## ANNALES POLONICI MATHEMATICI

### BibTeX

@MISC{_annalespolonici,

author = {},

title = {ANNALES POLONICI MATHEMATICI},

year = {}

}

### OpenURL

### Abstract

Some monotonicity and limit results for the regularised incomplete gamma function by Wojciech Chojnacki (Adelaide and Warszawa) Abstract. Letting P (u, x) denote the regularised incomplete gamma function, it is shown that for each α ≥ 0, P (x, x + α) decreases as x increases on the positive real semiaxis, and P (x, x + α) converges to 1/2 as x tends to infinity. The statistical significance

### Citations

848 | A course of Modern Analysis - Whittaker, Watson - 1978 |

38 | On some inequalities for the gamma and psi function - Alzer - 1997 |

28 | Some completely monotonic functions involving polygamma functions and an application - Qi, Cui, et al. |

14 | The incomplete Gamma functions since Tricomi,” in Tricomi’s ideas and contemporary applied mathematics, Atti dei Convegni Lincei, no. 147, Accademia Nazionale dei Lincei
- Gautschi
- 1998
(Show Context)
Citation Context ...> 0, x ≥ 0) = 1 − P (u, x) also appear in many different contexts and applications. An extended and highly readable overview on the incomplete gamma function and the related functions can be found in =-=[2]-=-. For a sample of more recent work, see [3]. The aim of this paper is to prove that for each α ≥ 0, (i) P (x, x + α) decreases as x increases on the positive real semi-axis; and (ii) P (x, x + α) tend... |

12 |
Some inequalities for the gamma function
- Kečlić, Vasić
(Show Context)
Citation Context ...le represents P (x, x) as P (x, x) = p1(x)p2(x), where p1(x) ∆ = xx−1e−x /Γ (x) and p2(x) ∆ = γ(x, x)x1−xex , and claims that both p1 and p2 are decreasing. But while the first function is decreasing =-=[4]-=-, the second is not. Figure 1 illustrates the (a) (b) Fig. 1. Contrasting behaviours of p1 and p2: (a) graph of p1; (b) graph of p2. x = linspace(0,10); % 100 equally % spaced values % between 0 and 1... |

4 | Remarks on some completely monotonic functions - Koumandos |

3 |
Functional inequalities for incomplete gamma and related functions
- ISMAIL, LAFORGIA
(Show Context)
Citation Context ...any different contexts and applications. An extended and highly readable overview on the incomplete gamma function and the related functions can be found in [2]. For a sample of more recent work, see =-=[3]-=-. The aim of this paper is to prove that for each α ≥ 0, (i) P (x, x + α) decreases as x increases on the positive real semi-axis; and (ii) P (x, x + α) tends to 1/2 as x → ∞. 2000 Mathematics Subject... |

2 |
Some inequalities for the chi square distribution function and the exponential function
- Merkle
- 1993
(Show Context)
Citation Context ... − 1)} ∞ n=1 increases, with 1/2 being the common limit of both sequences. Van de Lune [10] and, independently, Temme [8] proved that the function x ↦→ P (x, x − 1) increases to 1/2 on [1, ∞). Merkle =-=[6]-=- asserted that the function x ↦→ P (x, x) is decreasing on (0, ∞), but his argument to validate the statement is incorrect. Merkle represents P (x, x) as P (x, x) = p1(x)p2(x), where p1(x) ∆ = xx−1e−x... |

2 |
Some problems in connection with the incomplete gamma functions
- Temme
- 1980
(Show Context)
Citation Context ...[12] proved that the sequence {P (n, n)} ∞ n=1 decreases and the sequence {P (n, n − 1)} ∞ n=1 increases, with 1/2 being the common limit of both sequences. Van de Lune [10] and, independently, Temme =-=[8]-=- proved that the function x ↦→ P (x, x − 1) increases to 1/2 on [1, ∞). Merkle [6] asserted that the function x ↦→ P (x, x) is decreasing on (0, ∞), but his argument to validate the statement is incor... |

2 |
Dritter Beweis der die unvollstandige Gammafunktion betreenden Lochschen Ungleichungen
- Vietoris
- 1983
(Show Context)
Citation Context ...t’s result is insufficient to infer that the sequence {P(S′2 n ≤ σ2 )} ∞ n=1 decreases. However, as was already alluded to earlier, this latter result follows immediately from our Theorem 1. Vietoris =-=[12]-=- proved that the sequence {P (n, n)} ∞ n=1 decreases and the sequence {P (n, n − 1)} ∞ n=1 increases, with 1/2 being the common limit of both sequences. Van de Lune [10] and, independently, Temme [8] ... |

1 | Functions: An Introduction to the Classical Functions of Mathematical Physics, A Wiley-Interscience Publication - Special - 1996 |

1 |
de Lune, A note on Euler’s (incomplete) gamma function
- van
- 1975
(Show Context)
Citation Context ...y from our Theorem 1. Vietoris [12] proved that the sequence {P (n, n)} ∞ n=1 decreases and the sequence {P (n, n − 1)} ∞ n=1 increases, with 1/2 being the common limit of both sequences. Van de Lune =-=[10]-=- and, independently, Temme [8] proved that the function x ↦→ P (x, x − 1) increases to 1/2 on [1, ∞). Merkle [6] asserted that the function x ↦→ P (x, x) is decreasing on (0, ∞), but his argument to v... |

1 |
der Vaart, Some extensions of the idea of bias
- van
- 1961
(Show Context)
Citation Context ...≤ x) = 2 (n−1)/2Γ ( 1 t 2 (n − 1)) 0 (n−1)/2−1 e −t/2 dt = P ( 1 1 2 (n − 1), 2x) (x ≥ 0), with P(A) denoting the probability of the event A. Furthermore, in accordance with a result of van der Vaart =-=[11]-=-, S ′2 n is a negatively median-biased estimator of σ2 in the sense that (1) P(S ′2 n ≤ σ 2 ) > 1 2 for each n. Starting from the identities (2) P(S ′2 n ≤ σ 2 ) = P((n − 1)S ′2 n /σ 2 ≤ n − 1) = P ( ... |

1 | Lectures in Probability Theory and Mathematical Statistics - Zubrzycki - 1973 |