Catalan-like numbers and Stieltjes moment sequences
aa r X i v : . [ m a t h . C O ] J un Catalan-like numbers and Stieltjes moment sequences
Huyile Liang, Lili Mu, Yi Wang ∗ School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China
Abstract
We provide sufficient conditions under which the Catalan-like numbers are Stielt-jes moment sequences. As applications, we show that many well-known countingcoefficients, including the Bell numbers, the Catalan numbers, the central binomialcoefficients, the central Delannoy numbers, the factorial numbers, the large and lit-tle Schr¨oder numbers, are Stieltjes moment sequences in a unified approach.
MSC:
Keywords:
Stieltjes moment sequence; Catalan-like number; Recursive matrix; Ri-ordan array; Hankel matrix; Totally positive matrix
A sequence ( m n ) n ≥ of numbers is said to be a Stieltjes moment sequence if it has theform m n = Z + ∞ x n dµ ( x ) , (1.1)where µ is a non-negative measure on [0 , + ∞ ). It is well known that ( m n ) n ≥ is a Stieltjesmoment sequence if and only if det[ m i + j ] ≤ i,j ≤ n ≥ m i + j +1 ] ≤ i,j ≤ n ≥ n ≥ A = [ a n,k ] n,k ≥ be a finite or an infinite matrix. It is totally positive ( TP for short),if its minors of all orders are nonnegative. Let α = ( a n ) n ≥ be an infinite sequence ofnonnegative numbers. Define the Hankel matrix H ( α ) of the sequence α by H ( α ) = [ a i + j ] i,j ≥ = a a a a · · · a a a a · · · a a a a · · · a a a a · · · ... ... ... ... . . . . ∗ Email address: [email protected] (Y. Wang) α is a Stieltjes moment sequence if and only if H ( α ) is totally positive (see [7,Theorem 4.4] for instance).Many counting coefficients are Stieltjes moment sequences. For example, the factorialnumbers n ! form a Stieltjes moment sequence since n ! = Z ∞ x n e − x dx. (1.2)The Bell numbers B n form a Stieltjes moment sequence since B n can be interpreted asthe n th moment of a Poisson distribution with expected value 1 by Dobinski’s formula B n = 1 e X k ≥ k n k ! . The Catalan numbers C n = (cid:0) nn (cid:1) / ( n + 1) form a Stieltjes moment sequence sincedet[ C i + j ] ≤ i,j ≤ n = det[ C i + j +1 ] ≤ i,j ≤ n = 1 , n = 0 , , , . . . (see Aigner [1] for instance). Bennett [3] showed that the central Delannoy numbers D n and the little Schr¨oder numbers S n form Stieltjes moment sequences by means of theirgenerating functions (see Remark 2.11 and Example 2.12). All these counting coefficientsare the so-called Catalan-like numbers. In this note we provide sufficient conditions suchthat the Catalan-like numbers are Stieltjes moment sequences by the total positivity ofthe associated Hankel matrices. As applications, we show that the Bell numbers, theCatalan numbers, the central binomial coefficients, the central Delannoy numbers, thefactorial numbers, the large and little Schr¨oder numbers are Stieltjes moment sequencesin a unified approach. Let σ = ( s k ) k ≥ and τ = ( t k ) k ≥ be two sequences of nonnegative numbers and definean infinite lower triangular matrix R := R σ,τ = [ r n,k ] n,k ≥ by the recurrence r , = 1 , r n +1 ,k = r n,k − + s k r n,k + t k +1 r n,k +1 , (2.1)where r n,k = 0 unless n ≥ k ≥
0. Following Aigner [2], we say that R σ,τ is the recursivematrix and r n, are the Catalan-like numbers corresponding to ( σ, τ ).The Catalan-like numbers unify many well-known counting coefficients, such as(1) the Catalan numbers C n if σ = (1 , , , . . . ) and τ = (1 , , , . . . );(2) the central binomial coefficients (cid:0) nn (cid:1) if σ = (2 , , , . . . ) and τ = (2 , , , . . . );(3) the central Delannoy numbers D n if σ = (3 , , , . . . ) and τ = (4 , , , . . . );(4) the large Schr¨oder numbers r n if σ = (2 , , , . . . ) and τ = (2 , , , . . . );25) the little Schr¨oder numbers S n if σ = (1 , , , . . . ) and τ = (2 , , , . . . );(6) the (restricted) hexagonal numbers h n if σ = (3 , , , . . . ) and τ = (1 , , , . . . );(7) the Bell numbers B n if σ = τ = (1 , , , , . . . );(8) the factorial numbers n ! if σ = (1 , , , , . . . ) and τ = (1 , , , , . . . ).Rewrite the recursive relation (2.1) as r , r , r , r , r , r , r , r , r , · · · . . . = r , r , r , r , r , r , · · · . . . s t s t s . . .. . . . . . , or briefly, R = RJ where R is obtained from R by deleting the 0th row and J is the tridiagonal matrix J := J σ,τ = s t s t s t s . . .. . . . . . . Clearly, the recursive relation (2.1) is decided completely by the tridiagonal matrix J .Call J the coefficient matrix of the recursive relation (2.1). Theorem 2.1.
If the coefficient matrix is totally positive, then the corresponding Catalan-like numbers form a Stieltjes moment sequence.Proof.
Let H = [ r n + k, ] n,k ≥ be the Hankel matrix of the Catalan-like numbers ( r n, ) n ≥ .We need to show that H is totally positive. We do this by two steps. We first show thetotal positivity of the coefficient matrix J implies that of the recursive matrix R . Let R n = [ r i,j ] ≤ i,j ≤ n be the n th leading principal submatrix of R . Clearly, to show that R is TP, it suffices to show that R n are TP for n ≥
0. We do this by induction on n .Obviously, R is TP. Assume that R n is TP. Then by (2.1), we have R n +1 = (cid:20) OO R n (cid:21) L n , where L n = s t s t s . . .. . . . . . 1 t n − s n − t n s n . R n implies that of (cid:20) OO R n (cid:21) . On the other hand, J isTP, so is its submatrix J n , as well as the matrix L n . Thus the product matrix R n +1 isTP, and R is therefore TP by induction.Secondly we show the total positivity of R implies that of H . Let T = 1 , T k = t · · · t k and T = T T T . . . . Then it is not difficult to verify that H = RT R t (see [2, (2.5)]). Now R is TP, and so isits transpose R t . Clearly, T is TP. Thus the product H is also TP. This completes theproof.We now turn to the total positivity of tridiagonal matrices. Such a matrix is oftencalled a Jacobi matrix . There are many well-known results about the total positivityof tridiagonal matrices. For example, a finite nonnegative tridiagonal matrix is totallypositive if and only if all its principal minors containing consecutive rows and columns arenonnegative [7, Theorem 4.3]; and in particular, an irreducible nonnegative tridiagonalmatrix is totally positive if and only if all its leading principal minors are positive [6,Example 2.2, p. 149]. Clearly, J σ,τ is irreducible. So, to show the total positivity of J σ,τ ,it suffices to show that all its leading principal minors are positive. Example 2.2.
For the Catalan-like numbers n !, we have s k = 2 k + 1 and t k = k . It isnot difficult to show that the n th leading principal minor of J σ,τ is equal to n !. Thus J σ,τ is totally positive, and so the factorial numbers n ! form a Stieltjes moment sequence, awell-known result. Lemma 2.3. If s ≥ and s k ≥ t k + 1 for k ≥ , then the tridiagonal matrix J σ,τ istotally positive.Proof. Let D n be the n th leading principal minor of J σ,τ . It suffices to show that all D n are nonnegative. We do this by showing the following stronger result:1 ≤ D ≤ D ≤ D n − ≤ D n − ≤ D n ≤ · · · . Obviously, D = s ≥ D = s s − t ≥ s = D since s ≥ t + 1. Assume nowthat D n − ≥ D n − ≥ n ≥
2. Then by expanding the determinant D n along the lastrow or column, we obtain D n = s n D n − − t n D n − ≥ ( s n − t n ) D n − ≥ D n − ≥ s n ≥ t n + 1 and the induction hypothesis, as required. The proof is complete.Combining Theorem 2.1 and Lemma 2.3, we obtain the following. Corollary 2.4. If s ≥ and s k ≥ t k + 1 for k ≥ , then the Catalan-like numberscorresponding to ( σ, τ ) form a Stieltjes moment sequence. xample 2.5. The Bell numbers B n , the Catalan numbers C n , the central binomialcoefficients (cid:0) nn (cid:1) , the (restricted) hexagonal numbers H n , and the large Schr¨oder numbers r n form a Stieltjes moment sequence respectively.In what follows we apply Theorem 2.1 to the recursive matrix R ( p, q ; s, t ) = [ r n,k ] n,k ≥ defined by (cid:26) r , = 1 , r n +1 , = pr n, + qr n, ,r n +1 ,k +1 = r n,k + sr n,k +1 + tr n,k +2 . (2.2)The coefficient matrix of (2.2) is J ( p, q ; s, t ) = p q s t s t s . . .. . . . . . . (2.3)The following result is a special case of [4, Proposition 2.5]. Lemma 2.6.
The Jacobi matrix J ( p, q ; s, t ) is totally positive if and only if s ≥ t and p ( s + √ s − t ) / ≥ q . On the other hand, R ( p, q ; s, t ) is also a Riordan array. A Riordan array , denoted by( d ( x ) , h ( x )), is an infinite lower triangular matrix whose generating function of the k thcolumn is x k h k ( x ) d ( x ) for k = 0 , , , . . . , where d (0) = 1 and h (0) = 0 [8]. A Riordanarray R = [ r n,k ] n,k ≥ can be characterized by two sequences ( a n ) n ≥ and ( z n ) n ≥ such that r , = 1 , r n +1 , = X j ≥ z j r n,j , r n +1 ,k +1 = X j ≥ a j r n,k + j (2.4)for n, k ≥ Z ( x ) = P n ≥ z n x n and A ( x ) = P n ≥ a n x n . Thenit follows from (2.4) that d ( x ) = 11 − xZ ( xh ( x )) , h ( x ) = A ( xh ( x )) . (2.5)Now R ( p, q ; s, t ) is a Riordan array with Z ( x ) = p + qx and A ( x ) = 1 + sx + tx . Let R ( p, q ; s, t ) = ( d ( x ) , h ( x )). Then by (2.5), we have d ( x ) = 11 − x ( p + qxh ( x )) , h ( x ) = 1 + sxh ( x ) + tx h ( x ) . It follows that h ( x ) = 1 − sx − p − sx + ( s − t ) x tx and d ( x ) = 2 t t − q + ( qs − pt ) x + q p − sx + ( s − t ) x (see [10] for details).Combining Theorem 2.1 and Lemma 2.6, we have the following.5 heorem 2.7. Let p, q, s, t be all nonnegative and X n ≥ d n x n = 2 t t − q + ( qs − pt ) x + q p − sx + ( s − t ) x . If s ≥ t and p ( s + √ s − t ) / ≥ q , then ( d n ) n ≥ is a Stieltjes moment sequence. Setting q = t in Theorem 2.7, we obtain Corollary 2.8.
Let p, s, t be all nonnegative and X n ≥ d n x n = 21 + ( s − p ) x + p − sx + ( s − t ) x . If s ≥ t and p ( s + √ s − t ) / ≥ t , then ( d n ) n ≥ is a Stieltjes moment sequence. In particular, taking p = s in Corollary 2.8, and noting s ≥ t implies that s ( s + √ s − t ) / ≥ t , we have Corollary 2.9.
Let s, t be nonnegative and X n ≥ d n x n = 21 − sx + p − sx + ( s − t ) x . If s ≥ t , then ( d n ) n ≥ is a Stieltjes moment sequence. On the other hand, taking p = s and q = 2 t in Theorem 2.7, we obtain Corollary 2.10.
Let s, t be nonnegative and X n ≥ d n x n = 1 p − sx + ( s − t ) x . If s ≥ t , then ( d n ) n ≥ is a Stieltjes moment sequence. Remark 2.11.
Corollaries 2.9 and 2.10 have occurred in Bennett [3, § Example 2.12.
The Catalan numbers C n , the central binomial coefficients (cid:0) nn (cid:1) , thecentral Delannoy numbers D n , the large Schr¨oder numbers r n , the little Schr¨oder numbers S n have generating functions X n ≥ C n x n = 21 + √ − x , X n ≥ (cid:18) nn (cid:19) x n = 1 √ − x , X n ≥ D n x n = 1 √ − x + x , X n ≥ r n x n = 21 − x + √ − x + x , X n ≥ S n x n = 21 + x + √ − x + x . respectively. Again we see that these numbers are all Stieltjes moment sequences.6 Remarks
A Stieltjes moment sequence ( m n ) is called determinate , if there is a unique measure µ on [0 , + ∞ ) such that (1.1) holds; otherwise it is called indeterminate . For example, ( n !)is a Stieltjes moment sequence of the exponential distribution by (1.2) and determinateby Stirling’s approximation and Carleman’s criterion which states that the divergence ofthe series X n ≥ n √ m n implies the determinacy of the moment sequence ( m n ) (see [9, Theorem 1.11] for instance).We have shown that many well-known Catalan-like numbers are Stieltjes moment se-quences. However, we do not know how to obtain the associated measures in general andwhether these moment sequences are determinate. Acknowledgement
This work was supported in part by the NSF of China (Grant No. 11371078).