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1 Indian J. Pure Appl. Math., 47(4: 77-73, December 6 c Indian National Science Academy DOI:.7/s SOME PROPERTIES OF THE SCHRÖDER NUMBERS Feng Qi,, Xiao-Ting Shi Bai-Ni Guo Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454, China College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 843, China Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 3387 China School of Mathematics Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454, China s: qifeng68@gmail.com; qifeng68@hotmail.com; qifeng68@qq.com; xiao-ting.shi@hotmail.com; xiao-ting.shi@qq.com; bai.ni.guo@gmail.com (Received 3 March 6; after final revision 4 April 6; accepted 3 July 6 In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, determinantal inequalities, of the large Schröder numbers find three relations between the Schröder numbers central Delannoy numbers. Moreover, the authors sketch generalizing the Schröder numbers central Delannoy numbers their generating functions. Key words : Large Schröder number; little Schröder numbers; convexity; complete monotonicity; product inequality; determinantal inequality; relation; Delannoy number; generating function; generalization.. INTRODUCTION In mathematics, there are two kinds of Schröder numbers, the large Schröder numbers S n the little Schröder numbers s n. They are named after the German mathematician Ernst Schröder.

2 78 FENG QI, XIAO-TING SHI AND BAI-NI GUO A large Schröder number S n describes the number of paths from the southwest corner (, of an n n grid to the northeast corner (n, n, using only single steps north, northeast, or east, that do not rise above the southwest-northeast diagonal. The first eleven large Schröder numbers S n for n are,, 6,, 9, 394, 86, 8558, 4586, 698, In [3, Theorem 8.5.7], it was proved that the large Schröder numbers S n have the generating function G(t = t t 6t + t = S n t n. (. n= The little Schröder numbers s n form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. The first eleven little Schröder numbers s n for n are,, 3,, 45, 97, 93, 479, 793, 349, In [3, Theorem 8.5.6], it was proved that the little Schröder numbers s n have the generating function g(t = + t t 6t + 4 = s n t n. (. n= Comparing (. with (., we see easily that t 6t + = + t 4 s n t n = t S n t n+, n= n= that is, s n t n = s n+ t n = S n t n. n= n= n= Accordingly, we acquire S n = s n+, n N. (.3 See also [3, Corollary 8.5.8]. Due to the identity (.3, it is sufficient to analytically study just the large Schröder numbers S n. Accordingly, in what follows, we will just state some results for the large Schröder numbers S n rather than those for the little Schröder numbers s n.

3 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 79 Recently, among other things, two explicit formulas ( n+ n+ 3 (l 3!! [(n l + 3]!! ( l S n = 3 ± (l!! [(n l + ]!! S n = ( n 6 n l= n+ k= (n+/ 6 k k! ( k k n k +, n N for the large Schröder numbers S n were established in [, 5], where t sts for the ceiling function which gives the smallest integer not less than t t n is the falling factorial defined by n t(t (t n +, n, t n = (t k =, n =. k= Recall from [, Chapter XIII], [9, Chapter ] [3, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies ( k f (k (t < on I for all k. It is known [3, p. 6, Theorem b] that a function f is completely monotonic on (, if only if it is a Laplace transform f(t = e ts dµ(s of a positive measure µ defined on [, such that the integral converges on (,. In [, 8], it was defined implicitly explicitly that an infinitely differentiable positive function f is said to be logarithmically completely monotonic on an interval I if the inequality ( k [ln f(x] (k holds on I for all k N. In [, Theorem.], [6, Theorem 4], [7, Theorem ] [8, Theorem 4], it was found verified once again that a logarithmically completely monotonic function must be completely monotonic. In [3], several integral representations, logarithmically complete monotonicity, other properties for the generating function G(t = G( t for the large Schröder numbers S n were obtained. Some of these conclusions are recited in Lemma. below some can be reformulated as where n S n = 3+ π 3 q(s = S n = π (n +! 3+ 3 q(ss n+ ds (.4 (u 3 + ( 3 + u u n+ du, (.5 (u 3 + ( 3 + u e su du (.6 is a completely monotonic function on (,. By the way, interchanging the order of integrals in (.4 utilizing s n+ e us ds = result in the integral representation (.5. u n+

4 7 FENG QI, XIAO-TING SHI AND BAI-NI GUO Recall also from [, 9, 3] that a function f is said to be absolutely monotonic on an interval I if it has derivatives of all orders f (k (t for t I k N, where N denotes the set of all positive integers. It is easy to see that a function f(x is completely monotonic on I if only if the function f( x is absolutely monotonic on I. In [, pp ], it was stated that, if f C (a, b f (k (x for k x [a, b], we say that f is an absolutely convex function. A sequence µ n for n is said to be absolutely monotonic if its elements are non-negative its successive differences k µ n = k ( k ( m µ n+k m m m= for n, k. If elements of a sequence µ n for n are non-negative k µ n for n, k, then we say that the sequence µ n for n is absolutely convex. The first aim of this paper is to present some properties, including the convexity, complete monotonicity, product inequalities, determinantal inequalities, of the large Schröder numbers S k. Theorem. Let the Schröder function S x be defined by replacing n by x in (.4 or (.5. Then the function (x +!S x is absolutely convex with respect to x [,. Consequently, the sequence {(n +!S n } n is absolutely convex, particularly, convex. The Schröder function S x is absolutely monotonic on (, completely monotonic on (,. Consequently, the sequence {Sn } n is absolutely monotonic, particularly, convex. Theorem. Let x k {} N p k N such that n l= p l = for n N. Then Sx n S n k= x n S k x +x k (.7 k= S n k= x k S/p x n k= S /p k x +p k x k. (.8 Theorem.3 Let m be a positive integer let a ij m denote a determinant of order m with elements a ij.. If a i for i m are non-negative integers, then ( a i+a j (a i + a j!s ai +a j m (.9 (ai + a j!s ai +a j m. (.

5 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 7. If a = (a, a,..., a n b = (b, b,..., b n are non-increasing n-tuples of non-negative integers such that k i= a i k i= b i for k n n i= a i = n i= b i, then n S bi S i= ai n i= a i! b i!. (. convex. Coorollary. The sequence n!s n for n is logarithmically convex, consequently, is Corollary. If l n k, then [ ] (n + k [ ] l!sn+l (k + n l!sk+l. l!s l l!s l Theorem.4 If l, n k m, k n, m n, then S k+l S n k+l (m + l!(n m + l! S m+l S n m+l (k + l!(n k + l!. (. In combinatorics, central Delannoy numbers D(n are the number of king walks from the (, corner of an n n square to the upper right corner (n, n. Central Delannoy numbers D(n have the generating function 6t + t = D(kt k = + 3t + 3t + 63t 3 +. (.3 k= For more information on central Delannoy numbers D(k, please refer to the papers [5,, ], the websites [3, 33], closely-related reference therein. The second aim of this paper is to find three relations between the the large Schröder numbers S k central Delannoy numbers D(k. Theorem.5 For k N, the large Schröder numbers S k central Delannoy numbers D(k

6 7 FENG QI, XIAO-TING SHI AND BAI-NI GUO have relations D( D( D( D(3 D( D( S k = ( k , (.4 D(k D(k D(k 3 D( D(k D(k D(k D( D( D(k + D(k D(k D(3 D( D( 3 3 S 3 S S D(k = k , (.5 3 S k 3 S k 4 S k 5 S k 6 3 S k S k 3 S k 4 S k 5 S 3 S k S k S k 3 S k 4 S S k D(k + = 3D(k + S l D(k l. (.6 l=. LEMMAS In order to prove our main results, we need the following notions lemma. In [7, Definition.], it was defined that an infinitely differentiable nonnegative function f : I [, is called a Bernstein function on an interval I if f (t is completely monotonic on I. Theorem 3. in [9] states that a function f : (, [, is a Bernstein function if only if it admits the Lévy-Khintchine representation f(x = a + bx + ( e xt dµ(t, (. where a, b µ is a Lévy measure on (,, with min{, t}dµ(t <. The Lévy triplet (a, b, µ determines f uniquely vice versa. In [4, pp. 6-6, Theorem 3] [9, Proposition 5.5], it was proved that the reciprocal of a Bernstein function is logarithmically completely monotonic.

7 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 73 If dµ(t = m(tdt m(t is a completely monotonic function on (,, then the function f in (. is said in [9, Definition 6.] to be a complete Bernstein function. In [9, Definition.], it was defined that a Stieltjes transform is a function f : (, [, which can be written in the form f(t = a t + b + u + t dµ(u, where a, b are nonnegative constants µ is a nonnegative measure on (, such that +s dµ(s <. In [, Theorem.], it was proved that a positive Stieltjes transform must be a logarithmically completely monotonic function on (,, but not conversely. Lemma. [3] The generating function G(t = G( t is logarithmically completely monotonic on ( 3, a Stieltjes transform with G(t = 3+ π 3 (u 3 + ( 3 + u u(u + t du. (. The negative of G(t = g( t is a complete Bernstein function on ( 3, G(t = t π 4 q(s( e ts ds, where q(s is defined by (.6. Lemma. For α >, β R, k N, the definite integral I(α, β; k = α /α ( u <, β > (a u lnk u 3 a u β du =, β = 3 >, β < 3 is valid.

8 74 FENG QI, XIAO-TING SHI AND BAI-NI GUO PROOF : A straightforward computation gives ( I(α, β; k = = = + + /α /α /α /α α ( u ( ( u a <, β > 3 ; =, β = 3 ; >, β < 3. ( u (a u lnk u a u β du (a u lnk u a u β du s α ( α s (a u ( k lnk s s β ( u β u 3 β ln k udu ( s ds The proof of Lemma. is complete. Lemma.3 ([6, Lemma.4] Let f(t = + k= a kt k g(t = + k= b kt k be formal power series such that f(tg(t =. Then a a a b k = ( k a 3 a a a k a k a k 3 a k 4 a k a k a k a k 3 a 3. PROOFS OF THEOREMS AND COROLLARIES PROOF OF THEOREM. : Let f (x = q(ss x+ ds, x. Then f (k (x = q(s(ln s k s x+ ds, x. This means that f (k (x for k the function f (x is absolutely convex on [,. Consequently, by the integral representation in (.4, we obtain that the function (x +!S x for x

9 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 75 is absolutely convex on [,. Consequently, the sequence {(n +!S n } n is absolutely convex then convex. It is obvious that S x for x is absolutely convex on [,. Consequently, the sequences S n s n+ for n are absolutely convex. Then Let f (x = 3+ 3 (u 3 + ( 3 + u u x+ du, x R. 3+ f (k (x = ( k 3 (u ( ln k u u du, k. ux+ It is clear that f (k (x > on (, for k. By Lemma. applied to α = 3 +, since 3 + = 3, it follows that for k N. >, x > (x =, x = <, x < f (k Consequently, the Schröder function S x is absolutely monotonic on (, completely monotonic on (,. Thus, the sequence {Sn } n is absolutely convex then convex. PROOF OF THEOREM. : In [8] [, pp ], it was obtained that, if f is an absolutely monotonic function on [,, then ( n [f(x ] n f x k k= ( n f x k [f(x ] /p k= n f(x + x k k= n [f(x + p k x k ] /p k, where x l [, p l > such that n l= p l =. Applying f(x = S x to these inequalities taking x l, p l {} N lead to the inequalities (.7 (.8. The proof of Theorem. is complete. k=

10 76 FENG QI, XIAO-TING SHI AND BAI-NI GUO PROOF OF THEOREM.3 : In [9] [, p. 367], it was obtained that, if f is completely monotonic on [,, then f (a i +a j (x m (3. ( a i+a j f (a i+a j (x m. (3. By Lemma., applying f(x = G(x in (. to (3. (3. taking the limit x + give lim x + G (a i +a j (x m = ( a i +a j (a i + a j!s ai +a j m lim ( a i+a j G (a i+a j (x x + m = ( a i+a j ( a i+a j (a i + a j!s m ai +a j. The determinant inequalities (.9 (. follow. In [, p. 367, Theorem ], it was stated that, if f is a completely monotonic function on [,, then n [ ( a i f (ai (x ] n [ ( b i f (bi (x ]. (3.3 i= i= By Lemma., replacing f(x by G(x in (3.3 taking the limit x + give lim x + i= n [ ( a i G (ai (x ] = n (a i!s ai lim i= x + i= n [ ] ( b i G (bi (x = n (b i!s bi. i= The product inequality (. follows. The proof of Theorem.3 is complete. PROOF OF COROLLARY. : In [, p. 369] [, p. 49, Remark], it was stated that, if f(t is a completely monotonic function such that f (k (t for k, then the sequence ln [ ( k f (k (t ], k is convex. By Lemma., applying this result to the function G(x figures out that the sequence ln [ ( k G (k (x ] ln[(k!s k ], x + for k is convex. Hence, the sequence n!s n for n is logarithmically convex. Alternatively, letting l, n =, a = l +, a = l, b = b = l +

11 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 77 in the inequality (. leads to (l!s l [(l +!S l+ ] [(l +!] S l+ which means that the sequence k!s k for k N is logarithmically convex. The proof of Corollary. is complete. PROOF OF COROLLARY. : This follows from taking k n k {}}{{}}{ a = ( n + l,..., n + l, l,..., l b = (k + l, k + l,..., k + l in the inequality (.. The proof of Corollary. is complete. PROOF OF THEOREM.4 : In [3, p. 397, Theorem D], it was recovered that if f(x is completely monotonic on (, if n k m, k n k, m n m, then ( n f (k (xf (n k (x ( n f (m (xf (n m (x. By Lemma., replacing f(x by the function ( l G (l (x in the above inequality leads to ( n G (k+l (xg (n k+l (x ( n G (m+l (xg (n m+l (x. Further taking x + finds ( n ( k+l (k + l!s k+l ( n k+l (n k + l!s n k+l ( n ( m+l (m + l!s m+l ( n m+l (n m + l!s n m+l, which can be arranged as (.. The proof of Theorem.4 is complete. PROOF OF THEOREM.5 : From (., we see easily that t 6t + = t S k t k+ = 3t S k t k. k= k= Multiplying this with (.3 yields ( + D(kt ( k 3t S k t k =. (3.4 k= k=

12 78 FENG QI, XIAO-TING SHI AND BAI-NI GUO By virtue of Lemma.3, it follows immediately that D( D( D( D(3 D( D( S k = ( k D(k D(k D(k 3 D(k D(k D(k D(k + D(k D(k D( (k+ (k+ 3 S 3 D(k = ( k S S S k S k 3 S k 4 S k 5 S k S k S k 3 S k 4 3 which can be rearranged as the forms (.4 (.5 respectively. On the other h, we can also exp the left-h side of the equation (3.4 into a series [ k D(k 3D(k D(k ls l ]x k =. k= l= k k The identity (.6 is thus proved. The proof of Theorem.5 is complete. 4. A SKETCH OF GENERALIZING THE SCHRÖDER NUMBERS From [7, Theorem.], [7, Theorems ], [8, Theorem.], we can derive the integral representation [ (z + a(z + b = ab + z + b (b t(t a π a t where b > a > z C \ (, a]. ] t + z dt, (4. as Motivated by the integral representation (.5, we can generalize the large Schröder numbers S n S a,b (n = b (u a(b u π a u n+ du, b > a >, n.

13 SOME PROPERTIES OF THE SCHRÖDER NUMBERS 79 Their generating function is ab t (t a(t b H a,b; ;; (t = t = S a,b (nt n, b > a >. n= For b > a >, β, κ R with κ, q =,, we can consider the more general functions ab + βt (t a(t b H a,b;β;κ;q (t = κt q the corresponding sequences Q a,b;β;κ;q (k generated by H a,b;β;κ;q (t = Q a,b;β;κ;q (kt k, (t a(t b <. k= The sequences Q a,b;β;κ;q (k for k N have the following properties Q a,b;β;κ; ( =, Q a,b;β;κ; ( = ( a + b κ ab + β, Q a,b;β;κ; (k = Q a,b;β;κ; (k, κ Q a,b;β ;κ ;( β = κ Q a,b;β ;κ ;( β, κ Q a,b;β ;κ ;(k = κ Q a,b;β ;κ ;(k +. By virtue of (4., the sequence Q a,b;β;κ; (k has the integral representation Q a,b;β;κ; ( = [ + β + b ] (b s(s a κ π a s ds Q a,b;β;κ; (m = b (b s(s a κπ a s m+ ds, m. In [, Theorem.3], it was established that D(k = π 3+ 3 for k. Then we can consider the sequence D a,b (k = π b a (t 3 + ( 3 + t t k+ dt dt, k, b > a > (t a(b t tk+, by [, Lemma.4], find that D a,b (k can be generated by (t + a(t + b = D a,b (kt k. k=

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