Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

István Mező

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 931-939
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

István Mező. "Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions." Open Mathematics 11.5 (2013): 931-939. <http://eudml.org/doc/269660>.

@article{IstvánMező2013,
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.},
author = {István Mező},
journal = {Open Mathematics},
keywords = {Harmonic numbers; Hyperharmonic numbers; Hypergeometric function; Stirling numbers; r-Stirling numbers; Bell numbers; Dobinski formula; Exponential integral; Digamma function; exponential generating function; harmonic numbers; hyperharmonic numbers; hypergeometric function; -Stirling numbers; exponential integral; digamma function},
language = {eng},
number = {5},
pages = {931-939},
title = {Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions},
url = {http://eudml.org/doc/269660},
volume = {11},
year = {2013},
}

TY - JOUR
AU - István Mező
TI - Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 931
EP - 939
AB - 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.
LA - eng
KW - Harmonic numbers; Hyperharmonic numbers; Hypergeometric function; Stirling numbers; r-Stirling numbers; Bell numbers; Dobinski formula; Exponential integral; Digamma function; exponential generating function; harmonic numbers; hyperharmonic numbers; hypergeometric function; -Stirling numbers; exponential integral; digamma function
UR - http://eudml.org/doc/269660
ER -

References

top
  1. [1] Aigner M., Combinatorial Theory, Classics Math., Springer, Berlin, 1997 http://dx.doi.org/10.1007/978-3-642-59101-3 
  2. [2] Andrews G.E., Askey R., Roy R., Special Functions, Encyclopedia Math. Appl., 71, Cambridge University Press, Cambridge, 2001 
  3. [3] Benjamin A.T., Gaebler D., Gaebler R., A combinatorial approach to hyperharmonic numbers, Integers, 2003, 3, #A15 
  4. [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/S0013091500019088 Zbl0819.40003
  5. [5] Boyadzhiev K.N., Exponential polynomials, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, Abstr. Appl. Anal., 2009, #168672 Zbl1237.11011
  6. [6] Broder A.Z., The r-Stirling numbers, Discrete Math., 1984, 49(3), 241–259 http://dx.doi.org/10.1016/0012-365X(84)90161-4 
  7. [7] Charalambides Ch.A., Combinatorial Methods in Discrete Distributions, Wiley Ser. Probab. Stat., Wiley-Interscience, Hoboken, 2005 http://dx.doi.org/10.1002/0471733180 Zbl1087.60001
  8. [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.2012.04.001 Zbl1246.05009
  9. [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/BF01387753 Zbl0343.05011
  10. [10] Comtet L., Advanced Combinatorics, Reidel, Dordrecht, 2010 
  11. [11] Conway J.H., Guy R.K., The Book of Numbers, Copernicus, New York, 1996 http://dx.doi.org/10.1007/978-1-4612-4072-3 
  12. [12] Corcino R.B., The (r; β)-Stirling numbers, Mindanao Forum, 1999, 14(2), 91–100 
  13. [13] Corcino R.B., Corcino C.B., Aldema R., Asymptotic normality of the (r; β)-Stirling numbers, Ars. Combin., 2006, 81, 81–96 Zbl1189.11013
  14. [14] Corcino R.B., Montero M.B., Corcino C.B., On generalized Bell numbers for complex argument, Util. Math., 2012, 88, 267–279 Zbl1320.11017
  15. [15] Crandall R.E., Buhler J.P., On the evaluation of Euler sums, Experiment. Math., 1994, 3(4), 275–285 http://dx.doi.org/10.1080/10586458.1994.10504297 Zbl0833.11045
  16. [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.2009.12.011 Zbl1185.33019
  17. [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.2007.07.021 Zbl1152.05306
  18. [18] Dobinski G., Summirung der Reihe Σn m/n! für m = 1,2, 3,…, Archiv der Mathematik und Physik, 1877, 61, 333–336 Zbl09.0178.04
  19. [19] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, 1, Robert E. Krieger, Melbourne, 1981 Zbl0064.06302
  20. [20] Flajolet P., Salvy B., Euler sums and contour integral representations, Experiment. Math., 1998, 7(1), 15–35 http://dx.doi.org/10.1080/10586458.1998.10504356 Zbl0920.11061
  21. [21] Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007 Zbl1208.65001
  22. [22] Graham R.L., Knuth D.E., Patashnik O., Concrete Mathematics, Addison-Wesley, Reading, 1994 Zbl0836.00001
  23. [23] Hansen E.R., A Table of Series and Products, Prentice-Hall, Englewood Cliffs, 1975 Zbl0438.00001
  24. [24] Mező I., Analytic extension of hyperharmonic numbers, Online J. Anal. Comb., 2009, 4, #1 Zbl1190.33021
  25. [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-z Zbl1237.11010
  26. [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/s11533-009-0008-5 Zbl1229.11043
  27. [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.2009.08.005 Zbl1225.11032
  28. [28] Pitman J., Some probabilistic aspects of set partitions, Amer. Math. Monthly, 1997, 104(3), 201–209 http://dx.doi.org/10.2307/2974785 Zbl0876.05005
  29. [29] Rucinski A., Voigt B., A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl., 1990, 35(2), 161–172 Zbl0727.60024
  30. [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/s11139-010-9228-3 Zbl1234.11022

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.