On signs of several Toeplitz--Hessenberg determinants whose elements contain central Delannoy numbers

Document Type : Short notes


1 Department of Science, Henan University of Animal Husbandry and Economy, Zhengzhou 450046, Henan, China

2 School of Economics, Technology and Media University of Henan Kaifeng, Henan, Kaifeng 475001, China

3 Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454003, Henan, China


In the paper, by virtue of Wronski's formula and Kaluza's theorem for the power series and its reciprocal, and with the aid of the logarithmic convexity of a sequence constituted by central Delannoy numbers, the authors present negativity of several Toeplitz--Hessenberg determinants whose elements contain central Delannoy numbers and combinatorial numbers.


Main Subjects

[1] M. Dağli and F. Qi, Several closed and determinantal forms for convolved Fibonacci numbers, Discrete Math. Lett. 7 (2021), 14–20.
[2] V.J.W. Guo and J. Zeng, New congruences for sums involving apéry numbers or central delannoy numbers, Int. J. Number Theory 8 (2012), no. 8, 2003–2016.
[3] P. Henrici, Applied and Computational Complex Analysis. Vol. 1, John Wiley & Sons, Inc., New York, 1988.
[4] R.A. Horn and C.R. Johnson, Matrix Analysis, second ed., Cambridge University Press, Cambridge, 2013.
[5] A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl., 63 (1978), no. 2, 347–353.
[6] Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Zeitschrift 28 (1928), no. 1, 161–170.
[7] Y.-W. Li, M.C. Dağlı, and F. Qi, Two explicit formulas for degenerate peters numbers and polynomials, Discrete Math. Lett. 8 (2022), 1–5.
[8] F. Qi, Three closed forms for convolved Fibonacci numbers, Results Nonlinear Anal. 3 (2020), no. 4, 185–195.
[9] F. Qi, A determinantal expression and a recursive relation of the Delannoy numbers, Acta Univ. Sapientiae Math. 13 (2021), no. 2, 442–449.
[10] F. Qi, Determinantal expressions and recursive relations of Delannoy polynomials and generalized Fibonacci polynomials, J. Nonlinear Convex Anal. 22 (2021), no. 7, 1225–1239.
[11] F. Qi, On negativity of Toeplitz–Hessenberg determinants whose elements contain large Schr¨oder numbers, Palestine J. Math. 11 (2022), no. 4, 373–378.
[12] F. Qi, On signs of certain Toeplitz–Hessenberg determinants whose elements involve Bernoulli numbers, Contrib. Discrete Math. (In press).
[13] F. Qi, V. Čerňanová, X.-T. Shi, and B.-N. Guo, Some properties of central Delannoy numbers, J. Comput. Appl. Math. 328 (2018), 101–115.
[14] F. Qi and R.J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory 159 (2016), 89–100.
[15] F. Qi, M.C. Dağlı, and W.-S. Du, Determinantal forms and recursive relations of the Delannoy two-functional sequence, Adv. Theory Nonlinear Anal. Appl. 4 (2020), no. 3, 184–193.
[16] F. Qi, P. Natalini, and P.E. Ricci, Recurrences of Stirling and Lah numbers via second kind Bell polynomials, Discrete Math. Lett. 3 (2020), 31–36.
[17] F. Qi, X.-T. Shi, and B.-N. Guo, Some properties of the Schr¨oder numbers, Indian J. Pure Appl. Math. 47 (2016), no. 4, 717–732.
[18] F. Qi, V. Čerňanová, and Y.S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), no. 1, 123–136.
[19] H. Rutishauser, Eine formel von Wronski und ihre bedeutung f¨ur den quotienten-differenzen-algorithmus, Z. Angew. Math. Phys. 7 (1956), 164–169.
[20] D. Serre, Matrices, Graduate Texts in Mathematics, vol. 216, Springer, New York, 2010.
[21] M.H. Wronski, Introduction á la Philosophie des Math´ematiques: Et Technie de l’Algorithmie, Chez COURCIER, Imprimeur-Libraire pour les Math´ematiques,
quai des Augustins, no 57, Paris, 1811 (French).