Hankel Determinant for a Sequence that Satisfies a Three-Term Recurrence Relation

Baghdadi Aloui Faculty of Sciences of Gabes Department of Mathematics City of Erriadh 6072 Gabes Tunisia

                   Baghdadi.Aloui@fsg.rnu.tn

Abstract

In this paper, we give an explicit expression for the Hankel determinants for a sequence satisfying a three-term recurrence relation. Our method of evaluation is based on the theory of orthogonal polynomials. In particular, we analyze the linear functional associated with such a sequence.

1 Introduction

The problem of evaluation of Hankel determinants has emerged in the 19th century. These classes of determinants are important in many mathematical areas, such as number theory, integrable systems, approximation theory, and linear system theory, among others. Several authors have focused on the evaluation of the Hankel determinants using different methods. Here, we can cite Radoux [11] who evaluated Hankel determinants by using methods based primarily on matrix decomposition techniques. Krattenthaler [6] used combinatorial techniques and graph theory to evaluate a large class of these determinants. Egecioglu et al. [4, 5] evaluated Hankel determinants with binomial coefficient entries by using a technique based on the solution of differential equations. Chammam et al. [2] dealt with Hankel determinants with polynomial coefficient entries by using a method based mainly on the theory of orthogonal polynomials.

For instance, let (a_{n})_{n≥0} be a sequence satisfying the three-term recurrence relation

a_{n+1} = −\frac{2n + 1}{n + 1} a_{n} − a_{n−1}, n ≥ 0,

with initial values a_{0} = 1, a_{−1} = 0.

In this work, we are interested in the evaluation of the Hankel determinants

H_{n} := det [a_{i+j}]_{i,j=0}^{n} = | \begin{matrix}a_{0} & a_{1} & ⋯ & a_{n} \\ a_{1} & a_{2} & ⋯ & a_{n+1} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{n} & a_{n+1} & ⋯ & a_{2n} \end{matrix} | , n ≥ 0.

Our method of evaluation of such determinants is based on the study of the linear functional L associated with the sequence (a_{n})_{n≥0}, given by (L)_{n} = a_{n}, n ≥ 0. Mainly, we show that L is a solution of a functional first-order linear differential equation. Under certain conditions, L is related to the classical linear functional of Bessel (with parameter α = 1) by the relation L = −x(h_{− \frac{1}{2}} ∘ τ_{2}B^{(1)})′. Sfaxi and Alaya [12] gave special kinds of linear functionals that allow to us to obtain a necessary and sufficient condition for the quasi-definiteness of the linear functional L. By using the theory of orthogonal polynomials, this allows us to give an explicit expression of our Hankel determinants.

The paper is organized as follows. In Section 2, we introduce the basic background and notations to be used throughout the paper. In Section 3, first we deduce a necessary and sufficient condition for the quasi-definiteness of the linear functional L. Finally, we give an explicit expression of the Hankel determinants H_{n} := det [a_{i+j}]_{i,j=0}^{n}, n ≥ 0.

2 Notation and preliminary results

2.1 Orthogonality and semi-classical character

Let ℙ be the linear space of polynomials in one variable with complex coefficients and ℙ′ its algebraic dual space. We denote by ⟨𝒰 , p⟩ the action of 𝒰 ∈ ℙ′ on p ∈ ℙ and by (𝒰 )_{n} := ⟨𝒰 , x^{n}⟩, n ≥ 0, the sequence of moments of 𝒰 with respect to the polynomial sequence {x^{n}}_{n≥0}. Let us define the following operations in ℙ′: for linear functionals 𝒰 and 𝒱 , any polynomial q, and any (a, b, c) ∈ ℂ^{∗} × ℂ^{2}, let D𝒰 = 𝒰′, q𝒰 , (x − c)^{−1}𝒰 , τ_{−b}𝒰 and h_{a}𝒰 be the linear functionals defined by duality [10]:

⟨𝒰′, p⟩ := −⟨𝒰 , p′⟩, ⟨q𝒰 , p⟩ := ⟨𝒰 , qp⟩,

⟨(x − c)^{−1}𝒰 , p⟩ := ⟨𝒰 , θ_{c}p⟩ = ⟨𝒰 , \frac{p(x) − p(c)}{x − c}⟩,

⟨τ_{−b}𝒰 , p⟩ := ⟨𝒰 , τ_{b}p⟩ = ⟨𝒰 , p(x − b)⟩,

⟨h_{a}𝒰 , p⟩ := ⟨𝒰 , h_{a}p⟩ = ⟨𝒰 , p(ax)⟩, p ∈ ℙ.

A linear functional 𝒰 is called normalized if it satisfies (𝒰 )_{0} = 1.

With any sequence of complex numbers (a_{n})_{n≥0}, we can associate a unique linear functional 𝒰 ∈ ℙ′ given by (𝒰 )_{n} = a_{n}, n ≥ 0.

The linear functional 𝒰 is said to be quasi-definite (regular) if the Hankel determinant H_{n}(𝒰 ) = det[a_{i+j}]_{i,j=0}^{n} ≠ 0, for every integer n ≥ 0. In such a case, there exists a unique sequence of monic polynomials (SMP) {P_{n}}_{n≥0}, i.e., deg P_{n} = n and their leading coefficients are equal to 1, such that

⟨𝒰 , x^{ν}P_{n}⟩ = 0, 0 ≤ ν ≤ n − 1 and ⟨𝒰 , x^{n}P_{n}⟩ ≠ 0.

The sequence {P_{n}}_{n≥0} is said to be the sequence of monic orthogonal polynomials (SMOP) with respect to 𝒰 .

Notice that P_{n}(x) can be represented by the determinantal formula

P_{0}(x) = 1, P_{n}(x) = \frac{1}{H_{n−1}(𝒰 )} | \begin{matrix}a_{0} & a_{1} & ⋯ & a_{n} \\ a_{1} & a_{2} & ⋯ & a_{n+1} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{n−1} & a_{n} & ⋯ & a_{2n−1} \\ 1 & x & ⋯ & x^{n} \end{matrix} | , n ≥ 1.

The orthogonality of {P_{n}}_{n≥0} can be characterized by a three-term recurrence relation (TTRR, in short) [3]

P_{n+1}(x) = (x − β_{n})P_{n}(x) − γ_{n}P_{n−1}(x), n ≥ 0,

with initial values P_{0}(x) = 1, P_{−1}(x) = 0, where {β_{n}}_{n≥0} and {γ_{n}}_{n≥0} are sequences of complex numbers such that γ_{n} ≠ 0, n ≥ 1, and the convention γ_{0} = (𝒰 )_{0}.

Furthermore,

β_{n} = \frac{⟨𝒰 , xP_{n}^{2}⟩}{⟨𝒰 , P_{n}^{2}⟩} , γ_{n+1} = \frac{⟨𝒰 , P_{n+1}^{2}⟩}{⟨𝒰 , P_{n}^{2}⟩} , n ≥ 0,  (1)
⟨𝒰 , P_{n}^{2}⟩ = ∏_{ν=0}^{n} γ_{ν} = \frac{H_{n}(𝒰 )}{H_{n−1}(𝒰 )} , n ≥ 0, (H_{−1}(𝒰 ) = 1),  (2)

H_{n}(𝒰 ) = ∏_{k=0}^{n} ∏_{ν=0}^{k} γ_{ν}, n ≥ 0.  (3)

When {P_{n}}_{n≥0} is a SMOP with respect to a linear functional 𝒰 , then the sequence of monic polynomials {P̃_{n}}_{n≥0}, where P̃_{n}(x) = a^{−n}P_{n}(ax + b), is also orthogonal with respect to 𝒰̃ = (h_{a^{−1}} ∘ τ_{−b})𝒰 , and satisfies the following TTRR [8, 9, 10]

P̃_{n+1}(x) = (x − β̃_{n})P̃_{n}(x) − γ̃_{n}P̃_{n−1}(x), n ≥ 0,

with initial values P̃_{0}(x) = 1, P̃_{−1}(x) = 0, and where

β̃_{n} = a^{−1}(β_{n} − b), n ≥ 0,
γ̃_{n+1} = a^{−2}γ_{n+1}, n ≥ 0, (γ̃_{0} = (𝒰 )_{0}).

As a consequence, we get the following result.

Lemma 1. Let {a_{n}}_{n≥0} be a sequence of complex numbers and 𝒰 ∈ ℙ′ such that (𝒰 )_{n} = a_{n}, n ≥ 0. Then, for any pair (a, b) ∈ ℂ^{∗} × ℂ, we have

H_{n}((h_{a^{−1}} ∘ τ_{−b})𝒰 ) = a^{−n(n+1)}H_{n}(𝒰 ), n ≥ 0,

where the moments of the shifted linear functional (h_{a^{−1}} ∘ τ_{−b})𝒰 are given by

((h_{a^{−1}} ∘ τ_{−b})𝒰 )_{n} = a^{−n} ∑_{k=0}^{n} \binom{n}{k}(−b)^{n−k}a_{k}, n ≥ 0.

A quasi-definite linear functional 𝒰 is said to be semi-classical if it satisfies a functional equation (Pearson equation)

(Φ𝒰 )′ + Ψ𝒰 = 0,  (4)

where Φ and Ψ are polynomials such that Φ is monic and deg(Ψ) ≥ 1.

The corresponding SMOP {P_{n}}_{n≥0} is said to be semi-classical (for more details, see [1, 8, 10] and the literature therein).

If for each zero c of Φ we have

| Φ′(c) + Ψ(c) | + | ⟨𝒰 , θ_{c}^{2}Φ + θ_{c}Ψ⟩ |> 0,  (5)

then the nonnegative integer s := max{deg(Φ) − 2, deg(Ψ) − 1} is said to be either the class of 𝒰 or the class of {P_{n}}_{n≥0}.

The semi-classical character of a linear functional is invariant by shifting. Indeed, if 𝒰 is a semi-classical linear functional of class s satisfying (4) and (5), then for any pair (a, b) ∈ ℂ^{2} with a ≠ 0, the shifted linear functional 𝒰̃ = (h_{a^{−1}} ∘ τ_{−b})𝒰 is also semi-classical of class s and satisfies (Φ̃𝒰̃)′ + Ψ̃𝒰̃ = 0 with Φ̃(x) = a^{−t}Φ(ax + b), Ψ̃(x) = a^{1−t}Ψ(ax + b) and where t = deg Φ.

We find in [1, 9] a full description of the class s = 0. This corresponds to the classical linear functionals (Hermite, Laguerre, Bessel and Jacobi).

Further, we need the following properties of the monic Bessel polynomials {B_{n}^{(α)}}_{n≥0}, (orthogonal with respect to the linear functional B^{(α)}) [3, 7].

B_{n}^{(α)}(x) = ∑_{ν=0}^{n} \binom{n}{ν}\frac{2^{n−ν}Γ(n + 2α + ν − 1)}{Γ(2n + 2α − 1)} x^{ν}, n ≥ 0, (α ≠ −\frac{n}{2}, n ≥ 0).

β_{0} = −\frac{1}{α}, β_{n} = \frac{1 − α}{(n + α − 1)(n + α)}, n ≥ 0,

γ_{n} = −\frac{n(n + 2α − 2)}{(2n + 2α − 3)(n + α − 1)^{2}(2n + 2α − 1)}, n ≥ 1.

Φ(x) = x^{2}, Ψ(x) = −2(αx + 1).

(B^{(α)})_{n} = \frac{(−2)^{n}Γ(2α)}{Γ(n + 2α)}, n ≥ 0.

H_{n}(B^{(α)}) = (−4)^{\frac{n(n+1)}{2}} ∏_{k=0}^{n} k!\frac{Γ(k + 2α − 1)Γ(2α)}{Γ(2k + 2α − 1)Γ(2k + 2α)}, n ≥ 0.

Table 1: Some basic characteristics of Bessel polynomials.

2.2 Quasi-definiteness condition of the linear functional −(x−c)𝒞′

Let {S_{n}}_{n≥0} be a classical sequence of monic polynomials, orthogonal with respect to a normalized linear functional 𝒞 that satisfies

(Φ𝒞)′ − (λ + Φ′)𝒞 = 0,  (6)

where Φ is monic, deg Φ = 2, and λ ≠ 0. In other words, 𝒞 is either a shifted Jacobi or a shifted Bessel linear functional.

We have the following results.

Lemma 2. [12] For any c ∈ ℂ where Φ(c) ≠ 0, the following statements are equivalent.

(i) The linear functional −(x − c)𝒞′ is quasi-definite.

(ii) A_{n+1}(c) ≠ 0, n ≥ 0, where A_{n+1}(x) = Φ(x)S′_{n}(x) + λS_{n}(x), n ≥ 0.

Lemma 3. [12] Let 𝒞 be a classical linear functional satisfying (6). When it is quasidefinite, the linear functional w_{0}(c) = −(x − c)𝒞′ is semi-classical of class one and satisfies (φw_{0}(c))′ + ψw_{0}(c) = 0, where φ(x) = (x − c)Φ(x) and ψ(x) = −(2Φ(x) + λ(x − c)).

Under the hypothesis of the previous lemmas, let {Q_{n}}_{n≥0} be the SMOP with respect to w_{0}(c). Suppose that the SMOP {S_{n}}_{n≥0} satisfies

S_{n+1}(x) = (x − ξ_{n})S_{n}(x) − ρ_{n}S_{n−1}(x), n ≥ 0,

with initial values S_{0}(x) = 1, S_{−1}(x) = 0, where ρ_{n} ≠ 0, n ≥ 1.

Then the SMOP {Q_{n}}_{n≥0} satisfies the TTRR [12]

Q_{n+1}(x) = (x − α_{n})Q_{n}(x) − η_{n}Q_{n−1}(x), n ≥ 0,  (7)

with initial values Q_{0}(x) = 1, Q_{−1}(x) = 0, and where

α_{0} = 2ζ_{0} − c, α_{1} = c − \frac{2λρ_{1}}{A_{2}(c)},

α_{n} = c − \frac{(n + 1)}{n} \frac{A_{n}(c)}{A_{n+1}(c)}ρ_{n} − \frac{n − 1}{n} \frac{A_{n+1}(c)}{A_{n}(c)}, n ≥ 2,

η_{1} = −A_{2}(c), η_{2} = λρ_{1} \frac{A_{3}(c)}{A_{2}^{2}(c)},

η_{n+1} = \frac{A_{n+2}(c)A_{n}(c)}{A_{n+1}^{2}(c)}ρ_{n}, n ≥ 2.

The following result is a straightforward consequence of (3) and (7).

Theorem 4. Let {S_{n}}_{n≥0} be a classical polynomial sequence, orthogonal with respect to a normalized linear functional 𝒞 satisfying (Φ𝒞)′−(λ+Φ′)𝒞 = 0, where Φ is monic, deg Φ = 2, and λ ≠ 0. Then, for any c ∈ ℂ such that Φ(c) ≠ 0 and A_{n+1}(c) := Φ(c)S′_{n}(c) + λS_{n}(c) ≠ 0, n ≥ 0, we get

H_{n}( − (x − c)𝒞′) = (−1)^{n}λ^{n−1}A_{n+1}(c)H_{n−1}(𝒞) ≠ 0, n ≥ 0, with H_{−1}(𝒞) = 1.

3 Expression of the Hankel determinants

Let (a_{n})_{n≥0} be the sequence given by

a_{n+1} = −\frac{2n + 1}{n + 1} a_{n} − a_{n−1}, n ≥ 0,  (8)

with initial values a_{0} = 1 and a_{−1} = 0.

We can associate with the sequence (a_{n})_{n≥0} a unique linear functional L given by its moments with respect to the monomial sequence {x^{n}}_{n≥0},

(L)_{n} = a_{n}, n ≥ 0.  (9)

The aim of this section is to evaluate the Hankel determinants of the linear functional L, i.e., H_{n}(L) = det [a_{i+j}]_{i,j=0}^{n}, n ≥ 0. To do it, we need, first, to establish some useful properties for the linear functional L.

Using (8) and (9), we get

(L)_{n+1} + \frac{2n + 1}{n + 1}(L)_{n} + (L)_{n−1} = 0, n ≥ 0,  (10)

with (L)_{0} = 1 and (L)_{−1} = 0.

From the above relation for the moments, the linear functional L satisfies

(φL)′ + ψL = 0,

where φ(x) = x(x + 1)^{2} and ψ(x) = −(2x^{2} + 3x + 2).

We have the following fondamental result.

Proposition 5. The linear functional L is related to a classical linear functional B^{(1)} (the Bessel form with parameter α = 1), by the following relation

L = −x((h_{− \frac{1}{2}} ∘ τ_{2})B^{(1)})′.

Proof. Let V = −x((h_{− \frac{1}{2}} ∘ τ_{2})B^{(1)})′. We show that (V)_{n}, n ≥ 0, also satisfies (10). Indeed, we have

(V)_{n} = (−1)^{n}2^{−n}(n + 1)⟨B^{(1)}, (x + 2)^{n}⟩, n ≥ 0.  (11)

By Table 1, the linear functional B^{(1)} satisfies (x^{2}B^{(1)})′ − 2(x + 1)B^{(1)} = 0. Then, we obtain

⟨(x^{2}B^{(1)})′ − 2(x + 1)B^{(1)}, (x + 2)^{n}⟩ = 0, n ≥ 0.

Equivalently

(n + 2)⟨B^{(1)}, (x + 2)^{n+1}⟩ − 2(2n + 1)⟨B^{(1)}, (x + 2)^{n}⟩ + 4n⟨B^{(1)}, (x + 2)^{n−1}⟩ = 0, n ≥ 0.

By using (11), it follows that

(V)_{n+1} + \frac{2n + 1}{n + 1} (V)_{n} + (V)_{n−1} = 0, n ≥ 0, with (V)_{0} = 1, (V)_{−1} = 0.

We, thus, obtain V = L.

Under the hypothesis of the previous result and from Lemma 2, we are going to establish a necessary and sufficient condition for the quasi-definiteness of the linear functional L. Next, according to Theorem 4, we will give an explicit expression of our Hankel determinants.

Let {S_{n}}_{n≥0} be the SMOP with respect to L, and let {A_{n+1}}_{n≥0} be the sequence of polynomials given by

A_{n+1}(x) := Φ(x)S′_{n}(x) + λS_{n}(x).

By Proposition 5, we have L = −x((h_{− \frac{1}{2}} ∘τ_{2})B^{(1)})′, that satisfies (6) with Φ(x) = (x+1)^{2}

and λ = −1 ≠ 0. In this case, S_{n}(x) = (\frac{−1}{2})^{n}B_{n}^{(1)}( − 2(x + 1)), n ≥ 0, where \{B_{n}^{(1)}(x)\}_{n≥0} is the Bessel SMOP with parameter α = 1. By using to Table 1, we get

S_{n}(x) = ∑_{ν=0}^{n}(−1)^{ν}\binom{n}{ν}\frac{(2n − ν)!}{(2n)!}(x + 1)^{n−ν}, n ≥ 0.

According to Lemma 2, the linear functional L is quasi-definite if and only if A_{n+1}(0) := S′_{n}(0) − S_{n}(0) ≠ 0, n ≥ 0, i.e., ∑_{ν=0}^{n+1} λ_{n+1,ν} ≠ 0, where

λ_{n+1,0} = n,

λ_{n+1,ν} = \frac{(−1)^{ν}(2n − ν + 1)!\binom{n}{ν−1}}{(2n)!}[\frac{(n − ν)(n − ν + 1)}{(2n − ν + 1)ν} + 1], 1 ≤ ν ≤ n + 1.

On the other hand, we can deduce the following result.

Theorem 6. The Hankel determinants with coefficient (a_{n})_{n≥0}, are given by

H_{n} := det [a_{i+j}]_{i,j=0}^{n} = (−1)^{\frac{(n−2)(n+1)}{2}} A_{n+1}(0) ∏_{k=0}^{n−1} \frac{(k!)^{2}}{(2k)!(2k + 1)!},

for any non-negative integer number n and the convention ∏_{k=0}^{−1} = 1.

Proof. From Table 1 and Lemma 1, we get for n ≥ 0,

H_{n}((h_{− \frac{1}{2}} ∘ τ_{2})B^{(1)}) = (\frac{−1}{2})^{n(n+1)}H_{n}(B^{(1)}) = (−1)^{\frac{n(n+1)}{2}} ∏_{k=0}^{n} \frac{(k!)^{2}}{(2k)!(2k + 1)!}.

According to Theorem 4, such that A_{n+1}(0) ≠ 0, n ≥ 0, we obtain the desired result.

4 Acknowledgements

The author would like to thank the anonymous referee for their valuable comments and suggestions for improving the original version of this manuscript.

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2010 Mathematics Subject Classification: Primary 33C45; Secondary 42C05. Keywords: Orthogonal polynomial, three-term recurrence relation, semi-classical linear functional, Hankel determinant, Bessel polynomial.

Received October 22 2014; revised versions received December 15 2014; December 18 2014. Published in Journal of Integer Sequences, January 7 2015.

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