Asymptotic ratios of Bessel functions of purely imaginary argument

Trlifaj, L.

doi:10.21136/am.1974.103508

Cited by 3 publications

(1 citation statement)

References 5 publications

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“…appeared in solving Schrödinger's equation with a rectangular potential well and related problems (see [16]). Kelker [6] and Ismail and Kelker [9] found that the Student t-distribution is infinitely divisible if and only if the ratio T n−1/2 ( √ x) for n ∈ N is completely monotonic on (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

High order monotonicity of a ratio of the modified Bessel function with applications

Hang Yang,

Tian

2024

Hacettepe Journal of Mathematics and Statistics

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Section: Introductionmentioning

confidence: 99%

High order monotonicity of a ratio of the modified Bessel function with applications

Hang Yang,

Tian

2024

Hacettepe Journal of Mathematics and Statistics

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Convexity of a ratio of the modified Bessel functions of the second kind with applications

Yang¹,

Tian²

2022

Proc. Amer. Math. Soc.

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Let K ν K_{\nu } be the modified Bessel functions of the second kind of order ν \nu . The ratio Q ν ( x ) = x K ν − 1 ( x ) / K ν ( x ) Q_{\nu }\left ( x\right ) =xK_{\nu -1}\left ( x\right ) /K_{\nu }\left ( x\right ) appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that ( − 1 ) n Q ν ( n ) ( x ) > ( > ) 0 \left ( -1\right ) ^{n}Q_{\nu }^{\left ( n\right ) }\left ( x\right ) >\left ( >\right ) 0 for x > 0 x>0 and n = 2 , 3 n=2,3 if | ν | > ( > ) 1 / 2 \left \vert \nu \right \vert >\left ( >\right ) 1/2 holds for n = 2 n=2 . This yields several new consequences and improves some known results. Finally, two conjectures are proposed.

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The Bessel Polynomials and the StudenttDistribution

Ismail¹,

Kelker²

1976

SIAM J. Math. Anal.

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The quotient 2Fo(-n + 1, n;-;-1/2x) P,_,(x),,/'-x2Fo(-n,n+ 1;-;-1/2x)-P.(x) arose in connection with the problem of the infinite divisibility of the Student distribution. It is shown that P,_,(x)/P,(,,Ux) is completely monotonic in [0, oo) for n 4, 5 and 6. This implies that the Student distribution is infinitely divisible for 9, 11 and 13 degrees of freedom. We show that certain power sums of the zeros of the simple Bessel polynomials are zero. This is then used to show that for every n =0, 1, 2,. ., there exists a 0,>0 such that the inverse Laplace transform of P,_,(qx)/P,,(4x) is nonnegative in [0,, oo). This supports our conjecture that P,_,(,,/-x)/P,(x) is completely monotonic in (0, oo) for all n, and that the Student distribution is infinitely divisible for odd degrees of freedom.

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Asymptotic ratios of Bessel functions of purely imaginary argument

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Cited by 3 publications

References 5 publications

High order monotonicity of a ratio of the modified Bessel function with applications

High order monotonicity of a ratio of the modified Bessel function with applications

Convexity of a ratio of the modified Bessel functions of the second kind with applications

The Bessel Polynomials and the StudenttDistribution

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