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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

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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, 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. 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 numbers 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 product; integral representation; monotonicity; absolute monotonicity; convexity; logarithmic convexity; 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, respectively) 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 , . Con- sequently, the sequence of central−∞ − Delannoy numbers D(k) for k 0−is increasing∞ and convex. ≥

Theorem 1.5. Let n N, x1, x2,...,xn 0 N, and p1,p2,...,pn N such that n 1 =1. Then∈ ∈ { } ∪ ∈ ℓ=1 pℓ P n n n 1 [D(x0)] − D xk D(x0 + xk) (1.11) ! ≥ kY=0 kY=1 and n n 1/p1 1/pk D xk [D(x1)] [D(x1 + pkxk)] . (1.12) ! ≤ kY=1 kY=2 The ﬁnal aim of this paper is, with the aid of complete monotonicity of the generating function G( x), to construct product and determinantal inequalities and, consequently, to derive− the logarithmic convexity, for the sequence n!D(n),n 0 . { ≥ } Theorem 1.6. Let n 1 be a positive integer and let a denote a determinant ≥ | pq|n of order n with elements apq. PROPERTIESOFCENTRALDELANNOYNUMBERS 5

(1) If a for 1 p n are non-negative integers, then p ≤ ≤ ( 1)ap+aq (a + a )!D(a + a ) 0 (1.13) − p q p q n ≥ and

(a + a )!D(a + a ) 0. (1.14) p q p q n ≥ (2) If a = (a1,a2,...,a n) and b = (b1,b2,...,b n) are non-increasing n-tuples of non-negative integers such that a b, then n D(a ) n b ! p p . (1.15) D(b ) ≥ a ! p=1 p p=1 p Y Y Corollary 1.1. If ℓ 0 and n k 0, then ≥ ≥ ≥ (n + ℓ)! k D(n + ℓ) k (k + ℓ)! n D(k + ℓ) n . (1.16) ℓ! D(ℓ) ≥ ℓ! D(ℓ) Corollary 1.2. The sequence n!D(n),n 0 is logarithmically convex and, consequently, is convex. { ≥ } Theorem 1.7. If ℓ 0, n k m, 2k n, and 2m n, then ≥ ≥ ≥ ≥ ≥ D(k)D(n k) D(m)D(n m) n − n − . (1.17) k ≥ m 2. Lemmas In order to prove our main results, we need more notation and lemmas. In combinatorial mathematics, the Bell polynomials of the second kind Bn,k are deﬁned by

n k+1 ℓ n! − xi i Bn,k(x1, x2,...,xn k+1)= − n k+1 − ℓi! i! ℓi 0 N i=1 i=1 n∈{X}∪ Y i=1 iℓi=n n ℓi=k Q P i=1 for n k 0. See [7, p. 134, TheoremP A]. In combinatorial analysis, the Fa`adi Bruno≥ formula≥ plays an important role and can be described by n n d (k) (n k+1) [f h(t)] = f (h(t))B h′(t),h′′(t),...,h − (t) (2.1) d tn ◦ n,k k=0 X in terms of the Bell polynomials of the second kind Bn,k. See [7, p. 139, Theorem C]. Lemma 2.1 ([7, p. 135] and [35, Lemma 1]). For n k 0, we have ≥ ≥ 2 n k+1 k n Bn,k abx1,ab x2,...,ab − xn k+1 = a b Bn,k(x1, x2,...,xn k+1), (2.2) − − where aand b are any complex numbers. Lemma 2.2 ([29, Theorem 4.1], [36, Eq. (2.8)], [37, Section 3], and [40, Lemma 2.5]). For n k 0, we have ≥ ≥ (n k)! n k 2k n B (x, 1, 0,..., 0) = − x − , (2.3) n,k 2n k k n k − − 6 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

p where =0 for q>p 0. More generally, for n k 0 and λ R, we have q ≥ ≥ ≥ ∈ n k k n 1 − ( 1)k k − B 1, 1 λ, (1 λ)(1 2λ),..., (1 ℓλ) = − ( 1)ℓ (ℓ qλ). n,k − − − − k! − ℓ − ! q=0 ℓY=0 Xℓ=0 Y k k Lemma 2.3 ([25, Lemma 2.4]). Let f(t)=1+ k∞=1 akt and g(t)=1+ k∞=1 bkt be formal power series such that f(t)g(t)=1. Then P P a1 1 0 0 0 a a 1 0 ··· 0 2 1 ··· a3 a2 a1 1 0 k ··· bk = ( 1) ...... − ......

ak 1 ak 2 ak 3 ak 4 1 − − − − ··· ak ak 1 ak 2 ak 3 a1 − − − ···

In [1, 28], it was defined implicitly and explicitly that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if the inequality ( 1)k[ln f(x)](k) 0 holds on I for all k N. In [2, Theorem 1.1], [11, Theorem 4],− [27, Theorem≥ 1], and [28, Theorem 4], it∈ was found and verified once again that a logarithmically completely monotonic function must be completely monotonic, but not conversely. In [38, Definition 2.1], it was defined that a Stieltjes transform is a function f : (0, ) [0, ) which can be written in the form ∞ → ∞ a ∞ 1 f(x)= + b + d µ(u), (2.4) x u + x Z0 where a,b are nonnegative constants and µ is a nonnegative measure on (0, ) ∞ such that the integral ∞ 1 d µ(s) < . In [2, Theorem 1.2], it was proved 0 1+s ∞ that a positive Stieltjes transform must be a logarithmically completely monotonic R function on (0, ), but not conversely. ∞ Let i = √ 1 be the imaginary unit. We will consider, as usual, the principal − ln z +i arg z branch of the square root function √z as exp | | 2 for z C and arg z ( π, π]. For b>a, let ∈ ∈ − 1 ha,b(z)= , z C ( , a]. (2.5) (z + a)(z + b) ∈ \ −∞ −

The principal branch of the function ha,b(z) is analytic on C ( , a]. p \ −∞ − Lemma 2.4. For b>a and z C ( , a], the principal branch of the function ∈ \ −∞ − ha,b(z) deﬁned by (2.5) can be represented as 1 1 b 1 1 = d t. (2.6) (z + a)(z + b) π a (t a)(b t) t + z Z − − Consequently, thep real function ha,b(x) isp a Stieltjes transform, a logarithmically completely monotonic function on ( a, ), and a completely monotonic function on ( a, ). − ∞ − ∞ First proof of Lemma 2.4. For r > 0 and x a, by virtue of Figure 1, we see that ≤ − π arg(x + ir + a) < π and 0 < arg(x + ir + b) < π. 2 ≤ Then a straightforward computation gives PROPERTIESOFCENTRALDELANNOYNUMBERS 7

p y

x + ir b + ir x + ir a + ir − −

b a − −

Figure 1.

1 ha,b(x + ir)= (x + ir + a)(x + ir + b) ln (x + ir + a)(x + ir + b) + i arg[(x + ir + a)(x + ir + b)] = exp | p | − 2 i exp (2π 2π) , x< b ln (x + a)(x + b) −2 − − exp | |  → − 2 × i exp π , b 0 and any ﬁxed point z0 = x0 + iy0 C ( , a], choose r > 0 and R>b>a> 0 such that ∈ \ −∞ − 0

1 h (ξ) h (z )= a,b d ξ a,b 0 2πi ξ z IC(r,R) − 0 1 h (ξ) h (ξ) h (ξ) h (ξ) = a,b d ξ + a,b d ξ + a,b d ξ + a,b d ξ , 2πi C(R) ξ z0 C(a,r) ξ z0 ℓ+ ξ z0 ℓ ξ z0 Z − Z − Z − Z − − where ha,b(z) is deﬁned by (2.5). By a straightforward computation, we have

π arctan r iθ h (ξ) − √R2 r2 ha,b Re a,b − iθ d ξ = iθ iRe d θ C(R) ξ z0 π+arctan r Re z0 Z Z− √R2 r2 − − − π 1 iθ i lim iθ lim ha,b Re d θ, R → π R 1 z0/Re R → ∞ Z− →∞ − →∞ π 1 = i lim d θ π R iθ iθ Z− →∞ Re + a Re + b =0, q π/2 iθ h (ξ) − ire 1 a,b d ξ = d θ iθ iθ iθ C(a,r) ξ z0 π/2 re a z0 re (re a + b) Z − Z − − − 0, r 0+, → → p and, if we denote R(r)= √R2 r2 , − a h (ξ) h (ξ) − h (x + ir) a,b d ξ + a,b d ξ = a,b d x ℓ+ ξ z0 ℓ ξ z0 R(r) x + ir z0 Z − Z − − Z− − R(r) a − h (x ir) − h (x + ir) h (x ir) + a,b − d x = a,b a,b − d x a x ir z0 R(r) x + ir z0 − x ir z0 Z− − − Z− − − − a − (x z0)[ha,b(x + ir) ha,b(x ir)] ir[ha,b(x + ir)+ ha,b(x ir)] = − − − 2 − 2 − d x R(r) (x z0) + r Z− − a − (x z0) ha,b(x + ir) ha,b x + ir ir ha,b(x + ir)+ ha,b x + ir = − − 2 − 2 d x R(r) (x z0) + r Z− − a − (x z ) h (x + ir) h (x + ir) ir h (x + ir)+ h (x + ir) − 0 a,b − a,b − a,b a,b = 2 2 d x R(r) (x z0) + r Z− − a − 2i(x z0) [ha,b(x + ir)] 2ir [ha,b(x + ir)] = − ℑ 2 − 2 ℜ d x R(r) (x z0) + r Z− − a − 1 2i lim [ha,b(x + ir)] d x → x z0 r 0+ ℑ Z−∞ − → a − 1 1 = 2i d x by the limit (2.7) − b x z0 (x + a)(x + b) Z− − | | a 1 1 =2i p d t b t z0 (a t)(b t) Z − − | − − | b 1 1 =2i p d t a t + z0 (t a)(b t) Z − − p PROPERTIESOFCENTRALDELANNOYNUMBERS 9 as r 0+ and R . In conclusion, we obtain → → ∞ 1 b 1 1 ha,b(z0)= d t. π a t + z0 (t a)(b t) Z − − The rest of the results can be derived readilyp from comparing the integral representations (2.4) and (2.6) and making use of the relations, which are mentioned around (2.4), among the Stieltjes transforms, logarithmically completely monotonic functions, and completely monotonic functions. The ﬁrst proof of Lemma 2.4 is complete. Second proof of Lemma 2.4. After knowing the integral representation (2.6), we accidently ﬁnd a simple proof below. In [10, p. 256, 3.121], it was stated that π q , > 1; p q q 1 p p p − 1 1 d x  π q =  q , < 0;  q q 0 q px x(1 x) −p 1 p Z −  p p −  − q p 0, q 0 < < 1.  p  q t a For b>a> 0 and p < 0, letting x =b−a in the above integral and simplifying result in − b 1 d t π = . q q q a t p (b a)+ a (t a)(b t) (b a) 1 Z − − − − − p p − q s+a p q Further taking p = b a < 0 reveals − − b 1 d t π = , s> a. a t + s (t a)(b t) (s + a)(s + b) − Z − − The range of s in thep above integral representationp can be analytically extended to the cut complex plane C ( , a]. The second proof of Lemma 2.4 is thus \ −∞ − complete. Lemma 2.5. Let α> 1 and β R. Then the improper integral ∈ 1 α 2k 1 < 0, β> 2 1 ln − t I(α, β, k)= d t =0, β = 1 (t 1/α)(α t) tβ  2 Z1/α  1 − − > 0, β< 2 p for all k N.  ∈  Proof. A straightforward computation yields 1 α 2k 1 1 ln − t I(α, β, k)= + β d t 1/α 1 (t 1/α)(α t) t Z Z − − 1 2k 1 1p ln − t = β d t 1/α (t 1/α)(α t) t Z − − 1/α 2k 1 p 1 ln − (1/s) 1 + β 2 d s 1 (1/s 1/α)(α 1/s) (1/s) −s Z − − p 10 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

1 2k 1 1 2k 1 1 ln − t 1 ln − s = β d t 1 β d s 1/α (t 1/α)(α t) t − 1/α (α s)(s 1/α) s − Z − − Z − − 1 p 1 1 1 p 2k 1 − = β 1 β ln t d t 1/α (t 1/α)(α t) t − t − Z − − 1 < 0,p β> 2 ; 1 =0, β = 2 ;  1 > 0, β< 2 . The proof of Lemma 2.5 is complete.

3. Proofs of main results Now we are in a position to prove our main results.

Proof of Theorem 1.1. A direct calculation gives 1 1 (k) h(k)(z)= a,b √z + a √z + b k (ℓ) (k ℓ) k 1 1 − = ℓ √z + a √z + b Xℓ=0 k k 1 1 1 1 = ℓ+1/2 k ℓ+1/2 ℓ −2 ℓ (z + a) −2 k ℓ (z + b) − Xℓ=0 − k k ℓ k ℓ (2ℓ 1)!! ( 1) − [2(k ℓ) 1]!! 1 = ( 1) −ℓ − ℓ+1/2 −k ℓ− k ℓ+1/2 ℓ − 2 (z + a) 2 − (z + b) − Xℓ=0 k ( 1)k z + b 1/2 1 k z + b ℓ = − (2ℓ 1)!![2(k ℓ) 1]!! 2k z + a (z + b)k+1 ℓ − − − z + a Xℓ=0 ( 1)k b 1/2 1 k k b ℓ − (2ℓ 1)!![2(k ℓ) 1]!! → 2k a bk+1 ℓ − − − a Xℓ=0 ( 1)kk! k (2ℓ 1)!! [2(k ℓ) 1]!! b ℓ+1/2 = − − − − bk+1 (2ℓ)!! [2(k ℓ)]!! a Xℓ=0 − as z 0, where → n 1 − x(x 1) (x n + 1), n 1 x n = (x k)= − ··· − ≥ h i − (1, n =0 kY=0 is called the falling factorial of x R. Consequently, since ∈ 1 ∞ G( x)= = ( 1)kD(k)xk =1 3x + 13x2 63x3 + ... (3.1) − √1+6x + x2 − − − kX=0 means 1 (k) ( 1)kk!D(k) = lim − x 0 √1+6x + x2 → PROPERTIES OF CENTRAL DELANNOY NUMBERS 11

1 1 (k) = lim , x 0 x +3 2√2 x +3+2√2 → − we can procure that p p ( 1)k 1 1 (k) D(k)= − lim k! x 0 x +3 2√2 x +3+2√2 → − k 1 p (2ℓ 1)!! p[2(k ℓ) 1]!! 3+2√2 ℓ+1/2 = k+1 − − − 3+2√2 (2ℓ)!! [2(k ℓ)]!! 3 2√2 Xℓ=0 − − and, by symmetry between a = 3 2√2 and b = 3+2√2 with respect to the deﬁnition of the generating function−G,

k 1 (2ℓ 1)!! [2(k ℓ) 1]!! 3 2√2 ℓ+1/2 D(k)= k+1 − − − − . 3 2√2 (2ℓ)!! [2(k ℓ)]!! 3+2√2 − Xℓ=0 − By virtue of the relation 3+2 √2 = 1 , further simplifying leads to the formulas 3 2√2 in (1.2). − By virtue of the Fa´adi Bruno formula (2.1), the identity (2.2), and the explicit formula (2.3), we have

k 1 (k) 1 1 = Bk,ℓ( 6+2x, 2, 0,..., 0) 2 ℓ+1/2 √1 6x + x −2 ℓ 1 6x + x2 − − Xℓ=0 − k ( 1)ℓ(2ℓ 1)!! = − − ℓ+1/2 Bk,ℓ(x 3, 1, 0,..., 0) 1 6x + x2 − Xℓ=0 − k ( 1)ℓ(2ℓ 1)!!B ( 3, 1, 0,..., 0), x 0 → − − k,ℓ − → Xℓ=0 k ℓ (k ℓ)! k ℓ 2ℓ k = ( 1) (2ℓ 1)!! k− ℓ ( 3) − − − 2 − ℓ k ℓ − Xℓ=0 − k ( 1)kk! (2ℓ 1)!! ℓ = − ( 1)ℓ18ℓ − . 6k − ℓ! k ℓ Xℓ=0 − Accordingly, from (1.1), it follows that

k 1 1 (k) ( 1)k (2ℓ 1)!! ℓ D(k)= lim = − ( 1)ℓ18ℓ − . x 0 √ 2 k k! → 1 6x + x 6 − ℓ! k ℓ − Xℓ=0 − The explicit formula (1.3) is thus proved. For x2 6x < 1, by the binomial theorem, it follows that | − | ∞ 1 1 k 1 6x + x2 = x2 6x − 2 k k! − p Xk=0 k ∞ 1 1 k k ℓ k ℓ k+ℓ = ( 1) − 6 − x 2 k k! ℓ − Xk=0 Xℓ=0 12 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

2k ∞ 1 1 k 2k m 2k m m = ( 1) − 6 − x 2 k! m k − k=0 m=k k − X Xm ∞ 1 1 k 2k m 2k m m = ( 1) − 6 − x 2 k! m k − m=0 k= m/2 k X X⌈ ⌉ − k ∞ 1 1 ℓ 2ℓ k 2ℓ k k = ( 1) − 6 − x 2 ℓ! k ℓ − k=0 ℓ= k/2 ℓ X X⌈ ⌉ − k ∞ ( 1)k+1 (2ℓ 3)!! ℓ = − ( 1)ℓ36ℓ − xk. 6k − (2ℓ)! k ℓ k=0 ℓ= k/2 X X⌈ ⌉ − By (1.1), it is clear that

∞ 1= 1 6x + x2 D(k)xk − p kX=0 k ∞ ( 1)k+1 (2ℓ 3)!! ℓ ∞ = − ( 1)ℓ36ℓ − xk D(k)xk. 6k − (2ℓ)!! k ℓ k=0" ℓ= k/2 # k=0 X X⌈ ⌉ − X Hence, regarding the quantities in the above bracket as ak and D(k) as bk in Lemma 2.3, and interchanging the roles of ak and bk in Lemma 2.3, we can procure

a1 1 0 0 0 a a 1 0 ··· 0 2 1 ··· a3 a2 a1 1 0 k ··· D(k) = ( 1) ...... , k N − ...... ∈

ak 1 ak 2 ak 3 ak 4 1 − − − − ··· ak ak 1 ak 2 ak 3 a1 − − − ··· and

D(1) 1 0 0 0 D(2) D(1) 1 0 ··· 0 ··· D(3) D(2) D(1) 1 0 ··· ......

D(k 1) D(k 2) D(k 3) D(k 4) 1 − − − − ··· D(k) D(k 1) D(k 2) D(k 3) D(1) − − − ··· k 1 (2ℓ 3)!! ℓ = ( 1)ℓ 36ℓ − −6k − (2ℓ)!! k ℓ ℓ= k/2 X⌈ ⌉ − for k N, which can be rearranged as the determinantal representation (1.4) and ∈ the equality (1.6). The proof of Theorem 1.1 is complete. Proof of Theorem 1.2. From the generating function (1.1), when x(x 6) < 1, it follows that | − | k 2 ∞ k ∞ k 1 ∞ k k D(ℓ)D(k ℓ) x = D(k)x = 2 = x (6 x) " − # " # 1 6x + x − Xk=0 Xℓ=0 kX=0 − Xk=0 PROPERTIES OF CENTRAL DELANNOY NUMBERS 13

k k ∞ k k ℓ ℓ k ℓ ∞ ℓ k ℓ k k+ℓ = x ( 1) x 6 − = ( 1) 6 − x " ℓ − # − ℓ Xk=0 Xℓ=0 Xk=0 Xℓ=0 2k m ∞ m k 2k m k m ∞ m k 2k m k m = ( 1) − 6 − x = ( 1) − 6 − x − m k − m k k=0 m=k m=0 k= m/2 X X − X X⌈ ⌉ − k ∞ k m 2m k m k = ( 1) − 6 − x , − k m k=0"m= k/2 # X X⌈ ⌉ − where x stands for the ceiling function which gives the smallest integer not less than x.⌈ Therefore,⌉ equating coeﬃcients in the last equation yields

k k ( 1)k m D(ℓ)D(k ℓ)= − ( 1)m62m − 6k − k m ℓ=0 m= k/2 X X⌈ ⌉ − which can be reformulated as (1.7). On the other hand, we have

k ∞ 1 6x + x2 D(ℓ)D(k ℓ) xk =1, (3.2) − " − # Xk=0 Xℓ=0 k k ∞ ∞ D(ℓ)D(k ℓ) xk 6 D(ℓ)D(k ℓ) xk+1 " − # − " − # kX=0 Xℓ=0 Xk=0 Xℓ=0 k ∞ + D(ℓ)D(k ℓ) xk+2 =1, " − # Xk=0 Xℓ=0 k k 1 ∞ ∞ − D(ℓ)D(k ℓ) xk 6 D(ℓ)D(k ℓ 1) xk " − # − " − − # Xk=0 Xℓ=0 Xk=1 Xℓ=0 k 2 ∞ − + D(ℓ)D(k ℓ 2) xk =1, " − − # Xk=2 Xℓ=0 k k 1 k 2 ∞ − − D(ℓ)D(k ℓ) 6 D(ℓ)D(k ℓ 1)+ D(ℓ)D(k ℓ 2) xk =0. " − − − − − − # kX=2 Xℓ=0 Xℓ=0 Xℓ=0 The identity (1.8) is thus proved. From the equality (3.2) and Lemma 2.3, we now see that the identity (1.9) is valid. The proof of Theorem 1.2 is complete. Proof of Theorem 1.3. From (3.1), it follows that

1 (k) ( 1)kk!D(k) = lim . (3.3) − x 0 √1+6x + x2 → In light of Lemma 2.4, we obtain

(k) 1 (k) 1 lim = lim x 0 √1+6x + x2 x 0   → → x +3 2√2 x +3+2√2 −   q 14 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

( 1)kk! 3+2√2 1 1 = − lim k+1 d t π x 0 3 2√2 (t + x) → Z − t 3+2√2 3+2√2 t − − √ q ( 1)kk! 3+2 2 1 1 = − k+1 d t. π 3 2√2 t Z − t 3+2√2 3+2√2 t − − q Therefore, the proof of Theorem 1.3 is complete.

Proof of Theorem 1.4. Let us consider the 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 It is clear that

3+2√2 2k (2k) 1 1 ln t D (x)= x+1 d t> 0 π 3 2√2 t Z − t 3+2√2 3+2√2 t − − q and

3+2√2 2k 1 (2k 1) 1 1 ln − t D − (x)= x+1 d t −π 3 2√2 t Z − t 3+2√2 3+2√2 t − − q for k . Setting α =3+2√2 = 1 in Lemma 2.5 discloses that N 3 2√2 ∈ − 1 < 0, x , 2 (2k 1) ∈ −∞1 − I 3+2√2 , x, k = D − (x) =0, x =  − 2 > 0, x 1 , ∈ − 2 ∞ for all k N. Consequently, it follows thatD(k)(x) > 0 on 1 , and that ∈ − 2 ∞ ( 1)kD(k)(x) > 0 on , 1 for all k 0. In other words, the function D(x) is − −∞ − 2 ≥ absolutely monotonic on 1 , and completely monotonic on , 1 . Hence, 2 2 the sequence D(k), k 0− is∞ increasing and convex. The proof−∞ of Theorem− 1.4 is { ≥ } complete.

Proof of Theorem 1.5. In [16] and [18, pp. 369–370], it was obtained that, if f is an absolutely monotonic function on [0, ), then ∞ n n n 1 [f(x0)] − f xk f(x0 + xk) ! ≥ kY=0 kY=1 and n n 1/p1 1/pk f xk [f(x1)] [f(x1 + pkxk)] , ! ≤ kY=1 kY=2 n 1 where xℓ [0, ) and pℓ > 0 are such that = 1. Taking f(x)= D(x) and ∈ ∞ ℓ=1 pℓ xℓ,pℓ 0 N leads to the inequalities (1.11) and (1.12). The proof of Theorem 1.5 ∈{ }∪ P is complete. PROPERTIES OF CENTRAL DELANNOY NUMBERS 15

Proof of Theorem 1.6. In [17] and [18, p. 367], it was obtained that, if f is completely monotonic on [0, ), then ∞ f (ap+aq )(x) 0 (3.4) n ≥ and

( 1)ap+aq f (ap+aq )(x) 0. (3.5) − n ≥ 1 By Lemma 2.4, the function ha,b(t) = is completely monotonic on √(t+a)(t +b) [0, ). Substituting h for h in (3.4) and (3.5) gives ∞ h(ap+aq )(x) 0 (3.6) a,b n ≥ and (a +a ) ( 1)ap+aq h p q (x) 0. (3.7) − a,b n ≥ In (3.6) and (3.7), taking a =3 2√2 and b =3+2 √2 , letting x 0, and making use of (3.3) procure the determinantal− inequalities (1.13) and (1.14→). In [9] and [18, p. 367, Theorem 2], it was stated that, when a b, if f is a completely monotonic function on [0, ), then ∞ n n ( 1)ap f (ap)(x) ( 1)bp f (bp)(x) . (3.8) − ≥ − p=1 p=1 Y Y Similarly as above, after replacing f by ha,b, taking a =3 2√2 and b =3+2√2 , letting x 0, and utilizing (3.3), the inequality (3.8) becomes− (1.15). The proof → of Theorem 1.6 is complete. Proof of Corollary 1.1. The inequality (1.16) follows from taking

k n k − a = (n + ℓ,...,n + ℓ, ℓ,...,ℓ) and b = (k + ℓ, k + ℓ,...,k + ℓ) in the inequalityz (1.15).}| { z }| { Proof of Corollary 1.2. Letting ℓ 1, n =2, a = ℓ +2, a = ℓ, and b = b = ℓ +1 ≥ 1 2 1 2 in the inequality (1.15) leads to [ℓ!D(ℓ)][(ℓ + 2)!D(ℓ + 2)] [(ℓ + 1)!D(ℓ + 1)]2 ≥ which means that the sequence n!D(n),n 1 is logarithmically convex. In [18, p. 369] and [21, p. 429,{ Remark], it≥ was} stated that, if f(t) is a completely monotonic function such that f (k)(t) =0 for k 0, then the sequence 6 ≥ k 1 (k 1) ln ( 1) − f − (t) , k 1 − ≥ n k 1 (k 1) o is convex, that is, the sequence ( 1) − f − (t), k 1 is logarithmically convex. { − ≥ } Once replacing f by ha,b, setting a =3 2√2 and b =3+2√2 , letting t 0, and using (3.3) produce the logarithmic convexity− of (k 1)!D(k 1), k →1 . The { − − ≥ } proof of Corollary 1.2 is complete. 16 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

Proof of Theorem 1.7. In [39, p. 397, Theorem D], it was recovered that if f(x) is completely monotonic on (0, ) and if n k m, k n k, and m n m, then ∞ ≥ ≥ ≥ − ≥ − n (k) (n k) n (m) (n m) ( 1) f (x)f − (x) ( 1) f (x)f − (x). (3.9) − ≥ − Similarly as above, if replacing f by ha,b, taking a = 3 2√2 and b =3+2√2 , letting x 0, and employing (3.3), then the inequality (3.9)− becomes (1.17). The → proof of Theorem 1.7 is complete.

4. Remarks Finally, we list several remarks. Remark 4.1. Since 2 2 2 3 2√2 = √2 +1 ± = √2 1 ∓ = √2 1 , ± − ± the explicit formulas in (1.2) can be rewritten as either k 2k (2ℓ 1)!! [2(k ℓ) 1]!! 4ℓ D(k)= √2 1 − − − √2 1 , ± (2ℓ)!! [2(k ℓ)]!! ∓ Xℓ=0 − k 2k (2ℓ 1)!! [2(k ℓ) 1]!! 4ℓ D(k)= √2 +1 ± − − − √2 +1 ∓ , (2ℓ)!! [2(k ℓ)]!! ℓ=0 X − or k 2k (2ℓ 1)!! [2(k ℓ) 1]!! 4ℓ D(k)= √2 1 ± − − − √2 1 ∓ − (2ℓ)!! [2(k ℓ)]!! − ℓ=0 − X for k 0. ≥ (2ℓ 1)!! [2(k ℓ) 1]!! Remark 4.2. The factors (2−ℓ)!! and [2(−k ℓ−)]!! in (1.2) and (1.3) are the Wallis − ratios Wℓ and Wk ℓ respectively, where − (2n 1)!! (2n)! 1 Γ n +1/2 Wn = − = 2n 2 = (2n)!! 2 (n!) √π Γ( n + 1) and Γ denotes the gamma function which can be deﬁned by

∞ z 1 t Γ(z)= t − e− d t, (z) > 0 ℜ Z0 or by n!nz Γ(z) = lim n n →∞ k=0(z + k) for z C 0, 1, 2,... . For moreQ information on the Wallis ratio, please refer to∈ the\{ papers− [3,− 4, 5,} 6, 12, 13, 33, 34] and closely-related references therein. For detailed information on the ratio of two gamma functions, please refer to the expository and survey articles [23, 24, 31, 32] and plenty of references cited therein. Remark 4.3. The formulas (1.3), (1.5), and (1.7) can be reformulated in terms of the the ceiling function x , which gives the smallest integer not less than x, as ⌈ ⌉ k ( 1)k (2ℓ 1)!! ℓ D(k)= − ( 1)ℓ62ℓ − , 6k − (2ℓ)!! k ℓ ℓ= k/2 X⌈ ⌉ − PROPERTIES OF CENTRAL DELANNOY NUMBERS 17

( 1)k+1 k (2ℓ 3)!! ℓ a = − ( 1)ℓ62ℓ − , k 6k − (2ℓ)!! k ℓ ℓ= k/2 X⌈ ⌉ − and k k ( 1)k ℓ D(ℓ)D(k ℓ)= − ( 1)ℓ62ℓ , k 0 − 6k − k ℓ ≥ ℓ=0 ℓ= k/2 X X⌈ ⌉ − respectively. Remark 4.4. The identities (1.7), (1.8), and (1.9) are equivalent to each other. For example, the identity (1.8) can be veriﬁed by making use of the identity (1.7) or by expanding the determinant in (1.9) according to the last column or row. Consequently, we can obtain the following by-product 61 00 0 0 0 −1 6 1 0 ··· 0 0 0 − ··· 0 1 6 1 0 0 0 k − ··· 1 ℓ ...... ℓ 2ℓ ...... = k ( 1) 6 , 6 − k ℓ 0 0 00 6 1 0 ℓ=0 − ··· − X 0 0 00 1 6 1 ··· − 0 0 00 0 1 6 ··· − k k × where p = 0 for q>p 0. Accordingly, we guess that q ≥ c 1 0 0 0 0 0 1 c 1 0 ··· 0 0 0 ··· 0 1 c 1 0 0 0 k ··· ( 1)k ℓ ...... ℓ 2ℓ ...... = −k ( 1) c c − k ℓ 0 00 0 c 1 0 ℓ=0 − ··· X 0 00 0 1 c 1 ··· 0 00 0 0 1 c k k ··· × k ( 1)m k m = ck − − c2m m m=0 X for all c C and k N. This identity can be straightforwardly veriﬁed by induction ∈ ∈ on k N. For more information on this identity, please refer to the paper [26] and closely-related∈ references therein. Remark 4.5. This paper is a revised version of the preprint [30].

References [1] R. D. Atanassov and U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), no. 2, 21–23. [2] C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439; Available online at http://dx.doi.org/10.1007/ s00009-004-0022-6. [3] J. Cao, D.-W. Niu, and F. Qi, A Wallis type inequality and a double inequality for probability integral, Aust. J. Math. Anal. Appl. 5 (2007), no. 1, Art. 3; Available online at http://ajmaa. org/cgi-bin/paper.pl?string=v4n1/V4I1P3.tex. [4] C.-P. Chen and F. Qi, Best upper and lower bounds in Wallis’ inequality, J. Indones. Math. Soc. (MIHMI) 11 (2005), no. 2, 137–141. 18 F.QI,V. CERˇ NANOVˇ A,´ X.-T. SHI, AND B.-N. GUO

[5] C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma function and proof of Wallis’ inequality, Tamkang J. Math. 36 (2005), no. 4, 303–307; Available online at http://dx.doi.org/10.5556/j.tkjm.36.2005.101. [6] C.-P. Chen and F. Qi, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc. 133 (2005), no. 2, 397–401; Available online at http://dx.doi.org/10.1090/ S0002-9939-04-07499-4. [7] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. [8] M. Dziemiańczuk, Generalizing Delannoy numbers via counting weighted lattice paths, Inte- gers 13 (2013), Paper No. A54, 33 pages. [9] A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), 251–258; Available online at http://dx.doi.org/10.1016/0022-247X(82)90057-9. [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015; Available online at http://dx.doi.org/10.1016/B978-0-12-384933-5.00013-8. [11] B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21–30. [12] B.-N. Guo and F. Qi, On the Wallis formula, Internat. J. Anal. Appl. 8 (2015), no. 1, 30–38. [13] S. Guo, J.-G. Xu, and F. Qi, Some exact constants for the approximation of the quantity in the Wallis’ formula, J. Inequal. Appl. 2013, 2013:67, 7 pages; Available online at http: //dx.doi.org/10.1186/1029-242X-2013-67. [14] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934. [15] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and its Applications, 2nd Ed., Springer Verlag, New York-Dordrecht-Heidelberg-London, 2011; Avail- able online at http://dx.doi.org/10.1007/978-0-387-68276-1. [16] D. S. Mitrinovićand J. E. Peˇcarić, On some inequalities for monotone functions, Boll. Un. Mat. Ital. B (7) 5 (1991), no. 2, 407–416. [17] D. S. Mitrinovićand J. E. Peˇcarić, On two-place completely monotonic functions, Anzeiger Oster.¨ Akad. Wiss. Math.-Natturwiss. Kl. 126 (1989), 85–88. [18] D. S. Mitrinović, J. E. Peˇcarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; Available online at http: //dx.doi.org/10.1007/978-94-017-1043-5. [19] C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics 23, Springer-Verlag, New York, 2005. [20] R. Noble, Asymptotics of the weighted Delannoy numbers, Int. J. Number Theory 8 (2012), no. 1, 175–188; Availvle online at http://dx.doi.org/10.1142/S1793042112500108. [21] J. E. Peˇcarić, Remarks on some inequalities of A. M. Fink, J. Math. Anal. Appl. 104 (1984), no. 2, 428–431; Available online at http://dx.doi.org/10.1016/0022-247X(84)90006-4. [22] J. E. Peˇcarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Sta- tistical Applications, Mathematics in Science and Engineering 187, Academic Press, 1992. [23] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058. [24] F. Qi, Bounds for the ratio of two gamma functions: from Gautschi’s and Kershaw’s inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (2014), no. 5, 152–164; Available online at http://dx.doi.org/10.12691/tjant-2-5-1. [25] F. Qi and R. J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory 159 (2016), 89–100; Available online at http://dx.doi.org/10.1016/j.jnt.2015.07.021. [26] F. Qi, V. Cerˇnanová,ˇ and Y. S. Semenov, On tridiagonal determinants and the Cauchy product of central Delannoy numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.1.3772.6967. [27] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), 603–607; Available online at http://dx.doi.org/10.1016/j.jmaa. 2004.04.026. PROPERTIES OF CENTRAL DELANNOY NUMBERS 19

[28] F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63–72; Available online at http://rgmia.org/v7n1.php. [29] F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Article 140, 14 pages; Available online at http://dx.doi.org/10.1007/s00009-017-0939-1. [30] F. Qi, B.-N. Guo, V. Cerˇnanová,ˇ and X.-T. Shi, Explicit expressions, Cauchy products, integral representations, convexity, and inequalities of central Delannoy numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.1.4889.6886. [31] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s as- ymptotic relation to Elezović-Giordano-Peˇcarić’s theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-542. [32] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132–158; Available online at http://dx.doi.org/10.15352/bjma/1342210165. [33] F. Qi and C. Mortici, Some best approximation formulas and inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368; Available online at http://dx.doi.org/ 10.1016/j.amc.2014.12.039. [34] F. Qi, C. Mortici, and B.-N. Guo, Some properties of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body, Comput. Appl. Math. 37 (2018), in press; Available online at http://dx.doi.org/10.1007/s40314-017-0448-7. [35] F. Qi, X.-T. Shi, and B.-N. Guo, Two explicit formulas of the Schröder numbers, Integers 16 (2016), Paper No. A23, 15 pages. [36] F. Qi, X.-T. Shi, F.-F. Liu, and D. V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, J. Appl. Anal. Comput. 7 (2017), no. 3, 857–871; Available online at http://dx.doi.org/10.11948/2017054. [37] F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597–607; Available online at http://dx.doi.org/ 10.1016/j.amc.2015.02.027. [38] R. L. Schilling, R. Song, and Z. Vondraˇcek, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; Available online at http://dx.doi.org/10.1515/9783110269338. [39] H. van Haeringen, Completely monotonic and related functions, J. Math. Anal. Appl. 204 (1996), no. 2, 389–408; Available online at http://dx.doi.org/10.1006/jmaa.1996.0443. [40] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015:219, 8 pages; Available online at http://dx.doi.org/10.1186/s13660-015-0738-9. [41] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton Univer- sity Press, Princeton, N. J., 1941.

(Qi) Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; Department of Mathematics, College of Science, Tianjin Poly- technic University, Tianjin City, 300387, China E-mail address: [email protected], [email protected], [email protected] URL: https://qifeng618.wordpress.com

(Cerˇnanov´a)ˇ Institute of Computer Science and Mathematics, Slovak University of Technology, Bratislava, Slovak Republic E-mail address: [email protected], [email protected]

(Shi) Department of Mathematics, College of Science, Tianjin Polytechnic Univer- sity, Tianjin City, 300387, China E-mail address: [email protected], [email protected]

(Guo) School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address: [email protected], [email protected] URL: http://www.researchgate.net/profile/Bai-Ni_Guo/