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<p>This paper derives novel finite-sum expressions for the modified Bessel function of the first kind using properties of the Marcum <i>Q</i> function. Limits to infinite sums give expressions that are equivalent but frequently more compact when compared with results in the literature. The finite-sum expressions are also used to re-phrase an inequality for the probability that a sum of independent symmetric random vectors lies in a symmetric convex set not as the sum of two modified Bessel functions of the first kind but via a single function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\phantom {0}}_{1}F_{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mphantom> <mn>0</mn> </mphantom> <mn>1</mn> </msub> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On Finite and Infinite Sums of the Modified Bessel Function of the First Kind

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月28日

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  • Dirk Veestraeten

摘要

This paper derives novel finite-sum expressions for the modified Bessel function of the first kind using properties of the Marcum Q function. Limits to infinite sums give expressions that are equivalent but frequently more compact when compared with results in the literature. The finite-sum expressions are also used to re-phrase an inequality for the probability that a sum of independent symmetric random vectors lies in a symmetric convex set not as the sum of two modified Bessel functions of the first kind but via a single function \({\phantom {0}}_{1}F_{1}\) 0 1 F 1 .