<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q_{\nu }(a,b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the generalized Marcum <i>Q</i>-function and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_{\nu }(a,b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the generalized Marcum function of the second kind of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\nu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper our aim is to present some new results related to the monotonicity and bounds on the ratio of the generalized Marcum functions of the first and second kinds <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${Q_{\nu }(a,b)}\big /{R_{\nu }(a,b)}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <msub> <mi>Q</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">/</mo> </mrow> <mrow> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We also present new closed form series and integral representations for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_{\nu }(a,b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and its higher order derivatives. The results on the ratio of the generalized Marcum functions of the first and second kinds are based on some earlier established results on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q_{\nu }(a,b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R_{\nu }(a,b),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and also on properties of modified Bessel functions of the first and second kind. Moreover, we show that the generalized Marcum function of the second kind is related to a distribution, which is not only infinitely divisible, but belongs also to the classes of self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities.</p>

The Ratio of the Generalized Marcum Functions of the First and Second Kind: Monotonicity and Bounds

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发表日期：

2026年8月4日

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Árpád Baricz,
Snehalata Sahoo,
Sanjeev Singh

摘要

Let $Q_{\nu }(a,b)$ $Q_{ν} (a, b)$ be the generalized Marcum Q-function and $R_{\nu }(a,b)$ $R_{ν} (a, b)$ be the generalized Marcum function of the second kind of order $\nu >0$ $ν > 0$ . In this paper our aim is to present some new results related to the monotonicity and bounds on the ratio of the generalized Marcum functions of the first and second kinds ${Q_{\nu }(a,b)}\big /{R_{\nu }(a,b)}.$ $Q_{ν} (a, b) / R_{ν} (a, b) .$ We also present new closed form series and integral representations for $R_{\nu }(a,b)$ $R_{ν} (a, b)$ and its higher order derivatives. The results on the ratio of the generalized Marcum functions of the first and second kinds are based on some earlier established results on $Q_{\nu }(a,b)$ $Q_{ν} (a, b)$ and $R_{\nu }(a,b),$ $R_{ν} (a, b),$ and also on properties of modified Bessel functions of the first and second kind. Moreover, we show that the generalized Marcum function of the second kind is related to a distribution, which is not only infinitely divisible, but belongs also to the classes of self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities.