Matrix Decomposition of the Unified Generalized Stirling Numbers and Inversion

of the Generalized Factorial Matrices

Jiaqiang Pan School of Biomedical Engineering and Instrumental Science Zhejiang University Hangzhou 210027 China

                       jqpan@zju.edu.cn

Abstract In this paper, we give a matrix decomposition method used to calculate unified generalized Stirling numbers in an explicit, non-recursive mode, and some of its applications. Then, we define generalized factorial matrices which may be regarded as a generalization in the form of the Vandermonde matrices, and presents some of their properties — in particular, triangular matrix factors of the inverse matrices of the generalized factorial matrices.

1 Introduction

The unified generalized Stirling numbers, defined by Hsu and Shuie [1], are the connection coefficients of linear relations between generalized factorial functions. The generalized factorial functions of a real or complex number x with real increment α, denoted by (x|α)_{n}, are special polynomials in x of degree n, as

(x|α)_{0} = 1, and (x|α)_{n} = x(x − α) ⋯ (x − nα + α), n = 1, 2, ….  (1)

Thus, the unified generalized Stirling numbers with real parameters α, β, γ, denoted by S(n, k; α, β, γ), n, k = 0, 1, 2, …, may be defined as (see [1])

(x|α)_{n} = ∑_{k=0}^{∞} S(n, k; α, β, γ)(x − γ|β)_{k}.  (2)

We see from (2) that for any α, β, γ, S(n, n; α, β, γ) = 1; and S(n, k; α, β, γ) = 0 if k > n. Therefore, the upper limit ∞ of the summation in the right side of equality (2) may be replaced by n.

As Hsu and Shuie pointed, the definition (2) gives a more transparent unification of various Stirling-type numbers studied previously by other authors (see [1]). In particularly, we see from (2) that S(n, k; 0, 0, 0) = δ_{n,k} (the Kronecker symbols), S(n, k; 1, 0, 0) = s(n, k), S(n, k; 0, 1, 0) = S(n, k), and S(n, k; 0, 0, 1) = \binom{n}{k}, where s(n, k) and S(n, k) are the Stirling numbers of the first and second kind respectively.

Hsu, Shuie and other authors then investigated basic properties of the unified generalized Stirling numbers, such as recurrence relations, generating functions, convolution formulas, congruence properties, asymptotic expansion, q-analoque formulas and corresponding combinatorial interpretation (see [1, 2, 3]).

2 Matrix Decomposition of the Unified Generalized Stirling Numbers

Now let’s have the following definition.

Definition 1. the unified generalized Stirling (transform) matrix with real parameters α, β, γ, denoted by S_{α,β,γ}, is defined to be an infinite-dimensional lower triangular matrix, the (n, k)th entries of which are S(n, k; α, β, γ) (n, k = 0, 1, 2, …), such that

v_{α}(x) = S_{α,β,γ}v_{β}(x − γ)  (3)

where v_{α}(x) is the vector of the generalized factorials with real increment α, as follows

v_{α}(x) = (1, x, (x|α)_{2}, …, (x|α)_{n}, …)^{T} .  (4)

Remark 2. It can easily be shown that equations (2) and (3) are equivalent. We may find immediately that matrices S_{0,0,0} = E, S_{0,0,1} = B, S_{1,0,0} = S_{1}, S_{0,1,0} = S_{2} are, respectively, the infinite-dimensional unit matrix and the matrices of the binomial transform, the Stirling transforms of the first and second kind for integer sequences.

Remark 3. We see from the matrix definition of the unified generalized Stirling numbers, (3), that

v_{β}(x − γ) = S_{β,α,−γ}v_{α}(x).

This leads to that

v_{α}(x) = S_{α,β,γ}S_{β,α,−γ}v_{α}(x),

namely (S_{α,β,γ}S_{β,α,−γ} − E)v_{α}(x) = 0 (the infinite-dimensional zero matrix). Because x is an arbitrary real or complex number, we have S_{α,β,γ}S_{β,α,−γ} = E, namely matrix S_{β,α,−γ} is the inverse of S_{α,β,γ}.

We first investigate three basic types of the generalized Stirling matrices: S_{0,0,γ}, S_{α,0,0}, and S_{0,β,0}.

According to the binomial theorem, we have that

(x|0)_{n} = x^{n} = ∑_{k=0}^{n} \binom{n}{k}γ^{n−k}(x − γ)^{k} = ∑_{k=0}^{n} \binom{n}{k}γ^{n−k}(x − γ|0)_{k}.

Hence, the (n, k)th entry of S_{0,0,γ} is that

S(n, k; 0, 0, γ) = γ^{n−k}\binom{n}{k},  (n, k = 0, 1, 2, …).  (5)

Remark 4. In case γ is an integer, S_{0,0,γ} is just the γ-fold binomial transform matrix for an integer sequence, namely γ successive binomial transform for an integer sequence (see [4]).

Next, we look at the generalized Stirling numbers S(n, k; α, 0, 0) (α ≠ 0) defined in expression (x|α)_{n} = ∑_{k=0}^{n} S(n, k; α, 0, 0)x^{k}, (n, k = 0, 1, 2, …).

Because (x|α)_{n} = α^{n}(\frac{x}{α}|1)_{n} and x^{k} = α^{k}(\frac{x}{α})^{k},

α^{n}(\frac{x}{α}|1)_{n} = ∑_{k=0}^{n} S(n, k; α, 0, 0)α^{k}(\frac{x}{α})^{k}.

On the other hand, (\frac{x}{α}|1)_{n} = ∑_{k=0}^{n} S(n, k; 1, 0, 0)(\frac{x}{α})^{k}. Hence, we have S(n, k; α, 0, 0)α^{k} = α^{n}S(n, k; 1, 0, 0), which implies that

S(n, k; α, 0, 0) = α^{n−k}S(n, k; 1, 0, 0) = α^{n−k}s(n, k),  (n, k = 0, 1, 2, …).  (6)

Namely, the (n, k)th entry of matrix S_{α,0,0} is the Stirling number s(n, k) of the first kind, multiplied by α^{n−k}.

Similarly, we may obtain that when β ≠ 0,

S(n, k; 0, β, 0) = β^{n−k}S(n, k; 0, 1, 0) = β^{n−k}S(n, k),  (n, k = 0, 1, 2, …).  (7)

Namely, the (n, k)th entry of matrix S_{0,β,0} is the Stirling number S(n, k) of the second kind, multiplied by β^{n−k}.

Remark 5. We see from (5), (6), and (7) that, if θ is an integer, then each one of the generalized Stirling numbers S(n, k; θ, 0, 0), S(n, k; 0, θ, 0) and S(n, k; 0, 0, θ)) is an integer number.

Remark 6. In case α (or β) is an integer, matrix S_{α,0,0} (or S_{0,β,0}) defines a generalized Stirling transform matrix of the first (or second) kind for an integer sequence. They both are a pair of transform and inverse transform of integer sequences, if and only if α = β.

Now we may give a general decomposition formula of the unified generalized Stirling matrices, S_{α,β,γ}.

Theorem 7. Let S_{α,β,γ} be a generalized Stirling matrix with real parameters α, β, γ. Then

S_{α,β,γ} = S_{α,0,0}S_{0,0,γ}S_{0,β,0}.  (8)

Proof. We see from (3) that

v_{α}(x) = S_{α,0,0}v_{0}(x), v_{0}(x) = S_{0,0,γ}v_{0}(x − γ), v_{0}(x − γ) = S_{0,β,0}v_{β}(x − γ).

Hence we have

v_{α}(x) = S_{α,0,0}S_{0,0,γ}S_{0,β,0}v_{β}(x − γ).

On the other hand, we also have v_{α}(x) = S_{α,β,γ}v_{β}(x − γ). Thus,

(S_{α,β,γ} − S_{α,0,0}S_{0,0,γ}S_{0,β,0})v_{β}(x − γ) = 0.

Because x is an arbitrary real or complex number, the equality (8) holds.

Remark 8. In case all of α, β, γ are integer numbers, we may regard S_{α,β,γ} as a transform matrix of integer sequences. However, we see from (8) that, it is better to regard S_{α,β,γ} as a composition of three successive transforms: S_{0,β,0}, S_{0,0,γ} and S_{α,0,0}. Remark 9. As we know that, a linear homogeneous recurrent integer sequence a(n) of order q has the general-term with the following form: a(n) = ∑_{i=1}^{q} c_{i}λ_{i}^{n}, (n = 0, 1, 2, …), where λ_{i}, (i = 1, …, q) are its q real or complex characteristic values. We see from [5] that if γ is an integer, the q characteristic values of the γ-fold binomial transform b(n) of integer sequence a(n) are λ_{i} + γ, and the general-term of integer sequence b(n) is b(n) = ∑_{i=1}^{q} c_{i}(λ_{i} + γ)^{n}. Now, we may also obtain this conclusion by using the basic relation (3). Denoting the vectors corresponding to integer sequences a(n) and b(n) by a = (a(0), a(1), ⋯, a(n), ⋯)^{T} and b = (b(0), b(1), ⋯, b(n), ⋯)^{T}, we have a = ∑_{i=1}^{q} c_{i}v_{0}(λ_{i}), and

b = S_{0,0,γ}a = S_{0,0,γ} ∑_{i=1}^{q} c_{i}v_{0}(λ_{i}) = ∑_{i=1}^{q} c_{i}S_{0,0,γ}v_{0}(λ_{i}) = ∑_{i=1}^{q} c_{i}v_{0}(λ_{i} + γ).

Before we mention the next theorem, let us consider first the following definition.

Definition 10. Let α be an integer, and a(n) (n = 0, 1, 2, …) be a linear homogeneous recurrent integer sequence of order q. The α-generalized Stirling transform of the first kind of a(n) is defined by

b(n) = ∑_{k=0}^{n} S(n, k; α, 0, 0)a(k).

Now, let us give another property related to the recurrent integer sequences, as follows.

Theorem 11. The α-generalized Stirling transform of the first kind of a linear homogeneous recurrent integer sequence

a(n) = ∑_{i=1}^{q} c_{i}λ_{i}^{n}, n = 0, 1, 2, …,

is an integer sequence with the general-term

b(n) = ∑_{i=1}^{q} c_{i}(λ_{i}|α)_{n}, n = 0, 1, 2, …,

where λ_{i}, (i = 1, …, q) are q real or complex characteristic values of a(n).

Proof. Denoting the vectors corresponding to integer sequences a(n) and b(n) by a = (a(0), a(1), ⋯, a(n), ⋯)^{T} and b = (b(0), b(1), ⋯, b(n), ⋯)^{T}, we have a = ∑_{i=1}^{q} c_{i}v_{0}(λ_{i}), and

b = S_{α,0,0}a = S_{α,0,0} ∑_{i=1}^{q} c_{i}v_{0}(λ_{i}) = ∑_{i=1}^{q} c_{i}S_{α,0,0}v_{0}(λ_{i}) = ∑_{i=1}^{q} c_{i}v_{α}(λ_{i}),

that is, b(n) = ∑_{i=1}^{q} c_{i}(λ_{i}|α)_{n}, (n = 0, 1, 2, …). At the same time, we see from (6) that if α is integer, each of entries of S_{α,0,0} is also an integer, which implies that b(n) is an integer sequences.

Example 12. For example, the general-term of the recurrent integer sequence of order 2 of Lucas numbers L(n) = 2, 1, 3, 4, 7, 11, 18, … (A000032[6]) is L(n) = c_{1}λ_{1}^{n} + c_{2}λ_{2}^{n} = (\frac{1}{2} + \frac{\sqrt{5}}{2})^{n} + (\frac{1}{2} − \frac{\sqrt{5}}{2})^{n}. Hence, we find from Theorem 11 that the general-term of its Stirling transform of the first kind is

L_{1}(n) = (\frac{1}{2} + \frac{\sqrt{5}}{2}|1)_{n} + (\frac{1}{2} − \frac{\sqrt{5}}{2}|1)_{n}, n = 0, 1, 2, …,

that is, L_{1}(n) = 2, 1, 2, −3, 10, −45, 250, −1645, … (A213593[6]).

3 Generalized Factorial Matrices

We may see that if taking S_{α,β,γ} to be a p-dimensional (p = 1, 2, 3, …) matrix, which in fact is the p × p upper-left sub-matrix of the (original) infinite dimensional generalized Stirling matrix, then all of the conclusions presented in the preceding section still hold. We need to remember always this point of view while reading this section.

Definition 13. Let x_{1}, x_{2}, …, x_{p} be p distinct real or complex numbers, and α be a given real parameter. The Generalized Factorial Matrices of order p is a p × p matrix, denoted by V_{α}(x_{1}, x_{2}, …, x_{p}), whose ith column entries are the entries of the p-dimensional column vector v_{α}(x_{i}) (i = 1, …, p). That is

V_{α}(x_{1}, x_{2}, …, x_{p}) = [v_{α}(x_{1}), v_{α}(x_{2}), ⋯, v_{α}(x_{p})].  (9)

In particular, V_{0}(x_{1}, x_{2}, …, x_{p}) is just the Vandermonde matrix of p distinct parameters x_{1}, x_{2}, …, x_{p}. Hence we may regard the generalized factorial matrices as a generalization in form of the Vandermonde matrices.

Example 14. For example, the 6 × 6 matrices V_{α}(0, 1, 2, 3, 4, 5), (α = 0, 1, 2, 3) are, respectively,

V_{0}(0, 1, 2, 3, 4, 5) = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 4 & 9 & 16 & 25 \\ 0 & 1 & 8 & 27 & 64 & 125 \\ 0 & 1 & 16 & 81 & 256 & 625 \\ 0 & 1 & 32 & 243 & 1024 & 3125 \end{matrix} ) ,

V_{1}(0, 1, 2, 3, 4, 5) = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 2 & 6 & 12 & 20 \\ 0 & 0 & 0 & 6 & 24 & 60 \\ 0 & 0 & 0 & 0 & 24 & 120 \\ 0 & 0 & 0 & 0 & 0 & 120 \end{matrix} ) ,

V_{2}(0, 1, 2, 3, 4, 5) = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & −1 & 0 & 3 & 8 & 5 \\ 0 & 3 & 0 & −3 & 0 & 15 \\ 0 & −15 & 0 & 9 & 0 & −15 \\ 0 & 105 & 0 & −45 & 0 & 45 \end{matrix} ) ,

and

V_{3}(0, 1, 2, 3, 4, 5) = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & −2 & −2 & 0 & 4 & 10 \\ 0 & 10 & 8 & 0 & −8 & −10 \\ 0 & −80 & −56 & 0 & 40 & 40 \\ 0 & 880 & 560 & 0 & −320 & −280 \end{matrix} ) .

The next theorem gives basic properties of the generalized factorial matrices.

Theorem 15. Let x_{1}, x_{2}, …, x_{p} be p distinct real or complex numbers, and α,β,γ be three real parameters. Then the generalized factorial matrices have the following basic properties.

(i) Two generalized factorial matrices can be connected with a suitable generalized Stirling matrice, generally like

V_{α}(x_{1}, x_{2}, …, x_{p}) = S_{α,β,γ}V_{β}(x_{1} − γ, x_{2} − γ, …, x_{p} − γ).  (10)

(ii) Determinant of the generalized factorial matrix V_{α}(x_{1}, x_{2}, …, x_{p}) is

det V_{α}(x_{1}, x_{2}, …, x_{p}) = det V_{0}(x_{1}, x_{2}, …, x_{p}) = ∏_{1≤i<j≤p}(x_{j} − x_{i}).  (11)

This implies that matrix V_{α}(x_{1}, x_{2}, …, x_{p}) is invertible.

(iii) The inverse of the generalized factorial matrix V_{α}(x_{1}, x_{2}, …, x_{p}) is

V_{α}^{−1}(x_{1}, x_{2}, …, x_{p}) = V_{0}^{−1}(x_{1}, x_{2}, …, x_{p})S_{0,α,0},  (12)

namely, the inverse of the Vandermonde matrix with the same parameters, right-multiplied by the α-generalized Stirling matrix of the second kind.

Proof. The matrix equality (10) is the matrix form of p equalities v_{α}(x_{i}) = S_{α,β,γ}v_{β}(x_{i} − γ), (i = 1, 2, …, p). From (10), we have V_{α}(x_{1}, x_{2}, …, x_{p}) = S_{α,0,0}V_{0}(x_{1}, x_{2}, …, x_{p}). Then, we obtain

det V_{α}(x_{1}, x_{2}, …, x_{p}) = det S_{α,0,0} det V_{0}(x_{1}, x_{2}, …, x_{p}).

Because det S_{α,0,0} = 1 and det V_{0}(x_{1}, x_{2}, …, x_{p}) = ∏_{1≤i<j≤p}(x_{j} − x_{i}), the property (11) holds. Besides, we also see that

V_{α}^{−1}(x_{1}, …, x_{p}) = V_{0}^{−1}(x_{1}, …, x_{p})S_{α,0,0}^{−1} = V_{0}^{−1}(x_{1}, …, x_{p})S_{0,α,0},

that is, equality (12) holds.

Corollary 16. Let V_{α}(x_{1}, x_{2}, …, x_{p}) be a generalized factorial matrices of p distinct real or complex numbers x_{1}, x_{2}, …, x_{p}, with real parameter α. Then, we may express the inverse of matrix V_{α}(x_{1}, x_{2}, …, x_{p}) as

V_{α}^{−1}(x_{1}, x_{2}, …, x_{p}) = H_{p}L_{p}S_{0,α,0}.  (13)

where factor matrix L_{p} is a lower triangular matrix, the arbitrary entry L_{p}(n, k) (n, k = 0, 1, …, p − 1) of which is determined by the following linear relations

L_{p}(0, 0) = 1, and ∏_{i=1}^{n}(x − x_{i}) = ∑_{k=0}^{n} L_{p}(n, k)x^{k},  (14)

and the matrix factor H_{p} is a upper triangular matrix, the arbitrary entry h_{p}(n, k) (n, k = 0, 1, …, p − 1) of which is given by

h_{p}(0, 0) = 1, h_{p}(n, k) = \{ \begin{matrix}0, & if k < n, \\ ( ∏_{i=1,i≠n+1}^{k+1}(x_{n+1} − x_{i}))^{−1}, & if k ≥ n. \end{matrix}  (15)

Proof. By using a triangular-matrix decomposition method given by Hou and Hou [7], we may obtain

V_{0}^{−1}(x_{1}, x_{2}, …, x_{p}) = H_{p}L_{p},

where the lower triangular matrix L_{p} and the upper triangular matrix H_{p} are calculated by using (14) and (15). Hence, we see from (12) that the Corollary holds.

Example 17. In case x_{1} = 0, x_{2} = 1, x_{3} = 2,…, x_{p} = p − 1, we have L_{p} = S_{1,0,0}, and H_{p} = S_{0,0,−1}^{T}D_{p}, where matrix D_{p} is a diagonal matrix, the kth diagonal entry of which is \frac{1}{k!}, (k = 0, 1, …, p − 1). Hence,

V_{α}^{−1}(0, 1, …, p − 1) = H_{p}S_{1,0,0}S_{0,α,0} = S_{0,0,−1}^{T}D_{p}S_{1,0,0}S_{0,α,0}.  (16)

namely, V_{α}^{−1}(0, 1, …, p − 1) is a product of several triangular matrices. For example, we may obtain the inverse matrices of the 6 × 6 generalized factorial matrix V_{α}(0, 1, 2, 3, 4, 5) (α = 0, 1, 2, 3) (see Example 14), as follows

V_{0}^{−1}(0, 1, 2, 3, 4, 5) = H_{6}S_{1,0,0}S_{0,0,0} = H_{6}S_{1,0,0} = ( \begin{matrix}5 & −\frac{77}{12} & \frac{71}{24} & −\frac{7}{12} & \frac{1}{24} \\ −10 & \frac{107}{6} & −\frac{59}{6} & \frac{13}{6} & −\frac{1}{6} \\ 10 & −\frac{39}{2} & \frac{49}{4} & −3 & \frac{1}{4} \\ −5 & \frac{61}{6} & −\frac{41}{6} & \frac{11}{6} & −\frac{1}{6} \\ 1 & −\frac{25}{12} & \frac{35}{24} & −\frac{5}{12} & \frac{1}{24} \end{matrix}  ) ,

V_{1}^{−1}(0, 1, 2, 3, 4, 5) = H_{6}S_{1,0,0}S_{0,1,0} = H_{6} = ( \begin{matrix}1 & −1 & \frac{1}{2} & −\frac{1}{6} & \frac{1}{24} & −\frac{1}{120} \\ 0 & 1 & −1 & \frac{1}{2} & −\frac{1}{6} & \frac{1}{24} \\ 0 & 0 & \frac{1}{2} & −\frac{1}{2} & \frac{1}{4} & −\frac{1}{12} \\ 0 & 0 & 0 & \frac{1}{6} & −\frac{1}{6} & \frac{1}{12} \\ 0 & 0 & 0 & 0 & \frac{1}{24} & −\frac{1}{24} \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{120} \end{matrix} ) ,

V_{2}^{−1}(0, 1, 2, 3, 4, 5) = H_{6}S_{1,0,0}S_{0,2,0} = ( \begin{matrix}1 & −\frac{1}{2} & \frac{1}{8} & −\frac{1}{24} & −\frac{1}{24} & −\frac{1}{120} \\ 0 & 0 & 0 & \frac{1}{8} & \frac{1}{4} & \frac{1}{24} \\ 0 & \frac{1}{2} & −\frac{1}{4} & −\frac{1}{4} & −\frac{7}{12} & −\frac{1}{12} \\ 0 & 0 & 0 & \frac{5}{12} & \frac{2}{3} & \frac{1}{12} \\ 0 & 0 & \frac{1}{8} & −\frac{3}{8} & −\frac{3}{8} & −\frac{1}{24} \\ 0 & 0 & 0 & \frac{1}{8} & \frac{1}{12} & \frac{1}{120} \end{matrix} ) ,

and

V_{3}^{−1}(0, 1, 2, 3, 4, 5) = H_{6}S_{1,0,0}S_{0,3,0} = ( \begin{matrix}1 & −\frac{1}{3} & 0 & −\frac{1}{3} & −\frac{1}{8} & −\frac{1}{120} \\ 0 & 0 & \frac{1}{3} & \frac{11}{6} & \frac{2}{3} & \frac{1}{24} \\ 0 & 0 & −\frac{5}{6} & −\frac{25}{6} & −\frac{17}{12} & −\frac{1}{12} \\ 0 & \frac{1}{3} & 1 & \frac{29}{6} & \frac{3}{2} & \frac{1}{12} \\ 0 & 0 & −\frac{5}{6} & −\frac{17}{6} & −\frac{19}{24} & −\frac{1}{24} \\ 0 & 0 & \frac{1}{3} & \frac{2}{3} & \frac{1}{6} & \frac{1}{120} \end{matrix} ) ,

where matrices S_{0,2,0} and S_{0,3,0} are the 2- and 3-generalized Stirling matrices of the second kind, their arbitrary entries are 2^{n−k}S(n, k) and 3^{n−k}S(n, k) respectively.

4 Acknowledgement

The author would like to thank the referee for his/her very useful suggestions.

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2010 Mathematics Subject Classification: Primary 11B73; Secondary 11C20. Keywords: unified generalized Stirling numbers, generalized Stirling matrices, generalized factorial matrices, recurrence, integer sequence.

(Concerned with sequences \underline{A000032} and \underline{A213593}.)

Received February 15 2012; revised version received June 15 2012. Published in Journal of Integer Sequences, June 26 2012.

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