## MSC: Primary 33B15; Secondary 41A60 (2009)

### BibTeX

@MISC{Koum09msc:primary,

author = {Stamatis Koum and Henrik L. Pedersen and Keywords Gamma Function and Barnes G-function},

title = {MSC: Primary 33B15; Secondary 41A60},

year = {2009}

}

### OpenURL

### Abstract

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function

### Citations

255 | The Laplace Transform - Widder - 1941 |

76 | Special Functions. An Introduction to the Classical Functions of Mathematical Physics - Temme - 1996 |

64 | Ranjan Special functions Encyclopedia of Mathematics and its Applications 71 - Andrews, Askey, et al. - 1999 |

38 | On some inequalities for the gamma and psi function
- Alzer
- 1997
(Show Context)
Citation Context ... the logarithm of Euler’s gamma function ( log Γ(x) = x − 1 ) log x − x + 2 1 log(2π) + 2 + n∑ k=1 B2k (2k − 1)2k 1 x 2k−1 + (−1)n Rn(x), (2) where B2k are the Bernoulli numbers. It has been shown in =-=[2]-=- that for all n = 0,1,2,..., the remainder Rn(x) is completely monotonic on (0, ∞). We strengthen this result in Theorem 2.1 below. For t > 0 we write t et t = 1 − − 1 2 + n∑ k=1 B2k (2k)! t2k + (−1) ... |

11 |
M.: On Barnes’ multiple zeta and gamma functions
- Ruijsenaars
- 2000
(Show Context)
Citation Context ...em 1.3. n (0) = 0 for k ≤ n − 1. It follows (t) > 0 for t > 0, and the result follows from □ 4 Barnes double gamma function Concerning the notation of the double gamma function we follow Ruijsenaars, =-=[10]-=-. His asymptotic expansion of the logarithm of Barnes double gamma function (denoted Ψ2) is different from the expansion found by Ferreira and López and it is given in terms of double Bernoulli polyno... |

9 |
Error bounds for asymptotic expansions of the ratio of two gamma functions
- Frenzen
(Show Context)
Citation Context ...lt regarding complete monotonicity of the remainder of an asymptotic expansion of a ratio of gamma functions of the form Γ(x + a)/Γ(x + b) in terms of powers of 1/(x + c), is given in Frenzen’s paper =-=[5]-=-. Remark 2.2 In [7] it is proved that Vn(t) takes also the form Vn(t) = 1 (2n + 1)! 1 e t − 1 ∫ 1 e 0 tu (−1) n B2n+1(u)du, n ≥ 0, where B2n+1(u) are the Bernoulli polynomials. We observe that this fo... |

6 |
Completely monotonic and related functions
- Haeringen
- 1996
(Show Context)
Citation Context ...nd satisfies (−1) n f (n) (x) ≥ 0, for all x > 0 and n = 0,1,2,... (1) J. Dubourdieu [3] proved that if a non constant function f is completely monotonic then strict inequality holds in (1). See also =-=[6]-=- for a simpler proof 1of this result. A characterization of completely monotonic functions is given by Bernstein’s theorem, see [14, p. 161], which states that f is completely monotonic if and only i... |

5 |
Sur un theoreme de M. S. Bernstein relatif ala transformation de Laplace-Stieltjes
- DUBOURDIEU
(Show Context)
Citation Context ... preliminary results. A function f : (0, ∞) → R is called completely monotonic if f has derivatives of all orders and satisfies (−1) n f (n) (x) ≥ 0, for all x > 0 and n = 0,1,2,... (1) J. Dubourdieu =-=[3]-=- proved that if a non constant function f is completely monotonic then strict inequality holds in (1). See also [6] for a simpler proof 1of this result. A characterization of completely monotonic fun... |

4 |
Remarks on some completely monotonic functions
- Koumandos
(Show Context)
Citation Context ...te monotonicity of the remainder of an asymptotic expansion of a ratio of gamma functions of the form Γ(x + a)/Γ(x + b) in terms of powers of 1/(x + c), is given in Frenzen’s paper [5]. Remark 2.2 In =-=[7]-=- it is proved that Vn(t) takes also the form Vn(t) = 1 (2n + 1)! 1 e t − 1 ∫ 1 e 0 tu (−1) n B2n+1(u)du, n ≥ 0, where B2n+1(u) are the Bernoulli polynomials. We observe that this formula extends to t ... |

3 | An asymptotic expansion of the Double Gamma function - Ferreira, López |

3 |
Superadditive functions and a statistical application
- Trimble, Wells, et al.
- 1989
(Show Context)
Citation Context ...e −xt dµ(t), 0 where µ is a nonnegative measure on [0, ∞) such that the integral converges for all x > 0. Here we are interested in the class of strongly completely monotonic functions, introduced in =-=[13]-=-. A function f : (0, ∞) → R is called strongly completely monotonic if it has derivatives of all orders and (−1) n x n+1 f (n) (x) is nonnegative and decreasing on (0, ∞) for all n = 0,1,2,.... (It is... |

1 |
On Ruijsenaars’ asymptotic expansion of the logarithm of the double gamma function
- Koumandos
(Show Context)
Citation Context ... ∫ 1 e 0 tu (−1) n B2n+1(u)du, n ≥ 0, where B2n+1(u) are the Bernoulli polynomials. We observe that this formula extends to t = 0 because of the well-known property for n ≥ 0. ∫ 1 0 B2n+1(u)du = 0 In =-=[8]-=- it is proved that Vn is positive, decreasing and satisfies (t 2 Vn(t)) ′ > 0. We extend the last property in the following Lemma 2.3. Lemma 2.3 The function Vn has the following properties: (i) (t 2j... |

1 |
On the remainder in an asymptotic expansion of the double gamma function
- Pedersen
(Show Context)
Citation Context ...nsion (due to C. Ferreira and J. L. López, [4, Theorem 1]) of the logarithm of Barnes G-function. This function is defined as an infinite product and satisfies G(1) =1 and G(z+1) = Γ(z)G(z). See also =-=[9]-=- for details and additional considerations. The remainder in this expansion takes the form Pn(x) = (−1) n ∫ ∞ where Vn(t) is as above. 0 e −xt t 2n−1 Vn(t)dt, Theorem 3.1 The remainder (−1) n Pn(x) is... |

1 |
The remainder in Ruijsenaars’ asymptotic expansion of Barnes double gamma function
- Pedersen
(Show Context)
Citation Context ...−k + R2,M(w), k=3 where the remainder R2,M has the representation ∫ ∞ e R2,M(w) = −wt t3 ( t2 (1 − e−t M∑ (−1) − ) 2 k k! 0 k=0 B2,kt k Here, ℜw > 0 and M ≥ 2. See [11, (3.13) and (3.14)]. In [8] and =-=[10]-=- it was shown independently that (−1) n−1R2,2n(x) is a completely monotonic function. Below it is verified that it is indeed a completely monotonic function of order k for n ≥ k + 1. We briefly indica... |