Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 51 Issue: 3

Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers

Congruences with q- generalized Catalan Numbers and q-Harmonic Numbers

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Publication Date
2022-06-01
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.
📊
Key Findings & Data:In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.
💡
Conclusion & Impact:In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:congruence · q−analog · harmonic numbers

Abstract

In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.

Özet

In this paper, we give some congruences related to q- generalized Catalan numbers, q-harmonic numbers and alternating q-harmonic numbers.

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Cite This Article

Ömür, N., Gür, Z. B., Koparal, S. (2022). Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics, Volume: 51 Issue: 3. https://doi.org/10.15672/hujms.886839

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References (22)

  1. N.H. Abel, Untersuchungen über die Reihe $1+\frac{m}{1}x+\frac{m(m-1)}{1.2}x^{2}+\frac{m(m-1)(m-2)}{1.2.3}x^{3}+.$, J. Reine Angew. Math. 1, 311-339, 1826.
  2. G.E. Andrews, On the difference of successive Gaussian polynomials, J. Statist. Plann. Inference 34 (1), 19-22, 1993.
  3. G.E. Andrews, q−analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher, Discrete Math. 204, 15-25, 1999.
  4. E. Deutsch and L.W. Shapiro, A survey of the Fine numbers, Discrete Math. 241, 241-265, 2001.
  5. L. Elkhiri, S. Koparal and N. Ömür, New congruences with the generalized Catalan numbers and harmonic numbers, Bull. Korean Math. Soc. 58 (5), 1079-1095, 2021.
  6. J. Fürlinger and J. Hofbauer, q−Catalan numbers, J. Combin. Theory Ser. A 40 (2), 248-264, 1985.
  7. V.J.W. Guo and S-D.Wang, Factors of sums involving q−binomial coefficients and powers of q−integers, J. Difference Equ. Appl. 23 (10), 1670-1679, 2017.
  8. V.J.W. Guo and J. Zeng, Factors of binomial sums from the Catalan triangle, J. Number Theory, 130, 172-186, 2010.
  9. J.M. Gutiérrez, M.A. Hernández, P.J. Miana and N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (1), 52-61, 2008.
  10. B. He, On q−congruences involving harmonic numbers, Ukrainian Math. J. 69 (9), 1463-1472, 2018.
  11. B. He and K. Wang, Some congruences on q−Catalan numbers, Ramanujan J. 40, 93-101, 2016.
  12. P. Hilton and J. Pedersen, Catalan numbers, their generalization and their uses, Math. Intelligencer, 13, 64-75, 1991.
  13. S. Koparal and N. Ömür, On congruences involving the generalized Catalan numbers and harmonic numbers, Bull. Korean Math. Soc. 56 (3), 649-658, 2019.
  14. P.J. Miana and N. Romero, Computer proofs of new identities in the Catalan triangle, Bibl. Rev. Mat. Iberoamericana, Proceedings of the "Segundas jornadas de Teoriá de Números" 1-7, 2007.
  15. N. Ömür and S. Koparal, Some congruences involving numbers Bp,k−d, Util. Math. 95, 307-317, 2014.
  16. H. Pan, A q−analogue of Lehmer’s congruence, Acta Arith. 128, 303-318, 2007.
  17. H. Pan and H-Q. Cao, A congruence involving products of q−binomial coefficients, J. Number Theory, 121 (2), 224-233, 2006.
  18. L.W. Shapiro, A Catalan triangle, Discrete Math. 14, 83-90, 1976.
  19. L-L. Shi and H. Pan, A q−analogue of Wolstenholme’s harmonic series congruence, Amer. Math. Monthly 114 (6), 529-531, 2007.
  20. R. Tauraso, Some q−analogs of congruences for central binomial sums, Colloq. Math. 133, 133-143, 2013.
  21. J. Wolstenholme, On certain properties of prime numbers, Q. J. Math. 5, 35-39, 1862.
  22. N.H. Abel, Untersuchungen über die Reihe $1+\frac{m}{1}x+\frac{m(m-1)}{1.2}x^{2}+\frac{m(m-1)(m-2)}{1.2.3}x^{3}+.$, J. Reine Angew. Math. 1, 311-339, 1826.