Analytic computation of digamma function using some new identities

Qureshi, M. I.; Shadab, M.

doi:10.2478/jamsi-2020-0001

Abstract

Motivated by the substantial development in the theory of digamma function, we derive some new identities for the digamma function. These new identities enable us to compute the values of the digamma function for fractional orders in an analogous manner. Also, we mention two errata, found in Jensen’s article (An elementary exposition of the theory of the Gamma function, 1916), and present their correct forms.

References

[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.
Search in Google Scholar Back to article
[2] Alzer, H.; Sharp inequalities for digamma and polygamma functions, Forum Math., 16 (2004), 181-221.10.1515/form.2004.009
Search in Google Scholar Back to article
[3] Batir, N.; Some new inequalities for gamma and polygamma function, JIPAM. J. Inequal. Pure Appl. Math., 6(4) (2005), Article 103, 9 p.
Search in Google Scholar Back to article
[4] Böhmer, E.; Differenzengleichungen und bestimmte integrale, Leipzig, (1939).
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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).
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[11] Gradshteyn, I.S. and Ryzhik, I.M.; Table of integrals, series and products, 8th ed., Academic Press Inc., San Diego, CA. 2014.
Search in Google Scholar Back to article
[12] Grossman, N.; Polygamma functions of arbitrary order, SIAM J. Math. Anal., 7 (1976), 366-372.10.1137/0507030
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[14] Lehmer, D.H.; Euler constants for arithmetical progressions, Acta Arith., 27 (1975), 125-142.10.4064/aa-27-1-125-142
Search in Google Scholar Back to article
[15] Lewin, L.; Polygarithms and Associated Functions, Elsevier, Amsterdam, 1981.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[18] Nielsen, N.; Handbuch der theorie der gamma funktion, Leipzig Druck und Verleg Von B.G. Teubner, 1906.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[21] Qi, F. and Chen, Ch.-P.; A complete monotonicity of the gamma function, RGMIA Res. Rep. Coll, 7 (2007), Art. 1.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[26] Qureshi, M.I. Jabee, S. and Shadab, M.; Truncated Gauss hypergeometric series and its application in digamma function, (Communicated).
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article

Analytic computation of digamma function using some new identities

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