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 \(1/\sum_{k=1}^{n+1} (1-q)/(1-q^{k})\), 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.
Similar content being viewed by others
References
Akhiezer, N.I.: The Classical Moment Problem. Oliver and Boyd, Edinburgh (1965)
Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)
Berg, C., Beygmohammadi, M.: On a fixed point in the metric space of normalized Hausdorff moment sequences. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 82, 251–257 (2010)
Berg, C., Durán, A.J.: Some transformations of Hausdorff moment sequences and harmonic numbers. Can. J. Math. 57, 941–960 (2005)
Berg, C., Durán, A.J.: The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function. Math. Scand. 103, 11–39 (2008)
Berg, C., Durán, A.J.: Iteration of the rational function z−1/z and a Hausdorff moment sequence. Expo. Math. 26, 375–385 (2008)
Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 87. Springer, Berlin (1975)
Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence (1988)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990) (2nd edn. 2004)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, New York (2000)
Kingman, J.F.C.: Regenerative Phenomena. Wiley, London (1972)
Krattenthaler, C., Srivastava, H.M.: Summations for basic hypergeometric series involving a q-analogue of the Digamma function. Comput. Math. Appl. 32, 73–91 (1996)
Mansour, T., Shabani, A.S.: Some inequalities for the q-Digamma function. J. Inequal. Pure Appl. Math. 10, Art. 12 (2009), 8 pp.
Schilling, R., Song, R., Vondraček, Z.: Bernstein Functions: Theory and Applications. de Gruyter, Berlin (2010)
Acknowledgements
The first author wants to thank Fethi Bouzzefour, Tunisia for having raised the question of determining the measure ν q .
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Rights and permissions
About this article
Cite this article
Berg, C., Petersen, H.B. On an Iteration Leading to a q-Analogue of the Digamma Function. J Fourier Anal Appl 19, 762–776 (2013). https://doi.org/10.1007/s00041-013-9271-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-013-9271-8