Hankel Matrices and Lattice Paths

Wen-jin Woan

Department of Mathematics Howard University Washington, D.C. 20059, USA

Email address: wwoan@howard.edu

Abstract

Let H be the Hankel matrix formed from a sequence of real numbers S = {a_{0} = 1, a_{1}, a_{2}, a_{3}, ...}, and let L denote the lower triangular matrix obtained from the Gaussian column reduction of H. This paper gives a matrix-theoretic proof that the associated Stieltjes matrix S_{L} is a tri-diagonal matrix. It is also shown that for any sequence (of nonzero real numbers) T = {d_{0} = 1, d_{1}, d_{2} , d_{3}, ...} there are infinitely many sequences such that the determinant sequence of the Hankel matrix formed from those sequences is T.

1. Introduction. In this paper we give a matrix-theoretic proof (Theorem 2.1) of one of the main theorems in [1]. In Section 2 we discuss the connection between the decomposition of a Hankel matrix and Stieltjes matrices, and in Section 3 we discuss the connection between certain lattice paths and Hankel matrices. Section 4 presents an explicit formula for the decomposition of a Hankel matrix.

Definition 1.1. Let S = {a_{0} = 1, a_{1}, a_{2}, a_{3}, ...} be a sequence of real numbers. The Hankel matrix generated by S is the infinite matrix

H = [ \begin{matrix}1 & a_{1} & a_{2} & a_{3} & a_{4} & . \\ a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & . \\ a_{2} & a_{3} & a_{4} & a_{5} & a_{6} & . \\ a_{3} & a_{4} & a_{5} & a_{6} & a_{7} & . \\ a_{4} & a_{5} & a_{6} & a_{7} & a_{8} & . \\ . & . & . & . & . & . \end{matrix} ] .

Definition 1.2. A lower triangular matrix

L = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ l_{10} & 1 & 0 & 0 & 0 & . \\ l_{20} & l_{21} & 1 & 0 & 0 & . \\ l_{30} & l_{31} & l_{32} & 1 & 0 & . \\ l_{40} & l_{41} & l_{42} & l_{43} & 1 & . \\ . & . & . & . & . & . \end{matrix} ] .

is said to be a Riordan matrix if there exist Taylor series g(x) = 1 + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n} + ... and f(x) = x + b_{2}x^{2} + b_{3}x^{3} + ... + b_{n}x^{n} + .... such that for every k ≥ 0 the k-th column has ordinary generating function g(x)(f(x))^{k}.

Definition 1.3. The Stieltjes matrix of a lower triangular matrix L is the matrix S_{L} which satisfies LS_{L} = L^{r} where L^{r} is the matrix obtained from L by deleting the first row of L.

Thus

[ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ l_{10} & 1 & 0 & 0 & 0 & . \\ l_{20} & l_{21} & 1 & 0 & 0 & . \\ l_{30} & l_{31} & l_{32} & 1 & 0 & . \\ l_{40} & l_{41} & l_{42} & l_{43} & 1 & . \\ . & . & . & . & . & . \end{matrix} ] S_{L} = [ \begin{matrix}l_{10} & 1 & 0 & 0 & 0 & . \\ l_{20} & l_{21} & 1 & 0 & 0 & . \\ l_{30} & l_{31} & l_{32} & 1 & 0 & . \\ l_{40} & l_{41} & l_{42} & l_{43} & 1 & . \\ . & . & . & . & . & . \end{matrix} ]

and so

S_{L} = L^{−1}L^{r} = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ −l_{10} & 1 & 0 & 0 & 0 & . \\ × & −l_{21} & 1 & 0 & 0 & . \\ × & × & −l_{32} & 1 & 0 & . \\ × & × & × & −l_{43} & 1 & . \\ . & . & . & . & . & . \end{matrix} ] [ \begin{matrix}l_{10} & 1 & 0 & 0 & 0 & . \\ l_{20} & l_{21} & 1 & 0 & 0 & . \\ l_{30} & l_{31} & l_{32} & 1 & 0 & . \\ l_{40} & l_{41} & l_{42} & l_{43} & 1 & . \\ . & . & . & . & . & . \end{matrix} ]

= [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ × & c_{1} & b_{2} & 1 & 0 & . \\ × & × & c_{2} & b_{3} & 1 & . \\ × & × & × & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

where

b_{0} = l_{10,} b_{k} = l_{k+1,k} − l_{k,k−1}, k > 0,

c_{0} = l_{2,0} − l_{1,0}^{2}, c_{k} = (l_{k,k−1}l_{k+1,k} − l_{k+1,k−1}) − l_{k+1,k}^{2} + l_{k+2,k}, k > 0.

Definition 1.4. Let L and S_{L} be as in Definition 1.3. We define

D_{L} = [ \begin{matrix}d_{0} & 0 & 0 & 0 & 0 & . \\ 0 & d_{1} & 0 & 0 & 0 & . \\ 0 & 0 & d_{2} & 0 & 0 & . \\ 0 & 0 & 0 & d_{3} & 0 & . \\ 0 & 0 & 0 & 0 & d_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

to be the diagonal matrix with diagonal entries given by d_{0} = 1, d_{k+1} = d_{k}c_{k} for k > 0.

2. Stieltjes and Hankel Matrices. The following two theorems are proved in [1]. Theorem 2.1. Let L be a lower triangular matrix and let D = D_{L} be the diagonal matrix

with nonzero diagonal entries {d_{i}} as in Definition 1.4. Then LDL^{t} is a Hankel matrix if and only if S_{L} is a tri-diagonal matrix, i.e. if and only if

S_{L} = [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ 0 & c_{1} & b_{2} & 1 & 0 & . \\ 0 & 0 & c_{2} & b_{3} & 1 & . \\ 0 & 0 & 0 & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

where b_{0} = l_{1,0} , c_{0} = d_{1} , b_{k} = l_{k+1,k} − l_{k,k−1} , c_{k} = \frac{d_{k+1}}{d_{k}} , k ≥ 1.

Proof. Let H = LDL^{t} be a Hankel matrix. Then L = H(DL^{t})^{−1}, L^{r} = (H(DL^{t})^{−1})^{r} = H^{r}(DL^{t})^{−1}, S_{L} = L^{−1}L^{r} = L^{−1}(H^{r}(DL^{t})^{−1}) = (L^{−1}H^{r})(DL^{t})^{−1}. Since H is a Hankel matrix, deleting the first row has the same effect as deleting the first

column.

L^{−1}H = DL^{t} = [ \begin{matrix}d_{0} & d_{0}l_{10} & d_{0}l_{20} & d_{0}l_{3,0} & d_{0}l_{4,0} & . \\ 0 & d_{1} & d_{1}l_{21} & d_{1}l_{31} & d_{1}l_{41} & . \\ 0 & 0 & d_{2} & d_{2}l_{32} & d_{2}l_{42} & . \\ 0 & 0 & 0 & d_{3} & d_{3}l_{43} & . \\ 0 & 0 & 0 & 0 & d_{4} & . \\ . & . & . & . & . & . \end{matrix} ] ,

L^{−1}H^{r} = L^{−1}H^{c} = (L^{−1}H)^{c} = [ \begin{matrix}d_{0}l_{10} & d_{0}l_{20} & d_{0}l_{30} & d_{0}l_{4,0} & . \\ d_{1} & d_{1}l_{21} & d_{1}l_{31} & d_{1}l_{41} & . \\ 0 & d_{2} & d_{2}l_{32} & d_{2}l_{42} & . \\ 0 & 0 & d_{3} & d_{3}l_{43} & . \\ 0 & 0 & 0 & d_{4} & . \\ . & . & . & . & . \end{matrix} ] ,

S_{L} = (L^{−1}H)^{c}(DL^{t})^{−1} = [ \begin{matrix}d_{0}l_{10} & d_{0}l_{20} & d_{0}l_{30} & d_{0}l_{4,0} & . \\ d_{1} & d_{1}l_{21} & d_{1}l_{31} & d_{1}l_{41} & . \\ 0 & d_{2} & d_{2}l_{32} & d_{2}l_{42} & . \\ 0 & 0 & d_{3} & d_{3}l_{43} & . \\ 0 & 0 & 0 & d_{4} & . \\ . & . & . & . & . \end{matrix} ] [ \begin{matrix}\frac{1}{d_{0}} & × & × & × & × & . \\ 0 & \frac{1}{d_{1}} & × & × & × & . \\ 0 & 0 & \frac{1}{d_{2}} & × & × & . \\ 0 & 0 & 0 & \frac{1}{d_{3}} & × & . \\ 0 & 0 & 0 & 0 & \frac{1}{d_{4}} & . \\ . & . & . & . & . & . \end{matrix} ]

= [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ 0 & c_{1} & b_{2} & 1 & 0 & . \\ 0 & 0 & c_{2} & b_{3} & 1 & . \\ 0 & 0 & 0 & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

where

b_{0} = l_{1,0} , c_{0} = \frac{d_{1}}{d_{0}} = d_{1} , b_{k} = l_{k+1,k} − l_{k,k−1} , c_{k} = \frac{d_{k+1}}{d_{k}} , k ≥ 1.

Conversely, let S_{L} be a tri-diagonal matrix and let H = LDL^{t}. Then L^{−1}H^{r} = L^{−1}(LDL^{t})^{r} = L^{−1}(L^{r}DL^{t}) = (L^{−1}L^{r})DL^{t} = S_{L}DL^{t}

= [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ 0 & c_{1} & b_{2} & 1 & 0 & . \\ 0 & 0 & c_{2} & b_{3} & 1 & . \\ 0 & 0 & 0 & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ] [ \begin{matrix}d_{0} & d_{0}l_{10} & d_{0}l_{20} & d_{0}l_{3,0} & d_{0}l_{4,0} & . \\ 0 & d_{1} & d_{1}l_{21} & d_{1}l_{31} & d_{1}l_{41} & . \\ 0 & 0 & d_{2} & d_{2}l_{32} & d_{2}l_{42} & . \\ 0 & 0 & 0 & d_{3} & d_{3}l_{43} & . \\ 0 & 0 & 0 & 0 & d_{4} & . \\ . & . & . & . & . & . \end{matrix} ] .

Therefore (L^{−1}H^{r})_{n,k} = c_{n−1}d_{n−1}l_{k,n−1} + b_{n}d_{n}l_{k,n} + d_{n+1}l_{k,n+1} = \frac{d_{n}}{d_{n−1}} d_{n−1}l_{k,n−1} + b_{n}d_{n}l_{k,n} + c_{n}d_{n}l_{k,n+1} = d_{n}(l_{k,n−1} + b_{n}l_{k,n} + c_{n}l_{k,n+1}) = d_{n}l_{k+1,n} = (DL^{t})_{n,k+1} = (DL^{t})_{n,k}^{c} = (L^{−1}H)_{n,k}^{c} = (L^{−1}H^{c})_{n,k}. We have shown that L^{−1}H^{r} = L^{−1}H^{c}, and so H^{r} = H^{c}. Hence H is a Hankel matrix.

Theorem 2.2. L is a Riordan matrix (i.e. b_{k} = b_{1} = b and c_{k} = c_{1} = c for k ≥ 1) if and only if f = x(1 + bf + cf^{2}) and

g = \frac{1}{1 − xb_{0} − xc_{0}f} ,

where f, g are as in Definition 1.2.

See [1] for the proof. Corollary 2.3. Let T = {d_{0} = 1, d_{1}, d_{2} , d_{3}, ...} be any sequence of (nonzero) real numbers.

Then there exists a sequence S = {a_{0} = 1, a_{1}, a_{2}, a_{3}, ...} such that T is equal to the sequence of diagonal entries of D in the decomposition H = LDL^{t} of the Hankel matrix generated by S .

Proof. As in Theorem 2.1, let c_{0} = d_{1} , c_{k} = \frac{d_{k+1}}{d_{k}} , k ≥ 1, and form the Stieltjes matrix

S_{L} = [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ 0 & c_{1} & b_{2} & 1 & 0 & . \\ 0 & 0 & c_{2} & b_{3} & 1 & . \\ 0 & 0 & 0 & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

where the b_{i}s are arbitrary. By Definition 1.3 there is a lower triangular matrix L such that LS_{L} = L^{r}. Let S be the sequence formed by the first column of L and let H denote the Hankel matrix generated by S. By Theorem 2.1 the diagonal entries of D in the decomposition H = LDL^{t} form the sequence T.

Example 2.4. Let T = \{1, 1, 2, 5, 14, 42, 132, ...\} be the Catalan sequence (\underline{A000108} in [2]) and let

S_{L} = [ \begin{matrix}0 & 1 & 0 & 0 & 0 & . \\ 1 & 0 & 1 & 0 & 0 & . \\ 0 & 2 & 0 & 1 & 0 & . \\ 0 & 0 & \frac{5}{2} & 0 & 1 & . \\ 0 & 0 & 0 & \frac{14}{5} & 0 & . \\ . & . & . & . & . & . \end{matrix} ] .

Then

L = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ 0 & 1 & 0 & 0 & 0 & . \\ 1 & 0 & 1 & 0 & 0 & . \\ 0 & 3 & 0 & 1 & 0 & . \\ 3 & 0 & \frac{11}{2} & 0 & 1 & . \\ . & . & . & . & . & . \end{matrix} ] ,

LDL^{t} = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ 0 & 1 & 0 & 0 & 0 & . \\ 1 & 0 & 1 & 0 & 0 & . \\ 0 & 3 & 0 & 1 & 0 & . \\ 3 & 0 & \frac{11}{2} & 0 & 1 & . \\ . & . & . & . & . & . \end{matrix} ] [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ 0 & 1 & 0 & 0 & 0 & . \\ 0 & 0 & 2 & 0 & 0 & . \\ 0 & 0 & 0 & 5 & 0 & . \\ 0 & 0 & 0 & 0 & 14 & . \\ . & . & . & . & . & . \end{matrix} ] [ \begin{matrix}1 & 0 & 1 & 0 & 3 & . \\ 0 & 1 & 0 & 3 & 0 & . \\ 0 & 0 & 1 & 0 & \frac{11}{2} & . \\ 0 & 0 & 0 & 1 & 0 & . \\ 0 & 0 & 0 & 0 & 1 & . \\ . & . & . & . & . & . \end{matrix} ]

= [ \begin{matrix}1 & 0 & 1 & 0 & 3 & . \\ 0 & 1 & 0 & 3 & 0 & . \\ 1 & 0 & 3 & 0 & 14 & . \\ 0 & 3 & 0 & 14 & 0 & . \\ 3 & 0 & 14 & 0 & \frac{167}{2} & . \\ . & . & . & . & . & . \end{matrix} ] = H.

3. Lattice Paths and Hankel Matrices

We consider those lattice paths in the Cartesian plane running from (0, 0) that use steps from S = {u = (1, 1), h = (1, 0), d = (1, −1)} with assigned weights 1 for u, w_{1} for h and w_{2} for d. Let L(n, k) be the set of paths that never go below the x-axis and end at (n, k). The weight of a path is the product of the weights of its steps. Let l_{n,k} be the sum of the weights of all the paths in L(n, k). See also [3], [4].

Theorem 3.1. Let L = (l_{n,k})_{n,k≥0}. Then L is a lower triangular matrix, the Stieltjes matrix of L is

S_{L} = [ \begin{matrix}w_{1} & 1 & 0 & 0 & 0 & . \\ w_{2} & w_{1} & 1 & 0 & 0 & . \\ 0 & w_{2} & w_{1} & 1 & 0 & . \\ 0 & 0 & w_{2} & w_{1} & 1 & . \\ 0 & 0 & 0 & w_{2} & w_{1} & . \\ . & . & . & . & . & . \end{matrix} ]

and H = LDL^{t} is the Hankel matrix generated by the first column of L and d_{k} = w_{2}^{k} for k > 0. Proof. From Theorem 2.1.

Example 3.2. For w_{1} = 0, w_{2} = 1, L is the Catalan matrix. For w_{1} = t, w_{2} = 1, L is the t-Motzkin matrix. In both cases D is the identity matrix. For example, when t = 1,

L = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & . \\ 1 & 1 & 0 & 0 & 0 & . \\ 2 & 2 & 1 & 0 & 0 & . \\ 4 & 5 & 3 & 1 & 0 & . \\ 9 & 12 & 9 & 4 & 1 & . \\ . & . & . & . & . & . \end{matrix} ] ,

LDL^{t} = [ \begin{matrix}1 & 1 & 2 & 4 & 9 & . \\ 1 & 2 & 4 & 9 & 21 & . \\ 2 & 4 & 9 & 21 & 51 & . \\ 4 & 9 & 21 & 51 & 127 & . \\ 9 & 21 & 51 & 127 & 323 & . \\ . & . & . & . & . & . \end{matrix} ] = H

where S = \{1, 1, 2, 4, 9, 21, 51, ...\} is the Motzkin sequence \underline{A001006}.

Theorem 3.3. If w_{1}, w_{2} depend on the height k, i.e. w_{1}(k) = b_{k} and w_{2}(k + 1) = c_{k}, then

S_{L} = [ \begin{matrix}b_{0} & 1 & 0 & 0 & 0 & . \\ c_{0} & b_{1} & 1 & 0 & 0 & . \\ 0 & c_{1} & b_{2} & 1 & 0 & . \\ 0 & 0 & c_{2} & b_{3} & 1 & . \\ 0 & 0 & 0 & c_{3} & b_{4} & . \\ . & . & . & . & . & . \end{matrix} ]

and H = LDL^{t} is the Hankel matrix generated by the first column of L and d_{k} = Π_{i≤k}c_{i}. Proof. From Theorem 2.1.

See Example 2.4 for an illustration.

4. Gaussian Column Reduction

Let S = {a_{0} = 1, a_{1}, a_{2}, a_{3}, ...} be a sequence of real numbers and let H denote the Hankel matrix generated by S. All the results in this section are well-known in matrix theory. We shall express the entries of L in term of S. We assume that H is positive definite.

Lemma 4.1. The decomposition of a positive definite Hankel matrix H = LDU is unique and U = L^{t}, where L is a lower triangular matrix with diagonal entries 1, D is a diagonal matrix and U is an upper triangular matrix with diagonal entries 1.

Proof. Let LDU = H = L_{1}D_{1}U_{1}. Then DUU_{1}^{−1} = L^{−1}L_{1}D_{1} is both an upper and lower triangular matrix, hence UU_{1}^{−1} = L^{−1}L_{1} = I is the infinite identity matrix.

Let H_{n} be the truncated submatrix of H with n ≥ 0 . For example,

H_{3} = [ \begin{matrix}1 & a_{1} & a_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} & a_{4} \\ a_{2} & a_{3} & a_{4} & a_{5} \\ a_{3} & a_{4} & a_{5} & a_{6} \end{matrix} ] , H_{4} = [ \begin{matrix}1 & a_{1} & a_{2} & a_{3} & a_{4} \\ a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{3} & a_{4} & a_{5} & a_{6} & a_{7} \\ a_{4} & a_{5} & a_{6} & a_{7} & a_{8} \end{matrix} ] .

Let H_{n}(k) be the matrix obtained from H_{n} by replacing the last column of H_{n} by a_{k}, a_{k+1}, a_{k+2}, ..., a_{k+n}. For example,

H_{3}(1) = [ \begin{matrix}1 & a_{1} & a_{2} & a_{1} \\ a_{1} & a_{2} & a_{3} & a_{2} \\ a_{2} & a_{3} & a_{4} & a_{3} \\ a_{3} & a_{4} & a_{5} & a_{4} \end{matrix} ] , H_{3}(5) = [ \begin{matrix}1 & a_{1} & a_{2} & a_{5} \\ a_{1} & a_{2} & a_{3} & a_{6} \\ a_{2} & a_{3} & a_{4} & a_{7} \\ a_{3} & a_{4} & a_{5} & a_{8} \end{matrix} ] .

.

Let h_{i} = det H_{i} and define an infinite upper triangular matrix R = (r_{n,k}) in term of (n, k)- cofactor of H_{k} by r_{n,k} = 0 for k < n, and

r_{n,k} = \frac{1}{h_{k−1}} (−1)^{n+k+2} det [ \begin{matrix}1 & a_{1} & a_{2} & . & a_{k−1} \\ a_{1} & a_{2} & a_{3} & .. & a_{k} \\ a_{2} & a_{3} & a_{4} & . & a_{k+1} \\ . & . & . & . & .. \\ a_{n−1} & a_{n} & a_{n+1} & . & a_{k+n−2} \\ a_{n+1} & a_{n+2} & a_{n+3} & . & a_{k+n} \\ . & . & . & . & . \\ a_{k} & a_{k+1} & a_{k+2} & . & a_{k+k} \end{matrix} ]

for k ≥ n. For example,

r_{2,4} = \frac{1}{h_{3}} (−1)^{(2+4)+2} det [ \begin{matrix}1 & a_{1} & a_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} & a_{4} \\ a_{3} & a_{4} & a_{5} & a_{6} \\ a_{4} & a_{5} & a_{6} & a_{7} \end{matrix} ] .

Remark 4.2. HR = LD, where L = (l_{n,k}) is the Gaussian column reduction of the Hankel matrix H and D is the diagonal matrix with diagonal entries \{d_{i}\}, R^{−1} = L^{t} with d_{i} = \frac{h_{i}}{h_{i−1}} and l_{n,k} = \frac{1}{h_{k−1}} det H_{k}(n).

Remark 4.3. If L is a Riordan matrix, then for i ≥ 1, c = c_{i} = \frac{d_{i+1}}{d_{i}} = \frac{h_{i+1}h_{i−1}}{h_{i}h_{i}} and b = b_{i} = l_{i+1,i} − l_{i,i−1} = \frac{1}{h_{i−1}} det H_{i}(i + 1) − \frac{1}{h_{i−2}} det H_{i−1}(i) is a recurrence relation for the sequence S.

Example 4.4. Let S = {1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ...} be the central Delannoy numbers \underline{A001850} and let H be the Hankel matrix generated by S. Then

H = [ \begin{matrix}1 & 3 & 13 & 63 & . \\ 3 & 13 & 63 & 321 & . \\ 13 & 63 & 321 & 1683 & . \\ 63 & 321 & 1683 & 8989 & . \\ . & . & . & . & . \end{matrix} ] ,

R = [ \begin{matrix}1 & −3 & 5 & −9 & . \\ 0 & 1 & −6 & 21 & . \\ 0 & 0 & 1 & −9 & . \\ 0 & 0 & 0 & 1 & . \\ . & . & . & . & . \end{matrix} ] ,

LD = HR = [ \begin{matrix}1 & 0 & 0 & 0 & . \\ 3 & 4 & 0 & 0 & . \\ 13 & 24 & 8 & 0 & . \\ 63 & 132 & 72 & 16 & . \\ . & . & . & . & . \end{matrix} ] ,

R^{t}HR = D = [ \begin{matrix}1 & 0 & 0 & 0 & . \\ 0 & 4 & 0 & 0 & . \\ 0 & 0 & 8 & 0 & . \\ 0 & 0 & 0 & 16 & . \\ . & . & . & . & . \end{matrix} ] ,

L = HRD^{−1} = [ \begin{matrix}1 & 0 & 0 & 0 & . \\ 3 & 1 & 0 & 0 & . \\ 13 & 6 & 1 & 0 & . \\ 63 & 33 & 9 & 1 & . \\ . & . & . & . & . \end{matrix} ] ,

S_{L} = L^{−1}L^{r} = R^{t}L^{r} = [ \begin{matrix}1 & 0 & 0 & 0 & . \\ −3 & 1 & 0 & 0 & . \\ 5 & −6 & 1 & 0 & . \\ −9 & 21 & −9 & 1 & . \\ . & . & . & . & . \end{matrix} ] [ \begin{matrix}3 & 1 & 0 & 0 & 0 & . \\ 13 & 6 & 1 & 0 & 0 & . \\ 63 & 33 & 9 & 1 & 0 & . \\ 321 & 180 & 62 & 12 & 1 & . \\ . & . & . & . & . & . \end{matrix} ]

= [ \begin{matrix}3 & 1 & 0 & 0 & . \\ 4 & 3 & 1 & 0 & . \\ 0 & 2 & 3 & 1 & . \\ 0 & 0 & 2 & 3 & . \\ . & . & . & . & . \end{matrix} ] ,

LDL^{t} = [ \begin{matrix}1 & 0 & 0 & 0 & . \\ 3 & 1 & 0 & 0 & . \\ 13 & 6 & 1 & 0 & . \\ 63 & 33 & 9 & 1 & . \\ . & . & . & . & . \end{matrix} ] [ \begin{matrix}1 & 0 & 0 & 0 & . \\ 0 & 4 & 0 & 0 & . \\ 0 & 0 & 8 & 0 & . \\ 0 & 0 & 0 & 16 & . \\ . & . & . & . & . \end{matrix} ] [ \begin{matrix}1 & 3 & 13 & 63 & . \\ 0 & 1 & 6 & 33 & . \\ 0 & 0 & 1 & 9 & . \\ 0 & 0 & 0 & 1 & . \\ . & . & . & . & . \end{matrix} ]

= [ \begin{matrix}1 & 3 & 13 & 63 & . \\ 3 & 13 & 63 & 321 & . \\ 13 & 63 & 321 & 1683 & . \\ 63 & 321 & 1683 & 8989 & . \\ . & . & . & . & . \end{matrix} ] = H.

Remark 4.5. If H is the Hankel matrix corresponding to a sequence S, then by Theorem 3.1 and Theorem 3.3 we may use lattice paths to find L, the Gaussian column reduction of H.

Acknowledgment. The author would like to thank Professor Ralph Turner for his help in rewriting the paper.

References

[1] P. Peart and W. J. Woan, Generating functions via Hankel and Stieltjes matrices, J. of Integer Sequences, 3 (2) (2000), Article 00.2.1.

[2] Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. Published electronically at

   http://www.oeis.org.

[3] R. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences 3 (1) (2000), Article 00.1.1.

[4] J. G. Wendel, Left-continuous random walk and the Lagrange expansion, Amer. Math. Monthly 82 (1975), 494–499.

(Mentions sequences \underline{A000108}, \underline{A001006}, and \underline{A001850}.)

Received September 19, 2000; published in Journal of Integer Sequences, April 24, 2001.

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