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Some Properties of the Psi and Polygamma Functions

Year 2010, Volume: 39 Issue: 2, 219 - 231, 01.02.2010

References

  • Abramowitz, M. and Stegun, I. A. (Eds), Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables(National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965).
  • Alzer, H. Inequalities for the volume of the unit ball in Rn, II, Mediterr. J. Math. 5 (4), –413, 2008.
  • Alzer, H. Sharp inequalities for the harmonic numbers, Expo. Math. 24 (4), 385–388, 2006.
  • Andrews, G. E., Askey, R. and Roy, R. Special Functions (Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999).
  • Atanassov, R. D. and Tsoukrovski, U. V. Some properties of a class of logarithmically com- pletely monotonic functions, C. R. Acad. Bulgare Sci. 41 (2), 21–23, 1988.
  • Batir, N. Inequalities for the gamma function, Arch. Math. 91, 554–563, 2008.
  • Batir, N. On some properties of digamma and polygamma functions, J. Math. Anal. Appl. (1), 452–465, 2007.
  • Batir, N. Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (4), Art. 103, 2005.
  • (Available online at http://jipam.vu.edu.au/article.php?sid=577) Berg, C. Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (4), 433–439, 2004.
  • Chen, Ch. -P. Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336, 812–822, 2007.
  • Chen, Ch. -P. and Qi, F. Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (1), 405–411, 2006.
  • (Available online at http://dx.doi.org/10.1016/j.jmaa.2005.08.056).
  • Gautschi, W. The incomplete gamma function since Tricomi, in: Tricomi’s Ideas and Con- temporary Applied Mathematics(Atti Convegni Lincei 147, Accademia Nazionale dei Lincei, Rome, 1998), 203–237.
  • Grinshpan, A. Z. and Ismail, M. E. H. Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134, 1153–1160, 2006.
  • Koumandos, S. Monotonicity of some functions involving the gamma and psi functions, Math. Comp. 77, 2261–2275, 2008.
  • Mitrinovi´c, D. S., Peˇcari´c, J. E. and Fink, A. M. Classical and New Inequalities in Analysis (Kluwer, Dordrecht, 1993).
  • Qi, F. A completely monotonic function involving divided difference of psi function and an equivalent inequality involving sums, ANZIAM J. 48 (4), 523–532, 2007.
  • Qi, F. Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (7), 503–509, 2007.
  • (Available online at http://dx.doi.org/10.1080/10652460701358976).
  • Qi, F. and Chen, Ch. -P. A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2), 603–607, 2004.
  • (Available online at http://dx.doi.org/10.1016/j.jmaa.2004.04.026).
  • Qi, F. and Guo, B. -N. Some properties of the psi and polygamma functions, preprint. (Available online at http://arxiv.org/abs/0903.1003).
  • Qiu, S. -L. and Vuorinen, M. Some properties of the gamma and psi functions, with appli- cations, Math. Comp. 74 (250), 723–742, 2005.
  • Wang, Zh. -X. and Guo, D. -R. Special Functions (Translated from the Chinese by D.-R. Guo and X.-J. Xia, World Scientific Publishing, Singapore, 1989).
  • Widder, D. V. The Laplace Transform (Princeton University Press, Princeton, 1946).

Some Properties of the Psi and Polygamma Functions

Year 2010, Volume: 39 Issue: 2, 219 - 231, 01.02.2010

References

  • Abramowitz, M. and Stegun, I. A. (Eds), Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables(National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965).
  • Alzer, H. Inequalities for the volume of the unit ball in Rn, II, Mediterr. J. Math. 5 (4), –413, 2008.
  • Alzer, H. Sharp inequalities for the harmonic numbers, Expo. Math. 24 (4), 385–388, 2006.
  • Andrews, G. E., Askey, R. and Roy, R. Special Functions (Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999).
  • Atanassov, R. D. and Tsoukrovski, U. V. Some properties of a class of logarithmically com- pletely monotonic functions, C. R. Acad. Bulgare Sci. 41 (2), 21–23, 1988.
  • Batir, N. Inequalities for the gamma function, Arch. Math. 91, 554–563, 2008.
  • Batir, N. On some properties of digamma and polygamma functions, J. Math. Anal. Appl. (1), 452–465, 2007.
  • Batir, N. Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (4), Art. 103, 2005.
  • (Available online at http://jipam.vu.edu.au/article.php?sid=577) Berg, C. Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (4), 433–439, 2004.
  • Chen, Ch. -P. Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336, 812–822, 2007.
  • Chen, Ch. -P. and Qi, F. Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (1), 405–411, 2006.
  • (Available online at http://dx.doi.org/10.1016/j.jmaa.2005.08.056).
  • Gautschi, W. The incomplete gamma function since Tricomi, in: Tricomi’s Ideas and Con- temporary Applied Mathematics(Atti Convegni Lincei 147, Accademia Nazionale dei Lincei, Rome, 1998), 203–237.
  • Grinshpan, A. Z. and Ismail, M. E. H. Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134, 1153–1160, 2006.
  • Koumandos, S. Monotonicity of some functions involving the gamma and psi functions, Math. Comp. 77, 2261–2275, 2008.
  • Mitrinovi´c, D. S., Peˇcari´c, J. E. and Fink, A. M. Classical and New Inequalities in Analysis (Kluwer, Dordrecht, 1993).
  • Qi, F. A completely monotonic function involving divided difference of psi function and an equivalent inequality involving sums, ANZIAM J. 48 (4), 523–532, 2007.
  • Qi, F. Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (7), 503–509, 2007.
  • (Available online at http://dx.doi.org/10.1080/10652460701358976).
  • Qi, F. and Chen, Ch. -P. A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2), 603–607, 2004.
  • (Available online at http://dx.doi.org/10.1016/j.jmaa.2004.04.026).
  • Qi, F. and Guo, B. -N. Some properties of the psi and polygamma functions, preprint. (Available online at http://arxiv.org/abs/0903.1003).
  • Qiu, S. -L. and Vuorinen, M. Some properties of the gamma and psi functions, with appli- cations, Math. Comp. 74 (250), 723–742, 2005.
  • Wang, Zh. -X. and Guo, D. -R. Special Functions (Translated from the Chinese by D.-R. Guo and X.-J. Xia, World Scientific Publishing, Singapore, 1989).
  • Widder, D. V. The Laplace Transform (Princeton University Press, Princeton, 1946).
There are 25 citations in total.

Details

Primary Language Turkish
Journal Section Mathematics
Authors

Bai-ni Guo This is me

Feng Qi This is me

Publication Date February 1, 2010
Published in Issue Year 2010 Volume: 39 Issue: 2

Cite

APA Guo, B.-n., & Qi, F. (2010). Some Properties of the Psi and Polygamma Functions. Hacettepe Journal of Mathematics and Statistics, 39(2), 219-231.
AMA Guo Bn, Qi F. Some Properties of the Psi and Polygamma Functions. Hacettepe Journal of Mathematics and Statistics. February 2010;39(2):219-231.
Chicago Guo, Bai-ni, and Feng Qi. “Some Properties of the Psi and Polygamma Functions”. Hacettepe Journal of Mathematics and Statistics 39, no. 2 (February 2010): 219-31.
EndNote Guo B-n, Qi F (February 1, 2010) Some Properties of the Psi and Polygamma Functions. Hacettepe Journal of Mathematics and Statistics 39 2 219–231.
IEEE B.-n. Guo and F. Qi, “Some Properties of the Psi and Polygamma Functions”, Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 2, pp. 219–231, 2010.
ISNAD Guo, Bai-ni - Qi, Feng. “Some Properties of the Psi and Polygamma Functions”. Hacettepe Journal of Mathematics and Statistics 39/2 (February 2010), 219-231.
JAMA Guo B-n, Qi F. Some Properties of the Psi and Polygamma Functions. Hacettepe Journal of Mathematics and Statistics. 2010;39:219–231.
MLA Guo, Bai-ni and Feng Qi. “Some Properties of the Psi and Polygamma Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 2, 2010, pp. 219-31.
Vancouver Guo B-n, Qi F. Some Properties of the Psi and Polygamma Functions. Hacettepe Journal of Mathematics and Statistics. 2010;39(2):219-31.