewritten as γs,t(h, u) = γs,t(h, 0) + γs,t(0, u) − kγs,t(h, 0)γs,t(0, u)”