Central and local limit theorems for the weighted Delannoy numbers

Belovas, Igoris

doi:10.2478/auom-2022-0017

Abstract

In this research we generalize our result for numbers satisfying the Delannoy triangle. We obtain a central limit theorem and a local limit theorem for weighted numbers of the triangle and establish the rate of convergence to the limiting (normal) distribution.

References

[1] S. Amrouche, H. Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turk. J. Math., 44, (2020), 118 130. DOI: 10.3906/mat-1811-10910.3906/mat-1811-109
Search in Google Scholar Back to article
[2] S. Amrouche, H. Belbachir, J. L. Ramrez, Unimodality, linear recurrences and combinatorial properties associated to rays in the generalized Delannoy matrix, J. Differ. Equ., 25 (8), (2019), 1200-1215, DOI: 10.1080/10236198.2019.166241310.1080/10236198.2019.1662413
Search in Google Scholar Back to article
[3] P. Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Integ. Seq., Vol. 22 (5), Article 19.5.8 (2019), [1–50].
Search in Google Scholar Back to article
[4] I. Belovas, A central limit theorem for coefficients of the modified Borwein method for the calculation of the Riemann zeta-function, Lith. Math. J., vol. 59(1) (2019), 17–23. DOI: 10.1007/s10986-019-09421-410.1007/s10986-019-09421-4
Search in Google Scholar Back to article
[5] I. Belovas, A local limit theorem for coefficients of modified Borwein’s method, Glas. Mat. Ser. III, Vol. 54 (1) (2019), [1–9]. DOI: 10.3336/gm.54.1.0110.3336/gm.54.1.01
Search in Google Scholar Back to article
[6] I. Belovas, Central and local limit theorems for numbers of the Tribonacci triangle, Mathematics, vol. 9, no. 8, art. no. 880 (2021), [1-11]. DOI: 10.3390/math908088010.3390/math9080880
Search in Google Scholar Back to article
[7] I. Belovas, M. Sabaliauskas, Series with binomial-like coefficients for the evaluation and 3D visualization of zeta functions, Informatica, vol. 31(4), (2020), 659–680. DOI: 10.15388/20-INFOR43410.15388/20-INFOR434
Search in Google Scholar Back to article
[8] I. Belovas, L. Sakalauskas, Limit theorems for the coefficients of the modified Borwein method for the calculation of the Riemann zeta-function values, Colloq. Math., vol. 151, no. 2 (2018), 217–227. DOI: 10.4064/cm7086-2-201710.4064/cm7086-2-2017
Search in Google Scholar Back to article
[9] E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, Journal of combinatorial theory (A), 15 (1973), 91-111.10.1016/0097-3165(73)90038-1
Search in Google Scholar Back to article
[10] K.-W. Chen, Delannoy Numbers and Preferential Arrangements, Mathematics, vol. 7, no. 3, art. no. 238 (2019), 1–9. DOI: 10.3390/math703023810.3390/math7030238
Search in Google Scholar Back to article
[11] C. Cobeli, A. Zaharescu, Promenade around Pascal Triangle Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Vol. 56 (1), (2013), 73–98.
Search in Google Scholar Back to article
[12] H.-K. Hwang, On Convergence Rates in the Central Limit Theorems for Combinatorial Structures, European Journal of Combinatorics, 19(3) (1998), 329–343.10.1006/eujc.1997.0179
Search in Google Scholar Back to article
[13] L. Mu, S. Zheng, On the total positivity of Delannoy-like triangles, Journal of Integer Sequences, Vol. 20, (2017), Article 17.1.6
Search in Google Scholar Back to article
[14] R. Noble, Asymptotics of the weighted Delannoy numbers, Int. J. Number Theory, 8 (1), (2012), 175–188. DOI: 10.1142/S179304211250010810.1142/S1793042112500108
Search in Google Scholar Back to article
[15] Sloane, N. J. A. (ed.), Square array of Delannoy numbers D(i, j) (i ⩾ 0, j ⩾ 0) read by antidiagonals. OEIS Number sequence A008288, The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A008288
Search in Google Scholar Back to article
[16] Y. Wang, S. Zheng and X. Chen, Analytic aspects of Delannoy numbers, Discrete Math., Vol. 342 (8), (2019), 2270–2277. DOI: 10.1016/j.disc.2019.04.00310.1016/j.disc.2019.04.003
Search in Google Scholar Back to article
[17] S. Yang, S. Zheng, S. Yuan, T. He, Schröder matrix as inverse of Delannoy matrix, Linear Algebra Appl., 439 (11), (2013), 3605–3614. DOI: 10.1016/j.laa.2013.09.04410.1016/j.laa.2013.09.044
Search in Google Scholar Back to article

Central and local limit theorems for the weighted Delannoy numbers

Abstract

Paradigm

My account