PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 84(98) (2008), 49–60 DOI: 10.2298/PIM0898049M

ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT

Gradimir V. Milovanović, Aleksandar S. Cvetković, and Zvezdan M. Marjanović

Abstract. This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P → ℂ, where L = ∫_{−1}^{1} p(x) dμ(x), dμ(x) = (1 − x^{2})^{λ−1/2} exp(iζx) dx, and P is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational λ ∈ (−1/2, 0], give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation.

1. Introduction

In this paper we continue our investigation on orthogonality with respect to the oscillatory weight functions studied in [7], [8], and [9]. We are concerned with the following measure dμ(x) = (1 − x^{2})^{λ−1/2} exp(iζx) dx, supported on the interval [−1, 1], where λ > −1/2 and where ζ ∈ ℝ\{0}. Evidently, the corresponding weight function

w(x) = w^{ζ,λ}(x) = (1 − x^{2})^{λ−1/2} exp(iζx)

depends on two real parameters λ and ζ. We investigate the questions connected with the existence of polynomials π_{n} orthogonal with respect to the linear functional

(1.1)  L[p] = ∫_{−1}^{1} p(x) dμ(x), p ∈ P,

where P is the space of all algebraic polynomials. As it has been proved in [9] it is enough to consider only the case ζ > 0, and therefore we continue our discussion only for such values of ζ. As we proved in [9] it can be easily inferred that the orthogonal polynomials π_{n} exist in the case λ ∈ ℚ with λ > 0 and ζ is positive zero of the Bessel function of the first kind J_{λ−1}. The crucial ingredient in the proof is the result which states that non-trivial zeros of Bessel functions of the first kind are transcendental provided their index is rational. In this paper we extend the results from [9] for −1/2 < λ ⩽ 0, λ ∈ ℚ, and ζ being the zero of the Bessel function J_{λ−1}.

2000 Mathematics Subject Classification: Primary 30C10; Secondary 33C47.

The authors were supported in part by the Serbian Ministry of Science (Project: Orthogonal Systems and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320–111079 “New Methods for Quadrature”).

Also, in this paper, we investigate the possibility of computation of three-term recurrence coefficients for the given sequence of orthogonal polynomials {π_{n}}_{n∈ℕ_{0}}, as well as some consequences of the weight function w being semiclassical. We also find the second order differential equation for the corresponding orthogonal polynomials.

2. Existence of orthogonal polynomials

According to (1.1), the moments

(2.1)  μ_{k} = L[x^{k}] = ∫_{−1}^{1} x^{k}(1 − x^{2})^{λ−1/2} exp(iζx) dx, k ∈ ℕ_{0},

can be expressed in terms of Bessel functions J_{ν} of the order ν defined by

J_{ν}(z) = ∑_{m=0}^{+∞} \frac{(−1)^{m}(z/2)^{ν+2m}}{m!Γ(ν + m + 1)} .

Using the initial conditions

μ_{0} = AJ_{λ}(ζ), μ_{1} = iA (\frac{2λ}{ζ}J_{λ}(ζ) − J_{λ−1}(ζ)) ,
μ_{2} = A [(1 − \frac{2λ(2λ + 1)}{ζ^{2}}) J_{λ}(ζ) + \frac{2λ + 1}{ζ}J_{λ−1}(ζ)] ,

where A = (2/ζ)^{λ}\sqrt{π} Γ(λ + 1/2), we can see that the moments (2.1) satisfy the following recurrence relation

μ_{k+2} = −\frac{k + 2λ + 1}{iζ} μ_{k+1} + μ_{k} + \frac{k}{iζ} μ_{k−1}.

The following result is proved in [9].

Theorem 2.1. The moments μ_{k} can be expressed in the form

(2.2)  μ_{k} = \frac{A}{(iζ)^{k}} (P_{k}^{λ}(ζ)J_{λ}(ζ) + Q_{k}^{λ}(ζ)J_{λ−1}(ζ)) , k ∈ ℕ_{0},

where A = (2/ζ)^{λ}\sqrt{π} Γ(λ+1/2), and P_{k}^{λ} and Q_{k}^{λ} are polynomials in ζ, which satisfy the following four-term recurrence relation

y_{k+2} = −(k + 2λ + 1)y_{k+1} − ζ^{2}y_{k} − kζ^{2}y_{k−1},

with the initial conditions

P_{0}^{λ}(ζ) = 1, P_{1}^{λ}(ζ) = −2λ, P_{2}^{λ}(ζ) = 2λ(2λ + 1) − ζ^{2}

and

Q_{0}^{λ}(ζ) = 0, Q_{1}^{λ}(ζ) = ζ, Q_{2}^{λ}(ζ) = −(2λ + 1)ζ,

respectively.

Some obvious properties of the polynomials P_{n}^{λ} and Q_{n}^{λ}, n ∈ ℕ_{0}, are stated in the following lemma.

ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 51

Lemma 2.1. For the degree of the polynomials P_{n}^{λ} and Q_{n}^{λ} we have

deg(P_{2n}^{λ}, P_{2n+1}^{λ}) = 2n (n ∈ ℕ_{0}) and deg(Q_{2n}^{λ}, Q_{2n−1}^{λ}) = 2n (n ∈ ℕ).

The free terms in the polynomials P_{n}^{λ} and Q_{n}^{λ} are (−1)^{n}(2λ)_{n} and zero, respectively. The free term in the polynomial Q_{n}(ζ)/ζ, n ∈ ℕ, equals (−1)^{n−1}(2λ + 1)_{n−1}.

The leading coefficients in the polynomials P_{2n}^{λ}, P_{2n−1}^{λ}, Q_{2n+1}^{λ} and Q_{2n}^{λ} are (−1)^{n}, (−1)^{n}(2nλ + n − 1), (−1)^{n} and (−1)^{n}n(2λ + 1), respectively.

Now we are ready to state and prove a stronger version of the existence theorem than the one given in [9].

Theorem 2.2. Let λ ∈ ℚ, λ > −1/2 and let ζ ≠ 0 be a zero of the Bessel function J_{λ−1}. Then the sequence of monic polynomials {π_{n}}_{n∈ℕ_{0}} orthogonal with respect to the linear functional (1.1) exists.

Proof. As in [9] our result is based upon completely algebraic facts. Consider the sequence of Hankel determinants defined by

Δ_{n} = |\begin{matrix}μ_{0} & μ_{1} & ⋯ & μ_{n−1} \\ μ_{1} & μ_{2} & ⋯ & μ_{n} \\ ⋮ & ⋮ & ... & ⋮ \\ μ_{n−1} & μ_{n} & ⋯ & μ_{2n−2} \end{matrix}| , n ∈ ℕ.

The sequence of orthogonal polynomials {π_{n}}_{n∈ℕ_{0}} with respect to L exists provided Δ_{n} ≠ 0, n ∈ ℕ (see [4]). Thus, let ζ be a nontrivial zero of the Bessel function J_{λ−1}. Then we can express the moments of the linear functional L in the following form

μ_{k} = \frac{A}{(iζ)^{k}} P_{k}^{λ}(ζ)J_{λ}(ζ), k ∈ ℕ_{0}.

Obviously, from the Hankel determinant we can factor (AJ_{λ}(ζ))^{n+1}/(iζ)^{n(n+1)} and we are left with the sequence of Hankel determinants H_{n}, n ∈ ℕ, made from the new moment sequence μ′_{k} = P_{k}^{λ}(ζ), k ∈ ℕ_{0},

H_{n} = |\begin{matrix}P_{0}^{λ}(ζ) & P_{1}^{λ}(ζ) & … & P_{n−1}^{λ}(ζ) \\ P_{1}^{λ}(ζ) & P_{2}^{λ}(ζ) & … & P_{n}^{λ}(ζ) \\ ⋮ & ⋮ & & ⋮ \\ P_{n−1}^{λ}(ζ) & P_{n}^{λ}(ζ) & … & P_{2n−2}^{λ}(ζ) \end{matrix}| .

Accordingly, all H_{n}, n ∈ ℕ, are polynomials in ζ which have rational coefficients since λ ∈ ℚ. If some H_{n} = 0, then ζ is the zero of the polynomial H_{n} with rational coefficients and ζ is zero of the Bessel function J_{λ−1}. According to [13, p. 220], based on the results by [11], [12] (see also [6]), non-trivial zeros of the Bessel functions J_{λ}, λ ∈ ℚ, −λ ∉ ℕ, are transcendental and cannot be the zeros of the polynomials with rational coefficients unless polynomials are identically equal to zero.

We have to prove that determinants H_{n} are not identically equal to zero. To prove this fact we emphasize that, according to Lemma 2.1, the free coefficient in the polynomial P_{n}^{λ} equals (−1)^{n}(2λ)_{n}, n ∈ ℕ_{0}, so that the free coefficient in the polynomial H_{n} equals to the value of the Hankel determinant H̃_{n} made for the sequence of moments μ′′_{k} = (2λ)_{k}, k ∈ ℕ_{0}. For λ > 0, we can easily recognize the Hankel determinants H̃_{n} as being the same, up to multiplicative factor (Γ(2λ))^{n}, as the Hankel determinants for the generalized Laguerre weight x^{2λ−1}e^{−x} on [0, +∞). Hence, none of the determinants H̃_{n} cannot be equal according to the existence of the sequence of orthogonal polynomials with respect to the Laguerre weight function. This establishes the proof for λ > 0 (cf. [9]).

Actually, we can calculate the determinant H̃_{n} for λ > 0 in the form

H̃_{n} = ∏_{l=0}^{n−1}(n − l)! ∏_{i=0}^{l}(2λ + l − i), n ∈ ℕ.

We easily inspect that the same result about the values of the Hankel determinants H̃_{n} holds even in the case of −1/2 < λ ⩽ 0. This due to the fact that Hankel determinants H̃_{n} as being polynomials are analytic functions of λ as well as expressions representing their values. Inspecting values of the determinants H̃_{n}, n ∈ ℕ, we easily conclude that H̃_{n} ≠ 0, n ∈ ℕ, for λ ≠ 0. This in turn implies that Δ_{n} ≠ 0, n ∈ ℕ. Case λ = 0 requires a special attention.

Consider now the case λ = 0. As we easily inspect we have that H̃_{n} = 0, n ∈ ℕ, for λ = 0. Consider determinants H_{n}, we have H_{n} = (Γ(2λ))^{n}H̃_{n}, n ∈ ℕ. Since H_{n}, n ∈ ℕ, are analytic functions of λ we can compute values of H_{n}, n ∈ ℕ, for λ = 0, as the limiting values of the expressions on the right hand side. We have

lim_{λ→0} (Γ(2λ))^{n} ∏_{l=0}^{n−1}(n − l)! ∏_{i=0}^{l}(2λ + l − i) = n! ∏_{l=1}^{n−1}(n − l)! ∏_{i=0}^{l−1}(l − i) ≠ 0.

Thus, we have proved that if λ ∈ ℚ, λ > −1/2, and ζ is the zero of the Bessel function J_{λ−1}, then the sequence of Hankel determinants

Δ_{n} = (AJ_{λ}(ζ))^{n}/(iζ)^{n(n−1)}H_{n}, n ∈ ℕ,

contains no zero entry. Accordingly, the sequence of orthogonal polynomials with respect to the linear functional L exists. □

Regarding Theorem 2.2, we suppose such parameters λ and ζ which provide the existence of orthogonal polynomials π_{n} with respect to the linear functional (1.1). The quasi inner-product (p, q) := L[pq], in our case has the property (zp, q) = (p, zq), and because of that, the corresponding (monic) polynomials {π_{n}}_{n∈ℕ_{0}} satisfy the fundamental three-term recurrence relation

(2.3)  π_{n+1}(x) = (x − iα_{n})π_{n}(x) − β_{n}π_{n−1}(x), n ∈ ℕ,

with π_{0}(x) = 1, π_{−1}(x) = 0. The recursion coefficients α_{n} and β_{n} can be expressed in terms of Hankel determinants as (cf. [9]).

(2.4)  iα_{n} = \frac{Δ′_{n+1}}{Δ_{n+1}} − \frac{Δ′_{n}}{Δ_{n}} = \frac{1}{iζ} (\frac{H′_{n+1}}{H_{n+1}} − \frac{H′_{n}}{H_{n}} )
(2.5)  β_{n} = \frac{Δ_{n+1}Δ_{n−1}}{Δ_{n}^{2}} = \frac{1}{(iζ)^{2}} \frac{H_{n+1}H_{n−1}}{H_{n}^{2}} ,

where H_{n} is defined before and Δ′_{n} is the Hankel determinant Δ_{n+1} with the penultimate column and the last row removed. The corresponding determinant H′_{n} is given by

H′_{n} = |\begin{matrix}P_{0}^{λ}(ζ) & P_{1}^{λ}(ζ) & … & P_{n−2}^{λ}(ζ) & P_{n}^{λ}(ζ) \\ P_{1}^{λ}(ζ) & P_{2}^{λ}(ζ) & … & P_{n−1}^{λ}(ζ) & P_{n+1}^{λ}(ζ) \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ P_{n−1}^{λ}(ζ) & P_{n}^{λ}(ζ) & … & P_{2n−3}^{λ}(ζ) & P_{2n−1}^{λ}(ζ) \end{matrix}| .

Although β_{0} can be arbitrary, as usual it is convenient to take β_{0} = μ_{0} = AJ_{λ}(ζ).

In this case, however, the values of Hankel determinants cannot be found easily, but, it is clear that the recursion coefficients are rational functions in ζ. Using our software package [5] we can generate coefficients even in symbolic form for some reasonable small values of n (for n ⩽ 2 see Table 1).

Table 1. Recursion coefficients α_{n} and β_{n} for n ⩽ 2

\begin{matrix}n & α_{n} & β_{n} \\ 0 & \frac{2λ}{ζ} & (\frac{2}{ζ})^{λ} \sqrt{π} Γ (λ + \frac{1}{2}) J_{λ}(ζ) \\ 1 & \frac{4λ(1+λ)−ζ^{2}}{ζ(2λ−ζ^{2})} & \frac{ζ^{2}−2λ}{ζ^{2}} \\ 2 & \frac{(7+6λ)ζ^{6}−4λ(4+λ)(3+2λ)ζ^{4}+32λ^{2}(3+2λ)ζ^{2}−32λ^{3}(2+λ)}{2ζ(ζ^{2}−2λ)[ζ^{4}−λ(5+2λ)ζ^{2}+4λ^{2}]} & −\frac{2(1+2λ)[ζ^{4}−λ(5+2λ)ζ^{2}+4λ^{2}]}{ζ^{2}(−2λ+ζ^{2})^{2}} \end{matrix}

According to symbolic calculations we can conjecture that H_{n} is a polynomial in ζ^{2} of degree m,

H_{n} = S_{m}(ζ^{2}) = ∑_{ν=0}^{m} γ_{ν}(λ)ζ^{2ν},

where m = n^{2}/4 for even n and m = (n^{2} −1)/4 for odd n, so that, because of (2.5),

β_{2k} = −\frac{S_{k(k+1)}(ζ^{2})S_{k(k−1)}(ζ^{2})}{ζ^{2}S_{k^{2}}(ζ^{2})^{2}} , β_{2k+1} = −\frac{S_{(k+1)^{2}}(ζ^{2})S_{k^{2}}(ζ^{2})}{ζ^{2}S_{k(k+1)}(ζ^{2})^{2}} .

In a similar way, we can conjecture that the numerator in (2.4) is a polynomial in ζ^{2} of degree n(n + 1)/2,

H′_{n}H_{n+1} − H′_{n+1}H_{n} = V_{n(n+1)/2}(ζ^{2}),

so that

α_{2k} = \frac{V_{k(2k+1)}(ζ^{2})}{ζS_{k^{2}}(ζ^{2})S_{k(k+1)}(ζ^{2})^{2}} , α_{2k+1} = \frac{V_{(k+1)(2k+1)}(ζ^{2})}{ζS_{k(k+1)}(ζ^{2})S_{(k+1)^{2}}(ζ^{2})} .

Increasing n, the complexity of expressions for α_{n} and β_{n} increases quite rapidly. On the other side, using the Chebyshev algorithm, similarly as in [7], an efficient numerical construction of recursion coefficients can be done.

According to \overline{μ}_{k} = (−1)^{k}μ_{k}, k ∈ ℕ_{0} (see (2.2)) we can prove:

Lemma 2.2. If the sequence of monic orthogonal polynomials {π_{n}}_{n∈ℕ_{0}} exists, then π_{n}(z) = (−1)^{n}\overline{π_{n}(−\overline{z})}.

3. Differential-difference relations

In this section we consider some differential relations for (monic) polynomials {π_{n}}_{n∈ℕ_{0}} orthogonal with respect to the linear functional (1.1), i.e., to the the (quasi) inner-product product

(p, q) := L[pq] = ∫_{−1}^{1} p(x) q(x) w(x) dx,

supposing such parameters λ and ζ, which provide the existence of our orthogonal polynomials.

Our weight function w(x) = (1 − x^{2})^{λ−1/2}e^{iζx} satisfies the following Pearson type differential equation

(ϕw)′ = ψw, ϕ = 1 − x^{2}, ψ = −(2λ + 1)x + iζ(1 − x^{2}).

As before, we put dμ(x) = w(x) dx.

Theorem 3.1. For every n ∈ ℕ, we have

(3.1)  ϕπ′_{n} + ψπ_{n} = ∑_{k=n−1}^{n+2} θ_{n}^{k}π_{k},

where

θ_{n}^{n+2} = −iζ,
θ_{n}^{n+1} = ζ(α_{n+1} + α_{n}) − (n + 2λ + 1),
2θ_{n}^{n} = −(2λ + 1)iα_{n} + iζ(1 − β_{n+1} − β_{n} + α_{n}^{2}),
θ_{n}^{n−1} = (n − 1)β_{n},

and α_{n} and β_{n} are coefficients in the recurrence relation (2.3).

Proof. Since

∫_{−1}^{1}(ϕwπ_{n})′x^{k}dx = ϕwπ_{n}x^{k} |_{−1}^{1} −k ∫_{−1}^{1} ϕπ_{n}x^{k−1}dμ = −k ∫_{−1}^{1} ϕπ_{n}x^{k−1}dμ = 0

for k < n − 1, in that case we conclude that (ϕwπ_{n})′/w is orthogonal to x^{k}. Therefore, we have directly

(3.2)  \frac{(ϕwπ_{n})′}{w} = ϕπ′_{n} + ψπ_{n} = ∑_{k=n−1}^{n+2} θ_{n}^{k}π_{k},

according to the fact that deg(ψ) = 2. Since we are concerned with the monic polynomials it is readily seen that θ_{n}^{n+2} = −iζ.

For θ_{n}^{n−1}, after multiplying (3.2) with π_{n−1} and integrating with respect to dμ, we get

θ_{n}^{n−1}∥π_{n−1}∥^{2} = ∫_{−1}^{1}(ϕwπ_{n})′π_{n−1}dx = − ∫_{−1}^{1} ϕwπ_{n}π′_{n−1}dx = (n − 1)∥π_{n}∥^{2},

and accordingly θ_{n}^{n−1} = (n − 1)β_{n}.

We also have

θ_{n}^{k}∥π_{k}∥^{2} + θ_{k}^{n}∥π_{n}∥^{2} = ∫_{−1}^{1} [(ϕwπ_{n})′π_{k} + (ϕwπ_{k})′π_{n}] dx
= ∫_{1}^{1}[−ϕπ_{n}π′_{k} + ϕπ′_{k}π_{n} + ψπ_{k}π_{n}]dμ = ∫_{1}^{1} ψπ_{n}π_{k} dμ.

The last equality for k = n + 1 gives

θ_{n}^{n+1} = −\frac{θ_{n+1}^{n}}{β_{n+1}} + \frac{1}{∥π_{n+1}∥^{2}} ∫_{−1}^{1} [−(2λ + 1)x + iζ(1 − x^{2})] π_{n}π_{n+1} dμ

= −(n + 2λ + 1) − \frac{iζ}{∥π_{n+1}∥^{2}} ∫_{−1}^{1} x^{2}π_{n}π_{n+1} dμ

= −(n + 2λ + 1) + (α_{n+1} + α_{n})ζ,

because of xπ_{n} = π_{n+1} + iα_{n}π_{n} + β_{n}π_{n−1} and

∫_{−1}^{1}x^{2}π_{n}π_{n+1} dμ = ∫_{−1}^{1}(π_{n+1} + iα_{n}π_{n} + β_{n}π_{n−1})(π_{n+2} + iα_{n+1}π_{n+1} + β_{n+1}π_{n})dμ
= iα_{n+1}∥π_{n+1}∥^{2} + iα_{n}β_{n}∥π_{n}∥^{2}.

Similarly, for k = n, we get

2θ_{n}^{n} = \frac{1}{∥π_{n}∥^{2}} ∫_{−1}^{1} ψπ_{n}^{2}dμ

= \frac{1}{∥π_{n}∥^{2}} \{−(2λ + 1) ∫_{−1}^{1} xπ_{n}^{2}dμ + iζ (∥π_{n}∥^{2} − ∫_{−1}^{1} x^{2}π_{n}^{2}dμ)\}

= i[−(2λ + 1)α_{n} + ζ(1 − β_{n+1} − β_{n} + α_{n}^{2})],

because of

∫_{−1}^{1} x^{2}π_{n}^{2}dμ = ∫_{−1}^{1}(π_{n+1} + iα_{n}π_{n} + β_{n}π_{n−1})^{2}dμ
= ∥π_{n+1}∥^{2} − α_{n}^{2}∥π_{n}∥^{2} + β_{n}^{2}∥π_{n−1}∥^{2}.

Thus, the equality (3.1) is proved. □

Lemma 3.1. The monic polynomials orthogonal with respect to the linear functional (1.1) satisfy the following differential-difference equation

ϕπ′_{n} + ψπ_{n} = p_{2}^{n}π_{n} + q_{1}^{n}π_{n−1}, n ∈ ℕ,

where p_{2}^{n} and q_{1}^{n} are polynomials of the second and first degree, respectively.

Proof. Starting with equation (3.1) and using the three-term recurrence relation (2.3) we get

ϕπ′_{n} + ψπ_{n} = [(θ_{n}^{n+2}(x − iα_{n+1}) + θ_{n}^{n+1})(x − iα_{n}) − β_{n+1}θ_{n}^{n+2} + θ_{n}^{n}]π_{n}
+ [−β_{n}(θ_{n}^{n+2}(x − iα_{n+1}) + θ_{n}^{n+1}) + θ_{n}^{n−1}]π_{n−1}.

What is left to do is to read the respective polynomials. □

Theorem 3.2. The monic polynomials orthogonal with respect to the linear functional (1.1) satisfy the following equation

(3.3)  ϕπ′_{n} = p_{1}^{n}π_{n} + q_{1}^{n}π_{n−1},

where

p_{1}^{n} = −nx − \frac{iζ}{2} (1 − β_{n+1} + β_{n} + α_{n}^{2}) + \frac{iα_{n}}{2} (2n + 2λ + 1),
q_{1}^{n} = β_{n} [2(n + λ) + iζ(x + iα_{n})] .

Proof. Using expressions given in the proof of Lemma 3.1 and Theorem 3.1 we get exactly what is stated. □

Theorem 3.3. The polynomials p_{1}^{n} and q_{1}^{n}, which appear in (3.3), satisfy the following recurrence relations

p_{1}^{n+1} = −q_{1}^{n} \frac{x − iα_{n}}{β_{n}} + p_{1}^{n−1} + q_{1}^{n−1} \frac{x − iα_{n−1}}{β_{n−1}} ,
q_{1}^{n+1} = (x − iα_{n}) [p_{1}^{n} + q_{1}^{n} \frac{x − iα_{n}}{β_{n}} − p_{1}^{n−1} − q_{1}^{n−1} \frac{x − iα_{n−1}}{β_{n−1}} ] + ϕ + q_{1}^{n−1} \frac{β_{n}}{β_{n−1}} ,

respectively.

Proof. We can prove this result using almost the same arguments from the proof in [7]. Namely, we have

ϕπ′_{n+1} − ϕπ_{n} = (x − iα_{n})ϕπ′_{n} − β_{n}ϕπ′_{n−1}

= (x − iα_{n})(p_{1}^{n}π_{n} + q_{1}^{n}π_{n−1}) − β_{n}(p_{1}^{n−1}π_{n−1} + q_{1}^{n−1}π_{n−2})

= − (q_{1}^{n} \frac{x − iα_{n}}{β_{n}} − p_{1}^{n−1} − q_{1}^{n−1} \frac{x − iα_{n−1}}{β_{n−1}} ) π_{n+1}

+ \{(x − iα_{n})p_{1}^{n} + q_{1}^{n−1} \frac{β_{n}}{β_{n−1}}

+ (x − iα_{n}) (q_{1}^{n} \frac{x − iα_{n}}{β_{n}} − p_{1}^{n−1} − q_{1}^{n−1} \frac{x − iα_{n−1}}{β_{n−1}} )\} π_{n},

from where we just need to read corresponding polynomials. □

In the sequel we derive a system of nonlinear recurrence relations for the recursion coefficients α_{n} and β_{n}, which needs a few first coefficients as starting values (see Table 1).

Theorem 3.4. We have

(3.4)  p_{1}^{n+1} + p_{1}^{n} + q_{1}^{n} \frac{x − iα_{n}}{β_{n}} = (2λ − 1)x − iζ(1 − x^{2}), n ∈ ℕ.

Proof. Starting with the previous theorem we can prove that quantity

p_{1}^{n+1} + p_{1}^{n} + q_{1}^{n} \frac{x − iα_{n}}{β_{n}} ,

is independent of n. Hence, using Table 1, we can prove that

p_{1}^{2} + p_{1}^{1} + q_{1}^{1} \frac{x − iα_{1}}{β_{1}} = (2λ − 1)x − iζ(1 − x^{2}),

and hence finish the proof. □

Theorem 3.5. The three-term recurrence coefficients satisfy the following system of nonlinear recurrence equations

β_{n+2} = β_{n} + α_{n+1}^{2} − α_{n}^{2} − \frac{1}{ζ}[α_{n+1}(2n + 2λ + 3) − α_{n}(2n + 2λ − 1)],
α_{n+2} = \frac{1}{2β_{n+2}} [2α_{n}β_{n+1} + α_{n+1}(2β_{n+1} − 2β_{n} − β_{n+2} + α_{n+1}^{2} − α_{n}^{2})]

+ \frac{1}{ζβ_{n+2}} [α_{n+1}^{2}(n + λ + 1/2) − α_{n+1}α_{n}(n + λ − 1/2)

+ 2β_{n+2}(n + λ + 2) − 2β_{n+1}(n + λ) − 1]

Proof. The first equation we prove quite easily. Namely, simply compare coefficients with x^{0} in (3.4) and we have the stated equation.

For the proof of the second equation we have to use the second relation given in Theorem 3.3. In the same way, comparing coefficients with the term x^{0} in this relation we obtain the result. □

Now, we can prove a result concerning zeros of orthogonal polynomials π_{n}.

Theorem 3.6. The polynomials π_{2n+1}, n ∈ ℕ_{0}, have only simple zeros. The even polynomials π_{2n}, n ∈ ℕ, have only simple zeros except possibly one zero of the multiplicity two which is purely imaginary.

Proof. It is a trivial fact that polynomials π_{n} and π_{n−1} cannot have zeros in common. If x is multiple zero of π_{n}, using equation (3.3), it must be q_{1}^{n}(x) = 0. However, q_{1}^{n} is a polynomial of first degree and, therefore, the multiplicity of the zero can be at most two. Using a symmetry property of polynomials, we see that if x is a zero of the polynomial π_{n}, then also −\overline{x} is the zero of this polynomial π_{n}. Hence, we conclude that π_{n} cannot have a multiple zero outside of the imaginary axis. Finally, according to the symmetry we conclude that if polynomial of odd degree has zero on the imaginary axis it must be of odd multiplicity. Hence, all polynomials of odd degree have only simple zeros. □

We can now also state a result connected with a differential equation which solutions are polynomials orthogonal with respect to the functional (1.1).

Theorem 3.7. The polynomials π_{n} orthogonal with respect to the functional (1.1) satisfy the following second order differential equation

ϕq_{1}^{n}π′′_{n} + (ψq_{1}^{n} − iζβ_{n}ϕ)π′_{n} − [−nq_{1}^{n} − iζβ_{n}p_{1}^{n} − \frac{q_{1}^{n}(p̂_{1}^{n}q_{1}^{n} − p_{1}^{n}q̂_{1}^{n})}{ϕ}] π_{n} = 0,

where we used the notation from Theorem (3.2) and

p̂_{1}^{n} = −q_{1}^{n−1}/β_{n−1} and q̂_{1}^{n} = p_{2}^{n−1} + (x − iα_{n−1})q_{1}^{n−1}/β_{n−1}.

The term p̂_{1}^{n}q_{1}^{n} − p_{1}^{n}q̂_{1}^{n} must have the factor ϕ.

Proof. Starting with the equation (3.3) and the three-term recurrence relation, we compute easily

ϕπ′_{n−1} = −\frac{q_{1}^{n−1}}{β_{n−1}} π_{n} + (p_{1}^{n−1} + q_{1}^{n−1} \frac{x − iα_{n−1}}{β_{n−1}} ) π_{n−1} = p̂_{1}^{n}π_{n} + q̂_{1}^{n}π_{n−1}.

Combining this equation with the one for the polynomial π_{n}, i.e. with equation 3.3, we get

ϕ [\frac{ϕπ′_{n} − p_{1}^{n}π_{n}}{q_{1}^{n}} ]′ = p̂_{1}^{n}π_{n} + q̂_{1}^{n} \frac{ϕπ′_{n} − p_{1}^{n}π_{n}}{q_{1}^{n}} .

After some calculations we obtain the following differential equation

ϕq_{1}^{n}π′′_{n} + [q_{1}^{n}(ϕ′ − p_{1}^{n} − q̂_{1}^{n}) − ϕ(q_{1}^{n})′] π′_{n}

+ [−(p_{1}^{n})′q_{1}^{n} + p_{1}^{n}(q_{1}^{n})′ − q_{1}^{n} \frac{p̂_{1}^{n}q_{1}^{n} − p_{1}^{n}q̂_{1}^{n}}{ϕ} ] π_{n} = 0.

If we use Theorems 3.4 and 3.2, for the term with π′_{n} we got exactly what is stated. Similarly we have the same for the term with π_{n}.

Finally, it is clear that, since all other quantities are polynomials it must be that q_{1}^{n}(p̂_{1}^{n}q_{1}^{n} − p_{1}^{n}q̂_{1}^{n})/ϕ is polynomial, too. Obviously it cannot be that q_{1}^{n} has zeros in common with ϕ since it can have only purely imaginary zero. □

This theorem can be used to give the following result:

Theorem 3.8. Let n be an odd integer and let x_{ν}^{n}, ν = 1, …, n, be simple zeros of the polynomial π_{n}. Then

(3.5)  \frac{π′′_{n}(x_{ν}^{n})}{π′_{n}(x_{ν}^{n})} + \frac{λ + 1/2}{x_{ν}^{n} − 1} + \frac{λ + 1/2}{x_{ν}^{n} + 1} + iζ − \frac{1}{x_{ν}^{n} + iα_{n} + (2n + 2λ)/iζ} = 0,

for each ν = 1, …, n.

Proof. We start with the differential equation for polynomials π_{n}, derived in the previous theorem. We divide it with π′_{n}(x_{ν}^{n}) ≠ 0, ν = 1, …, n. This equality is fulfilled due to the fact that for odd n, according to Theorem 3.6, the polynomial π_{n} has only simple zeros. Thus, by division and using the fact that π_{n}(x_{ν}^{n}) = 0, ν = 1, …, n, we have

(\frac{π′′_{n}}{π′_{n}} + \frac{ψ}{ϕ} − \frac{iζβ_{n}}{q_{1}^{n}} ) (x_{ν}^{n}) = 0, ν = 1, …, n,

which after small calculation becomes exactly what is stated. □

This theorem enables the computation of zeros of orthogonal polynomial π_{n} of the high degree using technique similar as given in [10].

4. Zero distribution

Here we want to present some consequences of Theorem 3.8 in the light of some new results of Aptekarev and Van Assche (see [1]). Namely, provided that the sequence of polynomials π_{n} orthogonal with respect to the functional (1.1) exists, they have proved that for λ = 0 the three term recurrence coefficients satisfy the following asymptotic properties α_{n} → 0 and β_{n} → 1/4. Actually, we can use this fact to give some interesting observations connected to the zero distribution of polynomials π_{n}. A direct consequence of the mentioned theorem is the boundness of the zeros x_{ν}^{n}, ν = 1, …, n, of polynomials π_{n} uniformly in n ∈ ℕ (see [2], [3]).

Figure 1. Zero distribution of π_{n}, n = 10(10)100, for λ = 11 and ζ ≈ 14.47550068655454123845163765541315197630481 (a zero of J_{10})

Figure 2. Zero distribution of π_{n}, n = 10(10)100, for λ = 11 and ζ ≈ 100.71634047409274068436610723206019123535653 (a zero of J_{10})

According to the mentioned result and for n sufficiently large, our equation (3.5) becomes

\frac{π′′_{n}}{π′_{n}} (x_{ν}^{n}) + (λ + \frac{1}{2}) (\frac{1}{x_{ν}^{n} − 1} + \frac{1}{x_{ν}^{n} + 1}) + iζ + O(n^{−1}) ≈ 0,

for each ν = 1, …, n. If we compare it to the similar equation for the classical Gegenbauer weight w^{0,λ} (see [14]), we notice that there is only the difference in the term iζ + O(n^{−1}). Taking the imaginary part we get

2 ∑_{ν≠k} \frac{Im(x_{k} − x_{ν})}{|x_{k} − x_{ν}|^{2}} + (λ + \frac{1}{2}) (\frac{Im(x_{k})}{|x_{ν}^{n} − 1|^{2}} + \frac{Im(x_{k})}{|x_{k}^{n} + 1|^{2}} ) − ζ + O(n^{−1}) ≈ 0,

which describes a property of zeros. Namely, for a fixed n and bigger ζ, zeros are at the bigger distance from the x axis as presented in Figures 1 and 2.

References

[1] A. I. Aptekarev, W. V. Assche, Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Pade approximants and complex orthogonal polynomials with varying weight, J. Appox. Theory 129 (2004), 129–166.

[2] B. Beckermann, Complex Jacobi matrices, J. Comput. Appl. Math. 127 (2001), 17–65.

[3] B. Beckerman, M. Castro, On the determinacy of the comoplex Jacobi operators, Publication ANNO 446, Universite Lille, 2002.

[4] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

[5] A. S. Cvetković, G. V. Milovanović, The Mathematica package “OrthogonalPolynomials”, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36.

[6] N. I. Fel’dman, A. B. Šidlovskiĭ, The development and present state of the theory of transcendental numbers, Uspehi Mat. Nauk 22:3 (1967), (135) 3–81 (Russian) [Engl. transl. in Russian Math. Surveys 22:3 (1967), 1–79].

[7] G. V. Milovanović, A. S. Cvetković, Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions, J. Comput. Appl. Math. 179 (2005), 263–287.

[8] G. V. Milovanović, A. S. Cvetković, Orthogonal polynomials related to the oscillatory-Chebyshev weight function, Bull. Cl. Sci. Math. Nat. Sci. Math. 30 (2005), 47–60.

[9] G. V. Milovanović, A. S. Cvetković, M. P. Stanić, Orthogonal polynomial and quadrature rules realted to Oscillatory-Gegenabuer weight function (submitted; preprint version available at:

http://gauss-megatrend.edu.rs/radovi/GVM158.pdf).

[10] L. Pasquini, Accurate computation of the zeros of the generalized Bessel polynomials, Numer. Math. 86 (2000), 507–538.

[11] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss., Phys.-Math. 1 (1929), 1–70 [Gesammelte Abhandlungen. Springer-Verlag, Berlin–New York 1966 Band I, 209–266].

[12] C. L. Siegel, Transcendental Numbers, Annals of Mathematics Studies 16, Princeton University Press, Princeton, N. J., 1949.

[13] A. B. Šidlovskiĭ, Transcendental Numbers, Nauka, Moscow, 1987 (in Russian).

[14] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, 4th ed., Amer. Math. Soc., Providence, R. I., 1975.

Faculty of Computer Sciences, Megatrend University Beograd, Serbia

gvm@megatrend-edu.net

Faculty of Science and Mathematics, University of Niš Niš, Serbia

aca@elfak.ni.ac.yu

Faculty of Electronic Engineering, University of Niš Niš, Serbia

zvezdan@elfak.ni.ac.yu