On an Iteration Leading to a q-Analogue of the Digamma Function

Christian Berg; Helle Bjerg  Petersen

doi:10.1007/s00041-013-9271-8

On an Iteration Leading to a q-Analogue of the Digamma Function

Christian Berg, Helle Bjerg Petersen

Department of Mathematical Sciences

1 Citation (Scopus)

Abstract

We show that the q-Digamma function ψ _q for 0<q<1 appears in an iteration studied by Berg and Durán. This is connected with the determination of the probability measure ν _q on the unit interval with moments 1∑_k=1ⁿ⁺¹ (1-q)/(1-q^k), which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν _q can be expressed in terms of the q-Digamma function. It is shown that ν _q has a continuous density on ]0,1], which is piecewise C ^∞ with kinks at the powers of q. Furthermore, (1-q)e ^-x ν _q(e ^-x) is a standard p-function from the theory of regenerative phenomena.

Original language	English
Journal	Journal of Fourier Analysis and Applications
Volume	19
Issue number	4
Pages (from-to)	762-776
ISSN	1069-5869
DOIs	https://doi.org/10.1007/s00041-013-9271-8
Publication status	Published - Aug 2013

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title = "On an Iteration Leading to a q-Analogue of the Digamma Function",

abstract = "We show that the q-Digamma function ψ q for 0q on the unit interval with moments 1∑k=1n+1 (1-q)/(1-qk), which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν q can be expressed in terms of the q-Digamma function. It is shown that ν q has a continuous density on ]0,1], which is piecewise C ∞ with kinks at the powers of q. Furthermore, (1-q)e -x ν q(e -x) is a standard p-function from the theory of regenerative phenomena.",

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AU - Petersen, Helle Bjerg

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N2 - We show that the q-Digamma function ψ q for 0q on the unit interval with moments 1∑k=1n+1 (1-q)/(1-qk), which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν q can be expressed in terms of the q-Digamma function. It is shown that ν q has a continuous density on ]0,1], which is piecewise C ∞ with kinks at the powers of q. Furthermore, (1-q)e -x ν q(e -x) is a standard p-function from the theory of regenerative phenomena.

AB - We show that the q-Digamma function ψ q for 0q on the unit interval with moments 1∑k=1n+1 (1-q)/(1-qk), which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν q can be expressed in terms of the q-Digamma function. It is shown that ν q has a continuous density on ]0,1], which is piecewise C ∞ with kinks at the powers of q. Furthermore, (1-q)e -x ν q(e -x) is a standard p-function from the theory of regenerative phenomena.

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DO - 10.1007/s00041-013-9271-8

M3 - Journal article

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EP - 776

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 4

ER -

On an Iteration Leading to a q-Analogue of the Digamma Function

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