# A Qi formula for translated $r$-Dowling numbers

Volume 20, Issue 2, pp 88--100
Publication Date: October 19, 2019 Submission Date: June 28, 2019 Revision Date: September 07, 2019 Accteptance Date: September 10, 2019
• 100 Views

### Authors

Roberto B. Corcino - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines. Cristina B. Corcino - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines. Jeneveb T. Malusay - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines.

### Abstract

Another form of an explicit formula for translated $r$-Dowling numbers is derived using Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms of the translated $r$-Whitney numbers of the second kind and the ordinary Lah numbers, which is analogous to Qi formula. As a consequence, a relation between translated $r$-Dowling numbers and the sums of row entries of the product of two matrices containing the translated $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established.

### Keywords

• Qi formula
• Translated $r$-Dowling numbers
• Bell polynomials
• Lah numbers
• translated $r$-Whitney numbers
• Faa di Bruno's formula
• $r$-Whitney-Lah numbers

•  05A15
•  11B65
•  11B73

### References

• [1] I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Classical Discrete Orthogonal Polynomials, Lah Numbers, and Involutory Matrices, Appl. Math. Lett., 16 (2003), 383--387

• [2] H. Belbachir, I. E. Bousbaa, Translated Whitney and $r$-Whitney Numbers: A Combinatorial Approach, J. Integer Seq., 16 (2013), 7 pages

• [3] M. Benoumhani, On Whitney Numbers of Dowling Lattices, Discrete Math., 159 (1996), 13--33

• [4] K. N. Boyadzhiev, Lah Numbers, Laguerre Polynomials of Order Negative One, and the $n$th Derivative of $\exp(1/x)$, Acta Univ. Sapientiae Math., 8 (2016), 22--31

• [5] A. Z. Broder, The $r$-Stirling Numbers, Discrete Math., 49 (1984), 241--259

• [6] C. C. Chen, K.-M. Koh, Principles and Techniques in Combinatorics, World Scientific Publ. Co., Singapore (1992)

• [7] G.-S. Cheon, J.-H. Jung, $r$-Whitney numbers of Dowling Lattices, Discrete Math., 312 (2012), 2337--2348

• [8] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)

• [9] S. Daboul, J. Mangaldan, M. Z. Spivey, P. J. Taylor, The Lah Numbers and the $n$th Derivative of $e^\frac{1}{x}$, Math. Mag., 86 (2013), 39--47

• [10] T. A. Dowling, A Class of Geometric Lattices Based on Finite Groups, J. Combinatorial Theory Ser. B, 14 (1973), 61--86

• [11] B.-N. Guo, F. Qi, Six Proofs for an Identity of the Lah Numbers, Online J. Anal. Comb., 10 (2015), 5 pages

• [12] E. Gyimesi, G. Nyul, A comprehensive study of $r$-Dowling polynomials, Aequationes Math., 92 (2018), 515--527

• [13] E. Gyimesi, G. Nyul, New combinatorial interpretations of $r$-Whitney and r-Whitney-Lah numbers, Discrete Appl. Math., 255 (2019), 222--233

• [14] L. C. Hsu, P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math., 20 (1998), 366--384

• [15] I. Lah, IEine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, (German) Mitteilungsbl. Math. Statist., 7 (1955), 203--212

• [16] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando (1985)

• [17] M. Merca, A New Connection between $r$-Whitney numbers and Bernuolli Polynomials, Integral Transforms Spec. Funct., 25 (2014), 937--942

• [18] I. Mező, New Properties of $r$-Stirling Series, Acta Math. Hungar., 119 (2008), 341--358

• [19] I. Mező, On the Maximum of $r$-Stirling numbers, Adv. in Appl. Math., 41 (2008), 293--306

• [20] I. Mező, A New Formula for the Bernoulli Polynomials, Results Math., 58 (2010), 329--335

• [21] I. Mező, The $r$-Bell Numbers, J. Integer Seq., 14 (2011), 14 pages

• [22] F. Qi, An Explicit Formula for the Bell Numbers in terms of Lah and Stirling Numbers, Mediterr. J. Math., 13 (2016), 2795--2800

• [23] J. Riordan, Introduction to Combinatorial Analysis, John Wiley & Sons, New York (1958)

• [24] C. C. Wagner, Generalized Stirling and Lah numbers, Discrete Math., 160 (1996), 199--218

• [25] X.-J. Zhang, F. Qi, W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math., 5 (2012), 122--127