Integrals of logarithmic and hypergeometric functions

Anthony Sofo

Communications in Mathematics (2016)

  • Volume: 24, Issue: 1, page 7-22
  • ISSN: 1804-1388

Abstract

top
Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

How to cite

top

Sofo, Anthony. "Integrals of logarithmic and hypergeometric functions." Communications in Mathematics 24.1 (2016): 7-22. <http://eudml.org/doc/286696>.

@article{Sofo2016,
abstract = {Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.},
author = {Sofo, Anthony},
journal = {Communications in Mathematics},
keywords = {Logarithm function; Hypergeometric functions; Integral representation; Lerch transcendent function; Alternating harmonic numbers; Combinatorial series identities; Summation formulas; Partial fraction approach; Binomial coefficients},
language = {eng},
number = {1},
pages = {7-22},
publisher = {University of Ostrava},
title = {Integrals of logarithmic and hypergeometric functions},
url = {http://eudml.org/doc/286696},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Sofo, Anthony
TI - Integrals of logarithmic and hypergeometric functions
JO - Communications in Mathematics
PY - 2016
PB - University of Ostrava
VL - 24
IS - 1
SP - 7
EP - 22
AB - Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.
LA - eng
KW - Logarithm function; Hypergeometric functions; Integral representation; Lerch transcendent function; Alternating harmonic numbers; Combinatorial series identities; Summation formulas; Partial fraction approach; Binomial coefficients
UR - http://eudml.org/doc/286696
ER -

References

top
  1. Adamchik, V., Srivastava, H. M., 10.1524/anly.1998.18.2.131, Analysis, 18, 2, 1998, 131-144, (1998) Zbl0919.11056MR1625172DOI10.1524/anly.1998.18.2.131
  2. Borwein, J. M., Zucker, I. J., Boersma, J., 10.1007/s11139-007-9083-z, Ramanujan J., 15, 2008, 377-405, (2008) Zbl1241.11108MR2390277DOI10.1007/s11139-007-9083-z
  3. Choi, J., 10.5831/HMJ.2013.35.2.137, Honam Mathematical J, 35, 2, 2013, 137-146, (2013) Zbl1278.33002MR3112095DOI10.5831/HMJ.2013.35.2.137
  4. Choi, J., Cvijoviæ, D., 10.1088/1751-8113/40/50/007, J. Phys. A: Math. Theor., 40, 50, 2007, 15019-15028, Corrigendum, ibidem, 43 (2010), 239801 (1p). (2007) MR2442610DOI10.1088/1751-8113/40/50/007
  5. Choi, J., Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl., 49, 2013, 1-11, (2013) Zbl1283.11115MR3031578
  6. Choi, J., Srivastava, H. M., 10.1016/j.mcm.2011.05.032, Math. Comput. Modelling., 54, 2011, 2220-2234, (2011) Zbl1235.33006MR2834625DOI10.1016/j.mcm.2011.05.032
  7. Chu, W., 10.2298/FIL1201143C, Filomat, 26, 1, 2012, 143-152, (2012) Zbl1289.05019MR3086693DOI10.2298/FIL1201143C
  8. Ciaurri, O., Navas, L. M., Ruiz, F. J., Varano, J. L., 10.4169/amer.math.monthly.122.5.444, Amer. Math. Monthly., 122, 5, 2015, 444-451, (2015) MR3352803DOI10.4169/amer.math.monthly.122.5.444
  9. Coffey, M. W., Lubbers, N., On generalized harmonic number sums, Appl. Math. Comput., 217, 2010, 689-698, (2010) Zbl1202.33001MR2678582
  10. Dattoli, G., Srivastava, H. M., 10.1016/j.aml.2007.07.021, Appl. Math. Lett. , 21, 7, 2008, 686-693, (2008) Zbl1152.05306MR2423046DOI10.1016/j.aml.2007.07.021
  11. Devoto, A., Duke, D. W., Table of integrals and formulae for Feynman diagram calculation, La Rivista del Nuovo Cimento, 7, 6, 1984, 1-39, (1984) MR0781905
  12. Flajolet, P., Salvy, B., 10.1080/10586458.1998.10504356, Exp. Math., 7, 1, 1998, 15-35, (1998) Zbl0920.11061MR1618286DOI10.1080/10586458.1998.10504356
  13. Freitas, P., 10.1090/S0025-5718-05-01747-3, Math. Comp., 74, 251, 2005, 1425-1440, (2005) Zbl1086.33019MR2137010DOI10.1090/S0025-5718-05-01747-3
  14. Kölbig, K., The polygamma function ψ ( x ) for x = 1 / 4 and x = 3 / 4 , J. Comput. Appl. Math. , 75, 1996, 43-46, (1996) MR1424884
  15. Liu, H., Wang, W., 10.1080/10652469.2011.553718, Integral Transforms Spec. Funct., 23, 2012, 49-68, (2012) Zbl1269.33006MR2875570DOI10.1080/10652469.2011.553718
  16. Mez?, I, 10.2140/pjm.2014.272.201, Pacific J. Math. , 272, 1, 2014, 201-226, (2014) MR3270178DOI10.2140/pjm.2014.272.201
  17. Sitaramachandrarao, R., 10.1016/0022-314X(87)90012-6, J. Number Theory, 25, 1987, 1-19, (1987) Zbl0606.10032MR0871165DOI10.1016/0022-314X(87)90012-6
  18. Sofo, A., Computational Techniques for the Summation of Series, 2003, Kluwer Academic/Plenum Publishers, New York, (2003) Zbl1059.65002MR2020630
  19. Sofo, A., Integral identities for sums, Math. Commun., 13, 2, 2008, 303-309, (2008) Zbl1178.05002MR2488679
  20. Sofo, A., 10.1016/j.aam.2008.07.001, Adv. in Appl. Math., 42, 2009, 123-134, (2009) Zbl1220.11025MR2475317DOI10.1016/j.aam.2008.07.001
  21. Sofo, A., Integral forms associated with harmonic numbers, Appl. Math. Comput., 207, 2, 2009, 365-372, (2009) 
  22. Sofo, A., Srivastava, H. M., 10.1007/s11139-010-9228-3, Ramanujan J., 25, 1, 2011, 93-113, (2011) Zbl1234.11022MR2787293DOI10.1007/s11139-010-9228-3
  23. Sofo, A., 10.1007/s10476-011-0103-2, Anal. Math., 37, 1, 2011, 51-64, (2011) Zbl1240.33006MR2784242DOI10.1007/s10476-011-0103-2
  24. Sofo, A., 10.1016/j.jnt.2015.02.013, J. Number Theory, 154, 2015, 144-159, (2015) Zbl1310.05014MR3339570DOI10.1016/j.jnt.2015.02.013
  25. Srivastava, H. M., Choi, J., Series Associated with the Zeta and Related Functions, 530, 2001, Kluwer Academic Publishers, London, (2001) Zbl1014.33001MR1849375
  26. Srivastava, H. M., Choi, J., Zeta and q -Zeta Functions and Associated Series and Integrals, 2012, Elsevier Science Publishers, Amsterdam, London and New York, (2012) Zbl1239.33002MR3294573
  27. Wang, W., Jia, C., 10.1007/s11139-013-9511-1, Ramanujan J., 35, 2, 2014, 263-285, (2014) Zbl1306.05005MR3266481DOI10.1007/s11139-013-9511-1
  28. Wei, C., Gong, D., 10.1007/s11139-013-9510-2, Ramanujan J., 34, 3, 2014, 361-371, (2014) Zbl1301.33010MR3231317DOI10.1007/s11139-013-9510-2
  29. Wu, T. C., Tu, S. T., Srivastava, H. M., 10.1016/S0893-9659(99)00193-7, Appl. Math. Lett., 13, 3, 2000, 101-106, (2000) Zbl0953.33001MR1755751DOI10.1016/S0893-9659(99)00193-7
  30. Zheng, D. Y., 10.1016/j.jmaa.2007.02.002, J. Math. Anal. Appl., 335, 1, 2007, 692-706, (2007) Zbl1115.11054MR2340348DOI10.1016/j.jmaa.2007.02.002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.