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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 14, 2013

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

  • István Mező EMAIL logo
From the journal Open Mathematics

Abstract

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

MSC: 05A15

[1] Aigner M., Combinatorial Theory, Classics Math., Springer, Berlin, 1997 http://dx.doi.org/10.1007/978-3-642-59101-310.1007/978-3-642-59101-3Search in Google Scholar

[2] Andrews G.E., Askey R., Roy R., Special Functions, Encyclopedia Math. Appl., 71, Cambridge University Press, Cambridge, 2001 Search in Google Scholar

[3] Benjamin A.T., Gaebler D., Gaebler R., A combinatorial approach to hyperharmonic numbers, Integers, 2003, 3, #A15 Search in Google Scholar

[4] Borwein D., Borwein J.M., Girgensohn R., Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc., 1995, 38(2), 277–294 http://dx.doi.org/10.1017/S001309150001908810.1017/S0013091500019088Search in Google Scholar

[5] Boyadzhiev K.N., Exponential polynomials, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, Abstr. Appl. Anal., 2009, #168672 10.1155/2009/168672Search in Google Scholar

[6] Broder A.Z., The r-Stirling numbers, Discrete Math., 1984, 49(3), 241–259 http://dx.doi.org/10.1016/0012-365X(84)90161-410.1016/0012-365X(84)90161-4Search in Google Scholar

[7] Charalambides Ch.A., Combinatorial Methods in Discrete Distributions, Wiley Ser. Probab. Stat., Wiley-Interscience, Hoboken, 2005 http://dx.doi.org/10.1002/047173318010.1002/0471733180Search in Google Scholar

[8] Cheon G.-S., Jung J.-H., r-Whitney numbers of Dowling lattices, Discrete Math., 2012, 312(15), 2337–2348 http://dx.doi.org/10.1016/j.disc.2012.04.00110.1016/j.disc.2012.04.001Search in Google Scholar

[9] Chowla S., Nathanson M.B., Mellin’s formula and some combinatorial identities, Monatsh. Math., 1976, 81(4), 261–265 http://dx.doi.org/10.1007/BF0138775310.1007/BF01387753Search in Google Scholar

[10] Comtet L., Advanced Combinatorics, Reidel, Dordrecht, 2010 Search in Google Scholar

[11] Conway J.H., Guy R.K., The Book of Numbers, Copernicus, New York, 1996 http://dx.doi.org/10.1007/978-1-4612-4072-310.1007/978-1-4612-4072-3Search in Google Scholar

[12] Corcino R.B., The (r; β)-Stirling numbers, Mindanao Forum, 1999, 14(2), 91–100 Search in Google Scholar

[13] Corcino R.B., Corcino C.B., Aldema R., Asymptotic normality of the (r; β)-Stirling numbers, Ars. Combin., 2006, 81, 81–96 Search in Google Scholar

[14] Corcino R.B., Montero M.B., Corcino C.B., On generalized Bell numbers for complex argument, Util. Math., 2012, 88, 267–279 10.5402/2012/592818Search in Google Scholar

[15] Crandall R.E., Buhler J.P., On the evaluation of Euler sums, Experiment. Math., 1994, 3(4), 275–285 http://dx.doi.org/10.1080/10586458.1994.1050429710.1080/10586458.1994.10504297Search in Google Scholar

[16] Cvijovic D., The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers, Appl. Math. Comput., 2010, 215(11), 4040–4043 http://dx.doi.org/10.1016/j.amc.2009.12.01110.1016/j.amc.2009.12.011Search in Google Scholar

[17] Dattoli G., Srivastava H.M., A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett., 2008, 21(7), 686–693 http://dx.doi.org/10.1016/j.aml.2007.07.02110.1016/j.aml.2007.07.021Search in Google Scholar

[18] Dobinski G., Summirung der Reihe Σn m/n! für m = 1,2, 3,…, Archiv der Mathematik und Physik, 1877, 61, 333–336 Search in Google Scholar

[19] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, 1, Robert E. Krieger, Melbourne, 1981 Search in Google Scholar

[20] Flajolet P., Salvy B., Euler sums and contour integral representations, Experiment. Math., 1998, 7(1), 15–35 http://dx.doi.org/10.1080/10586458.1998.1050435610.1080/10586458.1998.10504356Search in Google Scholar

[21] Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007 Search in Google Scholar

[22] Graham R.L., Knuth D.E., Patashnik O., Concrete Mathematics, Addison-Wesley, Reading, 1994 Search in Google Scholar

[23] Hansen E.R., A Table of Series and Products, Prentice-Hall, Englewood Cliffs, 1975 Search in Google Scholar

[24] Mező I., Analytic extension of hyperharmonic numbers, Online J. Anal. Comb., 2009, 4, #1 Search in Google Scholar

[25] Mező I., A new formula for the Bernoulli polynomials, Results Math., 2010, 58(3–4), 329–335 http://dx.doi.org/10.1007/s00025-010-0039-z10.1007/s00025-010-0039-zSearch in Google Scholar

[26] Mező I., Dil A., Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence, Cent. Eur. J. Math., 2009, 7(2), 310–321 http://dx.doi.org/10.2478/s11533-009-0008-510.2478/s11533-009-0008-5Search in Google Scholar

[27] Mező I., Dil A., Hyperharmonic series involving Hurwitz zeta function, J. Number Theory, 2010, 130(2), 360–369 http://dx.doi.org/10.1016/j.jnt.2009.08.00510.1016/j.jnt.2009.08.005Search in Google Scholar

[28] Pitman J., Some probabilistic aspects of set partitions, Amer. Math. Monthly, 1997, 104(3), 201–209 http://dx.doi.org/10.2307/297478510.2307/2974785Search in Google Scholar

[29] Rucinski A., Voigt B., A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl., 1990, 35(2), 161–172 Search in Google Scholar

[30] Sofo A., Srivastava H.M., Identities for the harmonic numbers and binomial coefficients, Ramanujan J., 2011, 25(1), 93–113 http://dx.doi.org/10.1007/s11139-010-9228-310.1007/s11139-010-9228-3Search in Google Scholar

Published Online: 2013-3-14
Published in Print: 2013-5-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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