Some Combinatorial Identities of \(q\)-Harmonic and \(q\)-Hyperharmonic Numbers

Authors

  • Can Kızılateş Department of Mathematics, Bulent Ecevit University, Zonguldak
  • Naim Tuğlu Department of Mathematics, Gazi University University, Ankara

DOI:

https://doi.org/10.26713/cma.v6i2.318

Keywords:

Harmonic numbers, hyperharmonic numbers, \(q\)-harmonic numbers, \(q\)-hyperharmonic numbers, \(q\)-difference operator

Abstract

In this paper, by means of \(q\)-difference operator we derive \(q\)-analogue for several well known results for harmonic numbers. Also we give some identities concerning \(q\)-hyperharmonic numbers.

Downloads

Download data is not yet available.

References

R. Graham, D. Knuth and K. Patashnik, Concrete Mathematics, Addison Wesley (1989).

M.Z. Spivey, Combinatorial sums and finite difference, Discrete Math. 307 (2007), 3130–3146.

W. Chu, Summation formulae involving harmonic numbers, Filomat 26 (2012), 143–152.

J. Choi, Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl. 49 (2013), p. 11.

J.H. Comway and R.K. Guy, The Book of Numbers, Springer-Verlag, New York (1996).

A.T. Benjamin, D. Gaebler and R.A. Gaebler, Combinatorial approach to hyperharmonic numbers, Integers 3 (2003), 1–9.

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York (2001).

G.E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopaedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge (1999).

K. Dilcher, Determinant expressions for q-harmonic congruences and degenerate Bernoulli numbers, Electron. J. Combin. 15 (2008), #R63.

T. Mansour, M. Stattuck and C. Song, q-Analogs of identities involving harmonic numbers and binomial coefficients, Appl. Appl. Math. 7 (2012), 22–36.

T. Mansour, Identities on harmonic and q-harmonic number sums, Afr. Mat. 23 (2012), 135–143.

I. Mezö, Some possible q-generalizations of harmonic numbers, Arxiv:1106.5029.

W. Chu and Q. Yan, Combinatorial identities on q-harmonic numbers, J. Combin. Math. Combin. Comput. 77 (2011), 173–185.

T. Mansour and M. Shattuck, A q-analog of the hyperharmonic numbers, Afr. Math. 25 (2014), 147–160.

T. Ernst, q-Stirling numbers, and umbral approach, Adv. Dyn. Syst. Appl. 3 (2008), 251–282.

M.S. Kim and J.W. Son, A note on q-difference operators, Commun. Korean Math Soc. 17 (2002), 423–430.

A. Sofo, Harmonic numbers and double binomial coefficients, Integral Transforms Spec Funct. 20 (2009), 847–857.

A. Sofo and H.M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J. 25 (2011), 93–113.

J. Choi and H.M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Computer Modelling 54 (2011), 2220–2234.

G. Dattoli and H.M. Srivastava, A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett. 21 (2008), 686–693.

M. Bahsi and S. Solak, An application of hyperharmonic numbers in matrices, Hacet. J. Math. Stat. 42 (2013), 387–393.

N. Tuglu, C. Kızılates and S. Kesim, On the harmonic and hyperharmonic Fibonacci numbers, Adv. Difference Equ. 2015, p. 297, doi:10.1186/s13662-015-0635-z (2015).

X. Wang and Y. Qu, Some generalized q-harmonic number identities, Integral Transforms Spec. Funct. 26 (2015), 971–984.

R. Corcino, On p, q-binomial coefficients, Integers 8 (2008), #A29.

Downloads

Published

25-12-2015
CITATION

How to Cite

Kızılateş, C., & Tuğlu, N. (2015). Some Combinatorial Identities of \(q\)-Harmonic and \(q\)-Hyperharmonic Numbers. Communications in Mathematics and Applications, 6(2), 33–40. https://doi.org/10.26713/cma.v6i2.318

Issue

Vol. 6 No. 2 (2015)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.