Some properties of central Delannoy numbers

Abstract In the paper, by investigating the generating function of central Delannoy numbers, the authors establish several explicit expressions, including determinantal expressions, for central Delannoy numbers, present three identities involving the Cauchy products of central Delannoy numbers, discover an integral representation for central Delannoy numbers, find (absolute) monotonicity, convexity, and logarithmic convexity for the sequence of central Delannoy numbers, and construct several product and determinantal inequalities for central Delannoy numbers.

[1]  J. Gurland,et al.  On Wallis' Formula , 1956 .

[2]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[3]  J. Pečarić.  Remarks on some inequalities of A. M. Fink , 1984 .

[4]  Feng Qi (祁锋),et al.  Several closed expressions for the Euler numbers , 2015 .

[5]  Feng Qi (祁锋).  BOUNDS FOR THE RATIO OF TWO GAMMA FUNCTIONS-FROM WENDEL'S AND RELATED INEQUALITIES TO LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS , 2009, 0904.1048.

[6]  Jing Liu,et al.  Two spectral gradient projection methods for constrained equations and their linear convergence rate , 2015 .

[7]  Feng Qi (祁锋),et al.  Several formulas for special values of the Bell polynomials of the second kind and applications , 2017 .

[8]  Christian Berg,et al.  Integral Representation of Some Functions Related to the Gamma Function , 2004 .

[9]  H. van Haeringen,et al.  Completely Monotonic and Related Functions , 1996 .

[10]  Feng Qi (祁锋),et al.  Some properties of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body , 2018 .

[11]  Robin J. Chapman,et al.  Two closed forms for the Bernoulli polynomials , 2015, 1506.02137.

[12]  Feng Qi,et al.  Explicit expressions for a family of the Bell polynomials and applications , 2015, Appl. Math. Comput..

[13]  Josip Pečarić,et al.  Some Inequalities for Monotone Functions , 1993 .

[14]  Feng Qi (祁锋),et al.  The best bounds in Wallis' inequality , 2004 .

[15]  Rob Noble.  ASYMPTOTICS OF THE WEIGHTED DELANNOY NUMBERS , 2012 .

[16]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[17]  T. Neumann.  Advanced Combinatorics The Art Of Finite And Infinite Expansions , 2016 .

[18]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[19]  Feng Qi (祁锋),et al.  COMPLETELY MONOTONIC FUNCTION ASSOCIATED WITH THE GAMMA FUNCTIONS AND PROOF OF WALLIS' INEQUALITY , 2005 .

[20]  Feng Qi,et al.  Bounds for the Ratio of Two Gamma Functions , 2009, 0904.1048.

[21]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[22]  Cristinel Mortici,et al.  Some best approximation formulas and inequalities for the Wallis ratio , 2013, Appl. Math. Comput..

[23]  Feng Qi (祁锋),et al.  A PROPERTY OF LOGARITHMICALLY ABSOLUTELY MONOTONIC FUNCTIONS AND THE LOGARITHMICALLY COMPLETE MONOTONICITY OF A POWER-EXPONENTIAL FUNCTION , 2009, 0903.5038.

[24]  Constantin P. Niculescu,et al.  Convex Functions and Their Applications: A Contemporary Approach , 2005 .

[25]  Maciej Dziemiańczuk,et al.  Generalizing Delannoy Numbers via Counting Weighted Lattice Paths , 2013, Integers.

[26]  Feng Qi (祁锋),et al.  Some exact constants for the approximation of the quantity in the Wallis’ formula , 2013 .

[27]  Feng Qi (祁锋),et al.  Explicit Formulas for Special Values of the Bell Polynomials of the Second Kind and for the Euler Numbers and Polynomials , 2017 .

[28]  A. M. Fink.  Kolmogorov-Landau inequalities for monotone functions , 1982 .

[29]  Di Zhao,et al.  Notes on Greub-Rheinboldt inequalities , 2013, Journal of Inequalities and Applications.

[30]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[31]  Best upper and lower bounds for Wallis' inequality of higher order , 2016 .

[32]  Feng Qi,et al.  Two Explicit Formulas of the Schröder Numbers , 2016, Integers.

[33]  Feng Qi,et al.  Complete Monotonicities of Functions Involving the Gamma and Digamma Functions , 2004 .

[34]  Senlin Guo,et al.  A class of logarithmically completely monotonic functions , 2008, Appl. Math. Lett..

[35]  On the refinements of the integral Jensen-Steffensen inequality , 2013 .

[36]  Feng Qi (祁锋).  Bounds for the Ratio of Two Gamma Functions , 2009 .

[37]  Feng Qi,et al.  A complete monotonicity property of the gamma function , 2004 .

[38]  Chen Yi-jun.  REMARKS ON SOME INEQUALITIES , 2005 .