### Abstract

In this paper we consider the generalized Marcum Q-function of order ν > 0 real, defined byQν(a,b)=1/a^{ν-1}∫b∞tνe-t ^{2}+a^{2}/2I_{ν-1}(at)dt,where a > 0, b ≥ 0 and I_{ν} stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series.

Original language | English |
---|---|

Pages (from-to) | 2238-2250 |

Number of pages | 13 |

Journal | Applied Mathematics and Computation |

Volume | 217 |

Issue number | 5 |

DOIs | |

Publication status | Published - Nov 1 2010 |

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### Keywords

- Bounds
- Complementary error function
- Generalized Marcum Q-function
- Incomplete gamma function
- Modified Bessel functions

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*Applied Mathematics and Computation*,

*217*(5), 2238-2250. https://doi.org/10.1016/j.amc.2010.07.024

**Bounds for the generalized Marcum Q-function.** / Baricz, A.; Sun, Yin.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 217, no. 5, pp. 2238-2250. https://doi.org/10.1016/j.amc.2010.07.024

}

TY - JOUR

T1 - Bounds for the generalized Marcum Q-function

AU - Baricz, A.

AU - Sun, Yin

PY - 2010/11/1

Y1 - 2010/11/1

N2 - In this paper we consider the generalized Marcum Q-function of order ν > 0 real, defined byQν(a,b)=1/aν-1∫b∞tνe-t 2+a2/2Iν-1(at)dt,where a > 0, b ≥ 0 and Iν stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series.

AB - In this paper we consider the generalized Marcum Q-function of order ν > 0 real, defined byQν(a,b)=1/aν-1∫b∞tνe-t 2+a2/2Iν-1(at)dt,where a > 0, b ≥ 0 and Iν stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series.

KW - Bounds

KW - Complementary error function

KW - Generalized Marcum Q-function

KW - Incomplete gamma function

KW - Modified Bessel functions

UR - http://www.scopus.com/inward/record.url?scp=77957301678&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957301678&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2010.07.024

DO - 10.1016/j.amc.2010.07.024

M3 - Article

VL - 217

SP - 2238

EP - 2250

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 5

ER -