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

Year 2022, Volume: 51 Issue: 3, 712 - 724, 01.06.2022
https://doi.org/10.15672/hujms.886839

Abstract

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

References

  • [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.

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

Year 2022, Volume: 51 Issue: 3, 712 - 724, 01.06.2022
https://doi.org/10.15672/hujms.886839

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.

References

  • [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.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Neşe Ömür 0000-0002-3972-9910

Zehra Betül Gür 0000-0002-3685-4222

Sibel Koparal 0000-0001-9574-9652

Publication Date June 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 3

Cite

APA Ömür, N., Gür, Z. B., & Koparal, S. (2022). Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics, 51(3), 712-724. https://doi.org/10.15672/hujms.886839
AMA Ömür N, Gür ZB, Koparal S. Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):712-724. doi:10.15672/hujms.886839
Chicago Ömür, Neşe, Zehra Betül Gür, and Sibel Koparal. “Congruences With $q$-Generalized Catalan Numbers and $q$-Harmonic Numbers”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 712-24. https://doi.org/10.15672/hujms.886839.
EndNote Ömür N, Gür ZB, Koparal S (June 1, 2022) Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics 51 3 712–724.
IEEE N. Ömür, Z. B. Gür, and S. Koparal, “Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 712–724, 2022, doi: 10.15672/hujms.886839.
ISNAD Ömür, Neşe et al. “Congruences With $q$-Generalized Catalan Numbers and $q$-Harmonic Numbers”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 712-724. https://doi.org/10.15672/hujms.886839.
JAMA Ömür N, Gür ZB, Koparal S. Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics. 2022;51:712–724.
MLA Ömür, Neşe et al. “Congruences With $q$-Generalized Catalan Numbers and $q$-Harmonic Numbers”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 712-24, doi:10.15672/hujms.886839.
Vancouver Ömür N, Gür ZB, Koparal S. Congruences with $q$-generalized Catalan numbers and $q$-harmonic numbers. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):712-24.