Accepted Manuscript Some properties of central Delannoy numbers Feng Qi, Viera Cerˇ nanová,ˇ Xiao-Ting Shi, Bai-Ni Guo PII: S0377-0427(17)30354-0 DOI: http://dx.doi.org/10.1016/j.cam.2017.07.013 Reference: CAM 11224 To appear in: Journal of Computational and Applied Mathematics Received date : 29 June 2017 Please cite this article as: F. Qi, V. Cerˇ nanová,ˇ X. Shi, B. Guo, Some properties of central Delannoy numbers, Journal of Computational and Applied Mathematics (2017), http://dx.doi.org/10.1016/j.cam.2017.07.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As aserviceto our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Manuscript Click here to view linked References SOME PROPERTIES OF CENTRAL DELANNOY NUMBERS FENG QI, VIERA CERˇ NANOVˇ A,´ XIAO-TING SHI, AND BAI-NI GUO Abstract. In the paper, by investigating the generating function of central Delannoy numbers, the authors establish several explicit expressions, includ- ing determinantal expressions, for central Delannoy numbers, present three identities involving the Cauchy products of central Delannoy numbers, dis- cover 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 inequal- ities for central Delannoy numbers. 1. Main results The Delannoy numbers D(a,b) are the number of lattice paths from (0, 0) to (a,b) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed. They have the generating function 1 ∞ = D(p, q)xpyq. 1 x y xy p,q=0 − − − X Taking n = a = b gives central Delannoy numbers D(n) D(n,n), which are the number of “king walks” from the (0, 0) corner of an n n ≡square to the upper right corner (n,n). Central Delannoy numbers D(n) have the× generating function 1 ∞ G(x)= = D(n)xk =1+3x + 13x2 + 63x3 + . (1.1) √ 2 ··· 1 6x + x n=0 − X For more information on the Delannoy numbers D(a,b) and central Delannoy num- bers D(n), please refer to the papers [8, 20] and closely-related reference therein. It is well known [19, 22] that (1) a sequence ak, k 0 is said to be increasing if and only if ak ak+1 for all k 0; ≥ ≤ ≥ (2) a sequence ak, k 0 is said to be convex if and only if ak + ak+2 2ak+1 for all k 0; see≥ [19, p. 42, Exercise 1] and [22, Definition 1.11]; ≥ ≥ (3) a positive sequence ak, k 0 is said to be logarithmically convex if and only if a a a2 for all k≥ 0; see [19, p. 70, Exercise 5]. k k+2 ≥ k+1 ≥ (4) if the sequence ak, k 0 is increasing (or convex, logarithmically convex, respectively), then the≥ function whose graph is the polygonal line corner 2010 Mathematics Subject Classification. Primary 11Y55; Secondary 05A15, 05A19, 05A20, 11B75, 11B83, 11Y35, 26A48, 30E20, 33B99, 44A10, 44A15. Key words and phrases. central Delannoy number; explicit expression; identity; Cauchy prod- uct; integral representation; monotonicity; absolute monotonicity; convexity; logarithmic convex- ity; determinantal inequality; product inequality; majorization; Cauchy integral formula. This paper was typeset using -LATEX. AMS 1 2 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO points (k,ak) is also increasing (or convex, logarithmically convex, respec- tively) on [1, ); see [19, p. 43, Remark] and [22, Remark 1.12]. ∞ Recall from [18, Chapter XIII] and [41, Chapter IV] that a function f is said to be absolutely monotonic on an interval I if f has derivatives of all orders on I and 0 f (n)(x) < for x I and n 0. This notion has been generalized in [11]. Recall≤ from [18,∞ Chapter∈ XIII], [38,≥ Chapter 1], and [41, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if 0 ( 1)kf (k)(t) < on I for all k 0. It is easy to see that a function f(x) is≤ absolutely− monotonic∞ on an interval≥I if and only if the function f( x) is completely monotonic on the interval I. Theorem 12a in [41, p. 160] reads− that a function f is completely monotonic− on [0, ) if and only if it is a ∞ ∞ xt Laplace transform f(x)= 0 e− d µ(t) of a bounded and non-decreasing measure µ(t). R n n Let λ = (λ1, λ2,...,λn) R and µ = (µ1,µ2,...,µn) R . We say that λ is majorized by µ (in symbols∈λ µ) if ∈ k k n n λ µ , k =1, 2,...,n 1 and λ = µ , [ℓ] ≤ [ℓ] − ℓ ℓ Xℓ=1 Xℓ=1 Xℓ=1 Xℓ=1 where λ[1] λ[2] λ[n] and µ[1] µ[2] µ[n] are respectively the components≥ of λ and≥ ···µ in ≥ decreasing order.≥ We say≥ ··· that ≥ λ is strictly majorized by µ (in symbols λ µ) if λ is majorized by µ and is not a permutation of µ. For example, ≺ 1 1 1 1 1 1 ,..., ,..., , 0 , , 0,..., 0 (1, 0,..., 0). n n ≺ n 1 n 1 ≺ 2 2 ≺ − − n 2 n 1 n n 1 − − − | {z } | {z } For more| information{z } | on the{z theory of} majorization and its applications, please refer to monographs [14, 15] and closely related references therein. The first aim of this paper is, by investigating the generating function G(x), to establish several explicit expressions, including determinantal expressions, for central Delannoy numbers D(k). Theorem 1.1. For k 0, central Delannoy numbers D(k) can be computed by ≥ k k (2ℓ 1)!! [2(k ℓ) 1]!! 2ℓ D(k)= 3 2√2 − − − 3 2√2 (1.2) ± (2ℓ)!! [2(k ℓ)]!! ∓ ℓ=0 − X and k ( 1)k (2ℓ 1)!! ℓ D(k)= − ( 1)ℓ62ℓ − , (1.3) 6k − (2ℓ)!! k ℓ Xℓ=0 − where p = 0 for q >p 0 and the double factorial of negative odd integers q ≥ (2n + 1) is defined by − ( 1)n 2nn! ( 2n 1)!! = − = ( 1)n , n =0, 1,.... − − (2n 1)!! − (2n)! − PROPERTIESOFCENTRALDELANNOYNUMBERS 3 For k N, central Delannoy numbers D(k) satisfy ∈ a1 1 0 0 0 0 a a 1 ··· 0 0 0 2 1 ··· a3 a2 a1 0 0 0 ··· k . D(k) = ( 1) . .. , (1.4) − ak 2 ak 3 ak 4 a1 1 0 − − − ··· ak 1 ak 2 ak 3 a2 a1 1 − − − ··· ak ak 1 ak 2 a3 a2 a1 − − ··· where k ( 1)k+1 (2ℓ 3)!! ℓ a = − ( 1)ℓ62ℓ − , (1.5) k 6k − (2ℓ)!! k ℓ Xℓ=1 − and D(1) 1 0 0 0 0 D(2) D(1) 1 ··· 0 0 0 ··· D(3) D(2) D(1) 0 0 0 ··· . .. D(k 2) D(k 3) D(k 4) D(1) 1 0 − − − ··· D(k 1) D(k 2) D(k 3) D(2) D(1) 1 − − − ··· D(k) D(k 1) D(k 2) D(3) D(2) D(1) − − ··· k 1 (2ℓ 3)!! ℓ = ( 1)ℓ62ℓ − . (1.6) −6k − (2ℓ)!! k ℓ Xℓ=1 − The second aim of this paper is, by studying the generating function G(x), to present three identities involving the Cauchy products of central Delannoy numbers D(k). Theorem 1.2. The Cauchy products of central Delannoy numbers D(k) satisfy the following identities. (1) For k 0, the Cauchy products of central Delannoy numbers D(k) can be computed≥ by k ( 1)k k ℓ D(ℓ)D(k ℓ)= − ( 1)ℓ62ℓ , (1.7) − 6k − k ℓ Xℓ=0 Xℓ=0 − where p =0 for q>p 0; q ≥ (2) For k 2, the Cauchy products of central Delannoy numbers D(k) satisfy the identity≥ k k 1 k 2 − − D(ℓ)D(k ℓ) 6 D(ℓ)D(k ℓ 1)+ D(ℓ)D(k ℓ 2) = 0; (1.8) − − − − − − Xℓ=0 Xℓ=0 Xℓ=0 4 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO (3) For k 1, the Cauchy products of central Delannoy numbers D(k) can be represented≥ in terms of a tri-diagonal determinant by 61 00 0 0 0 −1 6 1 0 ··· 0 0 0 − ··· 0 1 6 1 0 0 0 k − ··· k . D(ℓ)D(k ℓ) = ( 1) . .. (1.9) − − ℓ=0 0 0 00 6 1 0 X ··· − 0 0 00 1 6 1 ··· − 0 0 00 0 1 6 ··· − k k × The third aim of this paper is, by representing the generating function G(x) asan integral representation, to discover an integral representation for central Delannoy numbers D(k). Theorem 1.3. For k 0, central Delannoy numbers D(k) can be represented by ≥ 1 3+2√2 1 1 D(k)= k+1 d t. (1.10) π 3 2√2 t Z − t 3+2√2 3+2√2 t − − q The fourth aim of this paper is, with the help of the integral representation (1.10), to find absolute monotonicity of the function D(x) on [0, ). Consequently, we deduce the monotonicity and convexity and construct two produc∞ t inequalities for the sequence of central Delannoy numbers D(k). Theorem 1.4. The central Delannoy function 1 3+2√2 1 1 D(x)= x+1 d t, x R π 3 2√2 t ∈ Z − t 3+2√2 3+2√2 t − − q 1 1 is completely monotonic on , 2 and absolutely monotonic on 2 , .