The Ramanujan Journal

, Volume 35, Issue 1, pp 131–139 | Cite as

q-Generalizations of Mortenson’s identities and further identities

  • Qinglun Yan
  • Chuanan Wei
  • Xiaona Fan


By means of partial fraction decomposition, we give simple proofs of Mortenson’s identities first. Then, inspired by them, we derive their q-generalizations and explore further identities of similar type.


Harmonic number Mortenson’s identities Supercongruences 

Mathematics Subject Classification (2000)

05A30 05A19 



The authors thank the anonymous referee for his/her valuable suggestions and comments that have contributed to the improvement of the paper’s presentation.


  1. 1.
    Ahlgren, S., Ekhad, S.B., Ono, K., Zeilberger, D.: A binomial coefficient identity associated to a conjecture of Beukers. Electron. J. Comb. 5, 10 (1998) (Research paper) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chu, W.: A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers. Electron. J. Comb. 11(1), 15 (2004) zbMATHGoogle Scholar
  4. 4.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004) CrossRefzbMATHGoogle Scholar
  5. 5.
    Mansour, T., Shattuck, M., Song, C.: q-Analogs of identities involving harmonic numbers and binomial coefficients. Appl. Appl. Math. 7(1), 22–36 (2012) MathSciNetzbMATHGoogle Scholar
  6. 6.
    McCarthy, D.: Binomial coefficient-harmonic sum identities associated to supercongruences. Integers 11, A37 (2011). 8 pp. MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    McCarthy, D.: Extending Gaussian hypergeometric series to the p-adic setting. Int. J. Number Theory 8(7), 1581–1612 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    McCarthy, D.: On a supercongruence conjecture of Rodriguez–Villegas. Proc. Am. Math. Soc. 140, 2241–2254 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mortenson, E.: On differentiation and harmonic numbers. Util. Math. 80, 53–57 (2009) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mortenson, E.: Supercongruences between truncated 2 F 1 hypergeometric functions and their Gaussian analogs. Trans. Am. Math. Soc. 335, 139–147 (2003) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Prodinger, H.: Mortenson’s identities and partial fraction decomposition. Util. Math., accepted.
  12. 12.
    Rodriguez-Villegas, F.: Hypergeometric Families of Calabi–Yau Manifolds. Calabi–Yau Varieties and Mirror Symmetry. Fields Inst. Commun., vol. 38. AMS, Providence (2003) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of ScienceNanjing University of Posts and TelecommunicationsNanjingChina
  2. 2.Department of Information TechnologyHainan Medical CollegeHaikouChina

Personalised recommendations