Determinantal forms and recursive relations of the Delannoy two-functional sequence

Yıl 2020, , 184 - 193, 31.08.2020

Feng Qi , Muhammet Cihat Dağlı , Wei-shih Du

https://doi.org/10.31197/atnaa.772734

Cited By: 5

Öz

In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers.

In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers.

Anahtar Kelimeler

Delannoy two-functional sequence, Delannoy one-functional sequence, Delannoy number, central Delannoy number

Destekleyen Kurum

Ministry of Science and Technology Republic of China

Proje Numarası

MOST-107-2115-M-017-004-MY2

Teşekkür

Thank a lot

Kaynakça

• \bibitem{banderier} C. Banderier and S. Schwer, \emph{Why Delannoy numbers?}, J. Statist. Plann. Inference \textbf{135} (2005), no.~1, 40\nobreakdash--54; available online at \url{https://doi.org/10.1016/j.jspi.2005.02.004}.
• \bibitem{closed-form-what-why-care} J. M. Borwein and R. E. Crandall, \emph{Closed forms: what they are and why we care}, Notices Amer. Math. Soc. \textbf{60} (2013), no.~1, 50\nobreakdash--65; available online at \url{https://doi.org/10.1090/noti936}.
• \bibitem{Bourbaki-Spain-2004} N. Bourbaki, \emph{Functions of a Real Variable, Elementary Theory}, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004; available online at \url{https://doi.org/10.1007/978-3-642-59315-4}.
• \bibitem{CollegeMJ-2002-Cahill} N. D. Cahill, J. R. D'Errico, D. A. Narayan, and J. Y. Narayan, \emph{Fibonacci determinants}, College Math. J. \textbf{33} (2002), no.~3, 221\nobreakdash--225; available online at \url{https://doi.org/10.2307/1559033}.
• \bibitem{M.C.Dagli-Accepted.tex} M. C. Da\u{g}l\i, \emph{A new generalization of Delannoy numbers}, accepted for publication in Indian Journal of Pure and Applied Mathematics.
• \bibitem{gould} H. W. Gould, \textit{Combinatorial Identities: A standardized set of tables listing 500 binomial coefficient summations}, Henry W. Gould, Morgantown, W.Va., 1972.
• \bibitem{guo} V. J. W. Guo, \emph{Proof of Sun's conjectures on integer-valued polynomials}, J. Math. Anal. Appl. \textbf{444} (2016), no.~1, 182\nobreakdash--191; available online at \url{https://doi.org/10.1016/j.jmaa.2016.06.028}.
• \bibitem{higgins} V. Higgins and C. Johnson, \emph{Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices}, Linear Algebra Appl. \textbf{489} (2016), 104\nobreakdash--122; available online at \url{https://doi.org/10.1016/j.laa.2015.10.004}.
• \bibitem{liu} J.-C. Liu, \emph{A supercongruence involving Delannoy numbers and Schr\"oder numbers}, J. Number Theory \textbf{168} (2016), 117\nobreakdash--127; available online at \url{https://doi.org/10.1016/j.jnt.2016.04.019}.
• \bibitem{liu1} J.-C. Liu, L. Li, and S.-D. Wang, \emph{Some congruences on Delannoy numbers and Schr\"oder numbers}, Int. J. Number Theory \textbf{14} (2018), no.~7, 2035\nobreakdash--2041; available online at \url{https://doi.org/10.1142/S1793042118501221}.
• \bibitem{martin} R. S. Martin and J. H. Wilkinson, \emph{Handbook Series Linear Algebra: Similarity reduction of a general matrix to Hessenberg form}, Numer. Math. \textbf{12} (1968), no.~5, 349\nobreakdash--368; available online at \url{https://doi.org/10.1007/BF02161358}.
• \bibitem{Delanoy-No.tex} F. Qi, \textit{A determinantal expression and a recursive relation of the Delannoy numbers}, Acta Univ. Sapientiae Math. \textbf{12} (2020), no.~2, in press; available online at \url{https://arxiv.org/abs/2003.12572}.
• \bibitem{ijaa753.tex} F. Qi and V. \v{C}er\v{n}anov\'a, \textit{Some discussions on a kind of improper integrals}, Int. J. Anal. Appl. \textbf{11} (2016), no.~2, 101\nobreakdash--109.
• \bibitem{Delannoy-Cent-P.tex} F. Qi, V. \v{C}er\v{n}anov\'a, X.-T. Shi, and B.-N. Guo, \textit{Some properties of central Delannoy numbers}, J. Comput. Appl. Math. \textbf{328} (2018), 101\nobreakdash--115; available online at \url{https://doi.org/10.1016/j.cam.2017.07.013}.
• \bibitem{Dagli.tex} F. Qi, M. C. Da\u{g}l\i, and W.-S. Du, \textit{Determinantal forms and recursive relations of the Delannoy two-functional sequence}, OSF Preprints (2020), available online at \url{https://doi.org/10.31219/osf.io/u683y}.
• \bibitem{Recipr-Sqrt-Geom-W.tex} F. Qi and B.-N. Guo, \textit{The reciprocal of the weighted geometric mean is a Stieltjes function}, Bol. Soc. Mat. Mex. (3) \textbf{24} (2018), no.~1, 181\nobreakdash--202; available online at \url{https://doi.org/10.1007/s40590-016-0151-5}.
• \bibitem{Schroder-Seq-3rd.tex} F. Qi, X.-T. Shi, and B.-N. Guo, \textit{Some properties of the Schr\"oder numbers}, Indian J. Pure Appl. Math. \textbf{47} (2016), no.~4, 717\nobreakdash--732; available online at \url{https://doi.org/10.1007/s13226-016-0211-6}.
• \bibitem{sun1} Z.-H. Sun, \emph{A kind of orthogonal polynomials and related identities}, J. Math. Anal. Appl. \textbf{456} (2017), no.~2, 912\nobreakdash--926; available online at \url{https://doi.org/10.1016/j.jmaa.2017.07.049}.
• \bibitem{sun2} Z.-W. Sun, \emph{Arithmetic properties of Delannoy numbers and Schr\"oder numbers}, J. Number Theory \textbf{183} (2018), 146\nobreakdash--171; available online at \url{https://doi.org/10.1016/j.jnt.2017.07.011}.
• \bibitem{sun3} Z.-W. Sun, \emph{On Delannoy numbers and Schr\"oder numbers}, J. Number Theory \textbf{131} (2011), no.~12, 2387\nobreakdash--2397; available online at \url{https://doi.org/10.1016/j.jnt.2011.06.005}.
• \bibitem{sun} Z.-W. Sun, \emph{Supercongruences involving dual sequences}, Finite Fields Appl. \textbf{46} (2017), 179\nobreakdash--216; available online at \url{https://doi.org/10.1016/j.ffa.2017.03.007}.
• \bibitem{MR3952588} Y. Wang, S.-N. Zheng, and X. Chen, \textit{Analytic aspects of Delannoy numbers}, Discrete Math. \textbf{342} (2019), no.~8, 2270\nobreakdash--2277; available online at \url{https://doi.org/10.1016/j.disc.2019.04.003}.