Congruences with q- generalized Catalan Numbers and q-Harmonic Numbers
Year 2022,
Volume: 51 Issue: 3, 712 - 724, 01.06.2022
Neşe Ömür
,
Zehra Betül Gür
,
Sibel Koparal
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
Neşe Ömür
,
Zehra Betül Gür
,
Sibel Koparal
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.