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Analytic computation of digamma function using some new identities Cover

Analytic computation of digamma function using some new identities

By: M. I. Qureshi and  M. Shadab  
Open Access
|Jul 2020

References

  1. [1] Al-Saqabi, B.N. Kalla, S.L. and Srivastava, H.M.; A certain family of infinite series associated with Digamma functions, J. Math. Anal. Appl., 159 (1991) 361-372.
  2. [2] Alzer, H.; Sharp inequalities for digamma and polygamma functions, Forum Math., 16 (2004), 181-221.10.1515/form.2004.009
  3. [3] Batir, N.; Some new inequalities for gamma and polygamma function, JIPAM. J. Inequal. Pure Appl. Math., 6(4) (2005), Article 103, 9 p.
  4. [4] Böhmer, E.; Differenzengleichungen und bestimmte integrale, Leipzig, (1939).
  5. [5] Borwein, D. and Borwein, J.M.; On an intriguing integral and some series related to ζ (4), Proc. Amer. Math. Soc., 123 (1995) 1191-1198.
  6. [6] Clark, W.E. and Ismail, M.E.H.; Inequalities involving gamma and psi function, Anal. Appl., 1(129) (2003), 129-140.10.1142/S0219530503000041
  7. [7] De Doelder, P.J.; On some series containing ψ(x) − ψ(y) and (ψ(x) − ψ(y))2 for certain values of x and y, J. Comput. Appl. Math., 37 (1991), 125-141.10.1016/0377-0427(91)90112-W
  8. [8] Erdélyi, A. Magnus, W. Oberhettinger, F. and Tricomi, F.G.; Higher Transcendental Functions,Vol.I(Bateman Manuscript Project), McGraw-Hill, Book Co. Inc., New York,Toronto and London, 1953.
  9. [9] Gauss, C.F.; Disquisitiones generales circa seriem infinitam etc., Comm. Soc. reg. Sci. Gott. rec., Vol II, (1813) pp. 1-46.; reprinted in Werke 3(1866).
  10. [10] Gosper, R.W.; ∫n/4m/6logΓ(z)dz\int_{n/4}^{m/6} {\log \Gamma \left( z \right)} dz, In special functions, q-series and related topics, Amer. Math. Soc., 14 (1997), 71-76.
  11. [11] Gradshteyn, I.S. and Ryzhik, I.M.; Table of integrals, series and products, 8th ed., Academic Press Inc., San Diego, CA. 2014.
  12. [12] Grossman, N.; Polygamma functions of arbitrary order, SIAM J. Math. Anal., 7 (1976), 366-372.10.1137/0507030
  13. [13] Jensen, J.L.W.V.; An elementary exposition of the theory of the Gamma function, Ann. Math., 17 (3) (1916), 124–166.10.2307/2007272
  14. [14] Lehmer, D.H.; Euler constants for arithmetical progressions, Acta Arith., 27 (1975), 125-142.10.4064/aa-27-1-125-142
  15. [15] Lewin, L.; Polygarithms and Associated Functions, Elsevier, Amsterdam, 1981.
  16. [16] Mahler, K.; Applications of a theorem of A. B. Shidlovski, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 305 (1968), 149-173.
  17. [17] Murty, M.R. and Saradha, N.; Transcendental values of the digamma function, J. Num. Theo., 125 (2007), 298-318.10.1016/j.jnt.2006.09.017
  18. [18] Nielsen, N.; Handbuch der theorie der gamma funktion, Leipzig Druck und Verleg Von B.G. Teubner, 1906.
  19. [19] Srivastava, H.M.; A simple algorithm for the evaluation of a class of generalized hypergeometric series, Stud. Appli. Math., 86 (1992), 79-86.10.1002/sapm199286179
  20. [20] Srivastava, H.M. and Choi, J.; Series Associated with the Zeta and Related Functions, Kluwer, Dordrecht, 2001.10.1007/978-94-015-9672-5
  21. [21] Qi, F. and Chen, Ch.-P.; A complete monotonicity of the gamma function, RGMIA Res. Rep. Coll, 7 (2007), Art. 1.
  22. [22] Qi, F. and Chen, Ch.-P.; A complete monotonicity property of the gamma function, J.Math. Anal. Appl., 296 (2004), 603-607.10.1016/j.jmaa.2004.04.026
  23. [23] Qi, F. and Guo, B.-P.; Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll, 7 (2004), 63-72, Art. 8.
  24. [24] Qi, F. Guo, B.-P. and Chen, Ch.-P.; Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll, 7 (2004), 31-36, Art.5.
  25. [25] Qiu, S.L. and Vuorinen, M.; Some properties of the gamma and psi functions with applications, Math. Comp., 74 (2005), 723-742.10.1090/S0025-5718-04-01675-8
  26. [26] Qureshi, M.I. Jabee, S. and Shadab, M.; Truncated Gauss hypergeometric series and its application in digamma function, (Communicated).
  27. [27] Wu, T.-C. Leu, S.-H. Tu, S.-T. and Srivastava, H.M. A certain class of infinite sums associated with Diagamma functions, Appl. Math. Comput., 105 (1999), 1-9.10.1016/S0898-1221(99)00232-1
DOI: https://doi.org/10.2478/jamsi-2020-0001 | Journal eISSN: 1339-0015 | Journal ISSN: 1336-9180
Language: English
Page range: 5 - 12
Published on: Jul 9, 2020
Published by: University of Ss. Cyril and Methodius in Trnava
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2020 M. I. Qureshi, M. Shadab, published by University of Ss. Cyril and Methodius in Trnava
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