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Keywords:
Logarithm function; Hypergeometric functions; Integral representation; Lerch transcendent function; Alternating harmonic numbers; Combinatorial series identities; Summation formulas; Partial fraction approach; Binomial coefficients.
Logarithm function; Hypergeometric functions; Integral representation; Lerch transcendent function; Alternating harmonic numbers; Combinatorial series identities; Summation formulas; Partial fraction approach; Binomial coefficients.
Summary:
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.
References:
[1] Adamchik, V., Srivastava, H. M.: Some series of the zeta and related functions. Analysis, 18, 2, 1998, 131-144, DOI 10.1524/anly.1998.18.2.131 | MR 1625172 | Zbl 0919.11056
[2] Borwein, J. M., Zucker, I. J., Boersma, J.: The evaluation of character Euler double sums. Ramanujan J., 15, 2008, 377-405, DOI 10.1007/s11139-007-9083-z | MR 2390277 | Zbl 1241.11108
[3] Choi, J.: Log-Sine and Log-Cosine Integrals. Honam Mathematical J, 35, 2, 2013, 137-146, DOI 10.5831/HMJ.2013.35.2.137 | MR 3112095 | Zbl 1278.33002
[4] Choi, J., Cvijoviæ, D.: Values of the polygamma functions at rational arguments. J. Phys. A: Math. Theor., 40, 50, 2007, 15019-15028, Corrigendum, ibidem, 43 (2010), 239801 (1p). DOI 10.1088/1751-8113/40/50/007 | MR 2442610
[5] Choi, J.: Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers. J. Inequal. Appl., 49, 2013, 1-11, MR 3031578 | Zbl 1283.11115
[6] Choi, J., Srivastava, H. M.: Some summation formulas involving harmonic numbers and generalized harmonic numbers. Math. Comput. Modelling., 54, 2011, 2220-2234, DOI 10.1016/j.mcm.2011.05.032 | MR 2834625 | Zbl 1235.33006
[7] Chu, W.: Summation formulae involving harmonic numbers. Filomat, 26, 1, 2012, 143-152, DOI 10.2298/FIL1201143C | MR 3086693 | Zbl 1289.05019
[8] Ciaurri, O., Navas, L. M., Ruiz, F. J., Varano, J. L.: A simple computation of $\zeta (2k)$. Amer. Math. Monthly., 122, 5, 2015, 444-451, DOI 10.4169/amer.math.monthly.122.5.444 | MR 3352803
[9] Coffey, M. W., Lubbers, N.: On generalized harmonic number sums. Appl. Math. Comput., 217, 2010, 689-698, MR 2678582 | Zbl 1202.33001
[10] Dattoli, G., Srivastava, H. M.: A note on harmonic numbers, umbral calculus and generating functions. Appl. Math. Lett. , 21, 7, 2008, 686-693, DOI 10.1016/j.aml.2007.07.021 | MR 2423046 | Zbl 1152.05306
[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, MR 0781905
[12] Flajolet, P., Salvy, B.: Euler sums and contour integral representations. Exp. Math., 7, 1, 1998, 15-35, DOI 10.1080/10586458.1998.10504356 | MR 1618286 | Zbl 0920.11061
[13] Freitas, P.: Integrals of polylogarithmic functions, recurrence relations and associated Euler sums. Math. Comp., 74, 251, 2005, 1425-1440, DOI 10.1090/S0025-5718-05-01747-3 | MR 2137010 | Zbl 1086.33019
[14] Kölbig, K.: The polygamma function $\psi (x)$ for $x=1/4$ and $x=3/4$. J. Comput. Appl. Math. , 75, 1996, 43-46, MR 1424884
[15] Liu, H., Wang, W.: Harmonic number identities via hypergeometric series and Bell polynomials. Integral Transforms Spec. Funct., 23, 2012, 49-68, DOI 10.1080/10652469.2011.553718 | MR 2875570 | Zbl 1269.33006
[16] Mez?, I: Nonlinear Euler sums. Pacific J. Math. , 272, 1, 2014, 201-226, DOI 10.2140/pjm.2014.272.201 | MR 3270178
[17] Sitaramachandrarao, R.: A formula of S. Ramanujan. J. Number Theory, 25, 1987, 1-19, DOI 10.1016/0022-314X(87)90012-6 | MR 0871165 | Zbl 0606.10032
[18] Sofo, A.: Computational Techniques for the Summation of Series. 2003, Kluwer Academic/Plenum Publishers, New York, MR 2020630 | Zbl 1059.65002
[19] Sofo, A.: Integral identities for sums. Math. Commun., 13, 2, 2008, 303-309, MR 2488679 | Zbl 1178.05002
[20] Sofo, A.: Sums of derivatives of binomial coefficients. Adv. in Appl. Math., 42, 2009, 123-134, DOI 10.1016/j.aam.2008.07.001 | MR 2475317 | Zbl 1220.11025
[21] Sofo, A.: Integral forms associated with harmonic numbers. Appl. Math. Comput., 207, 2, 2009, 365-372,
[22] Sofo, A., Srivastava, H. M.: Identities for the harmonic numbers and binomial coefficients. Ramanujan J., 25, 1, 2011, 93-113, DOI 10.1007/s11139-010-9228-3 | MR 2787293 | Zbl 1234.11022
[23] Sofo, A.: Summation formula involving harmonic numbers. Anal. Math., 37, 1, 2011, 51-64, DOI 10.1007/s10476-011-0103-2 | MR 2784242 | Zbl 1240.33006
[24] Sofo, A.: Quadratic alternating harmonic number sums. J. Number Theory, 154, 2015, 144-159, DOI 10.1016/j.jnt.2015.02.013 | MR 3339570 | Zbl 1310.05014
[25] Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions. 530, 2001, Kluwer Academic Publishers, London, MR 1849375 | Zbl 1014.33001
[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, MR 3294573 | Zbl 1239.33002
[27] Wang, W., Jia, C.: Harmonic number identities via the Newton-Andrews method. Ramanujan J., 35, 2, 2014, 263-285, DOI 10.1007/s11139-013-9511-1 | MR 3266481 | Zbl 1306.05005
[28] Wei, C., Gong, D.: The derivative operator and harmonic number identities. Ramanujan J., 34, 3, 2014, 361-371, DOI 10.1007/s11139-013-9510-2 | MR 3231317 | Zbl 1301.33010
[29] Wu, T. C., Tu, S. T., Srivastava, H. M.: Some combinatorial series identities associated with the digamma function and harmonic numbers. Appl. Math. Lett., 13, 3, 2000, 101-106, DOI 10.1016/S0893-9659(99)00193-7 | MR 1755751 | Zbl 0953.33001
[30] Zheng, D. Y.: Further summation formulae related to generalized harmonic numbers. J. Math. Anal. Appl., 335, 1, 2007, 692-706, DOI 10.1016/j.jmaa.2007.02.002 | MR 2340348 | Zbl 1115.11054
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