PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 73–83 DOI: https://doi.org/10.2298/PIM161115002M

THE REMAINDER TERM OF GAUSS–RADAU QUADRATURE RULE WITH SINGLE AND DOUBLE END POINT

Ljubica Mihić

Abstract. The remainder term of quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours for Gauss–Radau quadrature formula with the Chebyshev weight function of the second kind with double and single end point. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where the maximum modulus of the kernel is attained.

1. Gauss–Radau quadrature rule with double end point

In this section, we analyze the remainder term for the Gauss–Radau quadrature rule with the end point −1 of multiplicity r

(1.1)  ∫_{−1}^{1} f(t) ω(t) dt = ∑_{ρ=0}^{r−1} κ_{ρ}^{R}f^{(ρ)}(−1) + ∑_{ν=1}^{n} λ_{ν}^{R}f(τ_{ν}^{R}) + R_{n,r}^{R} (f),

where τ_{ν}^{R} are zeros of π_{n}(·; ω^{R}), orthogonal polynomial on [−1, 1], with respect to the weight function

ω^{R}(t) = (t + 1)^{r}ω(t).

Here, R_{n,r}^{R} (f) = 0 for all f ∈ ℙ_{2n+2r−1} (polynomials of degree ⩽ 2n + 2r − 1). Let Γ be a simple closed curve in the complex plane surrounding the interval [−1, 1] and let D = intΓ be its interior. If the integrand f is analytic in a domain D containing [−1, 1], then the remainder term R_{n,r}^{R} (f) admits the contour integral representation

(1.2)  R_{n,r}^{R} (f) = \frac{1}{2πi} ∮_{Γ} K_{n,r}^{R} (z; ω)f(z) dz.

2010 Mathematics Subject Classification: Primary 41A55; Secondary 65D30, 65D32.

Key words and phrases: Gauss–Radau quadrature formula; Chebyshev weight function; remainder term.

Communicated by Gradimir Milovanović.

The kernel is given by

K_{n,r}^{R} (z; ω) ≡ K_{n,r} (z, ω) = \frac{ϱ_{n,r}^{R} (z; ω)}{(z + 1)^{r}π_{n}(z; ω^{R})} , z ∉ [−1, 1],

where we denote w_{n,r} (z; ω) = (z + 1)^{r}π_{n}(z; ω^{R}). Also,

ϱ_{n,r}^{R} (z; ω) ≡ ϱ_{n,r} (z, ω) = ∫_{−1}^{1} \frac{w_{n,r} (z; ω)}{z − t} ω(t)dt.

Integral representation (1.2) leads to the error bound

|R_{n,r}^{R} (f)| ⩽ \frac{ℓ(Γ)}{2π}( max_{z∈Γ}|K_{n,r} (z; ω)|) ( max_{z∈Γ}|f(z)|) ,

where ℓ(Γ) is the length of the contour Γ. In this paper we take Γ = E_{ρ}, where the ellipse E_{ρ} is given by

(1.3)  E_{ρ} = \{z ∈ ℂ | z = \frac{1}{2}(u + u^{−1} ), 0 ⩽ θ ⩽ 2π\}, u = ρ e^{iθ}.

The upper bound on |R_{n,r}^{R} (f)| reduces to

|R_{n,r}^{R} (f)| ⩽ \frac{ℓ(E_{ρ})}{2π}( max_{z∈E_{ρ}}|K_{n,r} (z; ω)|) ( max_{z∈E_{ρ}}|f(z)|) .

Furthermore, we take r = 2, meaning we are dealing with a double end point. The goal is to determine the points where the kernel attains its maximum modulus along the contour of integration. In [2] Gautschi and Li considered the Gauss– Radau and the Gauss–Lobatto quadrature rules with multiple end points with respect to four Chebyshev weight functions and derived explicit expressions of the corresponding kernels K_{n,r} (z; ω_{j}) in terms of the variable u = ρe^{iθ}.

1.1. Maximum modulus of the kernel. Gautschi and Li [2, Section 3.3] analyzed the maximum modulus of the kernel K_{n,2} (z; ω_{2}). Based on numerical calculations, they made the conjectures that the maximum is attained on the negative real axis if (i) ρ > 1 and 1 ⩽ n ⩽ 11, and (ii) ρ ⩾ ρ_{n} and n ⩾ 12.

Here, ρ_{n} are numbers determined for 12 ⩽ n ⩽ 20. We can merge the cases from the previous conjectures. The maximum modulus of the kernel is attained on the negative real axis if ρ > ρ^{∗} and n ⩾ 1, where ρ^{∗} = 1 if 1 ⩽ n ⩽ 11, while ρ^{∗} = ρ_{n} if n ⩾ 12.

In this paper we prove the existence of the values ρ^{∗} from the previous conjecture. We give the strong numerical evidence for the precise values of ρ^{∗} for n ⩾ 12. Gautschi and Li [2, (2.7)] derived the explicit representation of the kernel

K_{n,2} (z; ω_{2}) = \frac{π(u^{2} − 1)}{u^{n+4}}

× \frac{u^{2} + αu + β}{β[u^{n+3} − u^{−(n+3)} ] + α[u^{n+2} − u^{−(n+2)} ] + [u^{n+1} − u^{−(n+1)} ]} ,

where α = \frac{4(n+1)}{2n+5} , β = \frac{(n+1)(2n+3)}{(n+3)(2n+5)} , z = (u + u^{−1} )/2 and u = ρe^{iθ}. We can determine the modulus of the kernel on E_{ρ}. We are also interested in the modulus of the kernel at θ = π. By introducing some substitutions, we can easily express the modulus of the kernel in the form

|K_{n,2} (z; ω_{2})| = ( \frac{π^{2}}{ρ^{2n+8}} \frac{ac}{δ})^{1/2} ,

where

a = |u^{2} − 1|^{2} = ρ^{4} − 2ρ^{2} cos2θ + 1,
c = |u^{2} + αu + β|^{2} = ρ^{4} + 2α cosθρ^{3} + (α^{2} + 2β cos2θ)ρ^{2} + 2αβ cosθρ + β^{2},
δ = |β[u^{n+3} − u^{−(n+3)} ] + α[u^{n+2} − u^{−(n+2)} ] + [u^{n+1} − u^{−(n+1)} ]|^{2} = \frac{d}{ρ^{2n+6}} ,

i.e.,

d = δ · ρ^{2n+6}
= |β[u^{n+3} − u^{−(n+3)} ] + α[u^{n+2} − u^{−(n+2)} ] + [u^{n+1} − u^{−(n+1)} ]|^{2} · ρ^{2n+6}
= β^{2} · ρ^{4n+12} + 2αβ cosθ · ρ^{4n+11} + [α^{2} + 2β cos2θ] · ρ^{4n+10}
+ 2α cosθ · ρ^{4n+9} + ρ^{4n+8} − 2β cos(2n + 4)θ · ρ^{2n+8}
− [2α cos(2n + 3)θ + 2αβ cos(2n + 5)θ] · ρ^{2n+7}
− [2 cos(2n + 2)θ + 2β^{2} cos(2n + 6)θ + 2α^{2} cos(2n + 4)θ] · ρ^{2n+6}
− [2αβ cos(2n + 5)θ + 2α cos(2n + 3)θ] · ρ^{2n+5} − 2β cos(2n + 4)θ · ρ^{2n+4}
+ ρ^{4} + 2α cosθ · ρ^{3} + [α^{2} + 2β cos2θ] · ρ^{2} + 2αβ cosθ · ρ + β^{2}.

In order to express d(ρ) as a polynomial function in ρ, the term δ is multiplied by ρ^{2n+6} , which reduces the expression for the square of the modulus of the kernel to

|K_{n,2} (z; ω_{2})|^{2} = \frac{π^{2}}{ρ^{2}} \frac{ac}{d}.

By letting A, C, D denote the values of a, c, d at the angle θ = π, the square of the modulus of the kernel at θ = π can be expressed as

|K_{n,2} (z; ω_{3})|^{2} = \frac{π^{2}}{ρ^{2}} \frac{AC}{D}.

The following substitutions are appropriate

A = ρ^{4} − 2ρ^{2} + 1,
C = ρ^{4} − 2α · ρ^{3} + (α^{2} + 2β) · ρ^{2} − 2αβ · ρ + β^{2},
D = β^{2} · ρ^{4n+12} − 2αβ · ρ^{4n+11} + (α^{2} + 2β) · ρ^{4n+10}
− 2α · ρ^{4n+9} + ρ^{4n+8} − 2β · ρ^{2n+8} + (2α + 2αβ) · ρ^{2n+7}
− (2 + 2β^{2} + 2α^{2}) · ρ^{2n+6} + (2αβ + 2α) · ρ^{2n+5} − 2β · ρ^{2n+4}
+ ρ^{4} − 2α · ρ^{3} + (α^{2} + 2β) · ρ^{2} − 2αβ · ρ + β^{2}.

According to Gautschi and Li’s conjecture, there exist some value ρ^{∗} such that the maximum modulus of the kernel is attained at θ = π for all ρ > ρ^{∗} and n ⩾ 1. In the case of conjecture (i) ρ^{∗} = 1, while in the case of conjecture (ii) ρ^{∗} = ρ_{n}.

We formulate the following theorem which states the existence of the value ρ^{∗}. Whereas that conjectures hold for all ρ from the interval [ρ^{∗}, ∞), we separately derive a detailed numerical study.

Theorem 1.1. For the Gauss–Radau quadrature formula with a double end point −1 (r = 2) with the Chebyshev weight function of the second kind, there exists a value ρ^{∗} ∈ [1, ∞), such that the modulus of the kernel |K_{n,2} (z; ω_{2})| attains its maximum value on the negative real axis (θ = π) for ρ > ρ^{∗} and n ⩾ 1, i.e.,

max_{z∈E_{ρ}}|K_{n,2} (z; ω_{2})| = |K_{n,2} ( − \frac{1}{2}(ρ + ρ^{−1} ), ω_{2})|;

for ρ > ρ^{∗}, n ⩾ 1.

Proof. i) Referring to the previously introduced notation, we have to show that \frac{ac}{d} ⩽ \frac{AC}{D} for each ρ > ρ^{∗} and n ⩾ 1. The previous inequality can be written as I(ρ) = [acD − ACd] ⩽ 0. We can easily see that I(ρ) is a polynomial in ρ, of degree equal to 4n + 19, whose coefficients depend only on θ, i.e.,

I = I(ρ) = ∑_{i=0}^{4n+19} a_{i}(θ)ρ^{i}. (1.4)

In order to show the existence of numbers ρ^{∗}, we use the well known fact that, starting from some value of ρ, the sign of the polynomial

I(ρ) = ρ^{4n+19} ( a_{4n+19} + \frac{a_{4n+18}}{ρ} + \frac{a_{4n+17}}{ρ^{2}} + ⋯+ \frac{a_{0}}{ρ^{4n+19}} )

coincides with the sign of the leading coefficient a_{4n+19} = 2αβ(1 + cosθ)(β − 1), where α = 4\frac{n+1}{2n+5} and β = \frac{(n+1)(2n+3)}{(n+3)(2n+5)} . Therefore,

a_{4n+19} < 0 iff β < 1 iff (n + 1)(2n + 3) < (n + 3)(2n + 5).

The previous inequality reduces to n > −2. We conclude that the term a_{4n+19} is negative for all n ⩾ 1, i.e., for all n ⩾ 1 there exists a number ρ^{∗} such that I(ρ) ⩽ 0 for all ρ > ρ^{∗}. □

1.2. Gautschi and Li’s conjecture. According to the conjecture, the maximum is attained at θ = π for all ρ > ρ^{∗} and n ⩾ 1. In order to ensure the nonpositivity of polynomial I(ρ) given by (1.4) for each ρ > ρ^{∗}, we can write the initial polynomial in the terms of positive differences ρ − ρ^{∗}, and show the nonpositivity of its new coefficients. We have

J(ρ) = ∑_{i=0}^{4n+19} b_{i}(θ, ρ^{∗})(ρ − ρ^{∗})^{i} for all ρ > ρ^{∗}.

Numerical calculations show that all functions b_{i}(θ, ρ^{∗}), i = 0, 1, …, 4n + 19 are strictly under the x-axis for all θ. In general, nonpositivity of the coefficients b_{i}(θ, ρ^{∗}) is not a necessary condition for nonpositivity of a polynomial for each ρ > ρ^{∗}, but in this case, it is obviously a sufficient condition.

Explicit formulae for b_{i}(θ, ρ^{∗}) can be given in the terms of the coefficients a_{j} (θ) by using the binomial formula, but in MatLab implementation it is more practical to use a Horner scheme. The new coefficients b_{0}(θ, ρ^{∗}), b_{1}(θ, ρ^{∗}), …, b_{4n+19} (θ, ρ^{∗}) are complicated trigonometric functions, inappropriate for further analytical consideration. The method has been tested for all values of n from 1 to 100 and it gives the optimal results. Some of the cases are displayed in Fig. 1.

Figure 1. The functions b_{0}(θ, ρ^{∗}), …, b_{31}(θ, ρ^{∗}), in the case n = 3 , ρ^{∗} = 1 (left) and the functions b_{0}(θ, ρ^{∗}), …, b_{139}(θ, ρ^{∗}), in the case n = 30, ρ^{∗} = 16.8838 (right).

1.3. The determination of ρ^{∗} in the case n ⩾ 12 . Our aim is to determine the minimal values of ρ^{∗} for n ⩾ 12 by using MatLab. For fixed n ⩾ 12, we treated the terms J(ρ) and tested the smallest possible values of ρ^{∗} such that the terms J(ρ) are nonpositive for each ρ > ρ^{∗} (Table 1).

Table 1. The values of ρ^{∗} for 12 ⩽ n ⩽ 47

\begin{matrix}n & ρ^{∗} & n & ρ^{∗} & n & ρ^{∗} & n & ρ^{∗} \\ 12 & 2.3455 & 21 & 10.5861 & 30 & 16.8838 & 39 & 23.0093 \\ 13 & 3.4034 & 22 & 11.3053 & 31 & 17.5691 & 40 & 23.6857 \\ 14 & 4.7165 & 23 & 12.0172 & 32 & 18.2529 & 41 & 24.3615 \\ 15 & 5.8433 & 24 & 12.7232 & 33 & 18.9354 & 42 & 25.0367 \\ 16 & 6.7473 & 25 & 13.4245 & 34 & 19.6167 & 43 & 25.7115 \\ 17 & 7.5731 & 26 & 14.1219 & 35 & 20.2969 & 44 & 26.3859 \\ 18 & 8.3575 & 27 & 14.8161 & 36 & 20.9762 & 45 & 27.0599 \\ 19 & 9.1162 & 28 & 15.5076 & 37 & 21.6547 & 46 & 27.7334 \\ 20 & 9.8575 & 29 & 16.1967 & 38 & 22.3323 & 47 & 28.4066 \end{matrix}

1.4. The error bounds. Let us consider the numerical calculation of integral (1.1) with a Chebyshev weight function ω = ω_{2}

I(f) = ∫_{−1}^{1} f(t)\sqrt{1 − t^{2}} dt.

According to the previously introduced notation, the error bound of the corresponding quadrature formula is given by |R_{n,2} (f)| ⩽ r_{n}(f), where

r_{n}(f) = inf_{ρ_{n} <ρ<ρ_{max}} [ \frac{ℓ(E_{ρ})}{2π}( max_{z∈E_{ρ}}|K_{n,2} (z)|) ( max_{z∈E_{ρ}}|f(z)|) ].

Here, ℓ(E_{ρ}) represents the length of the ellipse E_{ρ}, and can be estimated by

ℓ(E_{ρ}) ⩽ 2πa_{1}( 1 − \frac{1}{4}a_{1}^{−2} − \frac{1}{64}a_{1}^{−4} − \frac{5}{256}a_{1}^{−6} ) ,

where a_{1} = (ρ + ρ^{−1} )/2 [13]. According to the conjecture, the kernel attains its maximum value at θ = π, i.e., max_{z∈E_{ρ}} |K_{n,2} (z)| = \frac{π}{ρ} \sqrt{AC/D}, where A, C, D denote the values of the terms a, c, d for the fixed angle θ = π. The error bound r_{n}(f) reduces to

(1.5) r_{n}(f) = inf_{ρ_{n} <ρ<ρ_{max}} [a_{1} \frac{π}{ρ}( 1− \frac{1}{4}a_{1}^{−2} − \frac{1}{64}a_{1}^{−4} − \frac{5}{256}a_{1}^{−6} ) \sqrt{\frac{AC}{D}}( max_{z∈E_{ρ}}|f(z)|) ].

In order to check the proposed error bounds we made several tests and compared them with respect to the exact errors (“Error”) calculated by using a modified Gautschi MatLab code gradau.m (cf. [4, 5]) to a high precision arithmetic.

Example 1.1. Let f_{1}(z) = \frac{cos(z)}{z^{2} +ω^{2}} , ω > 0. For the function f_{1}(z) (see [15]) it holds that

max_{z∈E_{ρ}}|f_{1}(z)| = \frac{cos(b_{1})}{−b_{1}^{2} + ω^{2}} ,

where b_{1} = (ρ − ρ^{−1} )/2, and the infimum is calculated with respect to the interval ρ ∈ (ρ_{n}, ρ_{max} ), where ρ_{max} = ω + \sqrt{1 + ω^{2}} . The corresponding error bounds and actual errors are displayed in Table 2.

Table 2. Error bounds r_{n}(f_{1}) and actual errors, where f_{1}(z) = \frac{cos(z)}{z^{2} +ω^{2}}
n r_{n}, ω = 2  Error  r_{n}, ω = 5  Error  r_{n}, ω = 20  Error
1 1.082(−1)  2.544(−2)  2.800(−3)  1.549(−3)  8.426(−5)  6.903(−5)
2 5.601(−3)  1.051(−3)  4.087(−5)  1.671(−5)  6.819(−7)  4.226(−7)
3 3.224(−4)  4.858(−5)  5.367(−7)  1.572(−7)  3.089(−9)  1.665(−9)
4 1.909(−5)  2.395(−6)  6.547(−9)  1.417(−9) 9.374(−12) 4.491(−12)
5 1.142(−6)  1.225(−7) 7.637(−11) 1.285(−11) 2.045(−14) 8.808(−15)
10 8.600(−13) 5.335(−14) 1.215(−20) 9.504(−22) 4.433(−29) 1.046(−29)
13 1.761(−16) 2.714(−17) 1.432(−26) 8.481(−28) 2.155(−38) 3.104(−39)

The values r_{n}(f_{1}), and ρ_{opt} ∈ (ρ_{n}, ρ_{max} ), for the same values n and ω from Table 2, in which the expression within the brackets under the sign of inf in (1.5) attains its minimum, are presented in Table 3.

Example 1.2. Let f_{2}(z) = e^{e^{cos (ωz)}} , ω > 0. The function f_{2} is entire and it is

known (see [14]) that

max_{z∈E_{ρ}}|f_{2}(z)| = e^{e^{cosh (ωb_{1} )}} .

Table 4 displays some error bounds and actual errors.

Table 3. Error bounds r_{n}(f_{1}) and values ρ_{opt}, where f_{1}(z) = \frac{cos(z)}{z^{2} +ω^{2}}

n r_{n}(f_{1}), ω = 2  ρ_{opt}  r_{n}(f_{1}), ω = 5  ρ_{opt}  r_{n}(f_{1}), ω = 20  ρ_{opt}
1  1.082(−1)  3.3511  2.800(−3)  6.2311  8.426(−5)  8.2831
2  5.601(−3)  3.5666  4.087(−5)  7.4816  6.819(−7)  11.8126
3  3.224(−4)  3.7104  5.367(−7)  8.2394  3.089(−9)  15.4024
4  1.909(−5)  3.8087  6.547(−9)  8.6927  9.374(−12)  18.9057
5  1.142(−6)  3.8765  7.637(−11)  8.9795  2.045(−14)  22.2445
10 8.600(−13)  4.0397  1.215(−20)  9.5587  4.433(−29)  33.8827
13 1.761(−16)  4.0814  1.432(−26)  9.6884  2.155(−38)  36.4644

Table 4. Error bounds r_{n}(f_{2}) and actual errors, where f_{2}(z) = e^{e^{cos (wz)}}

n r_{n}, ω = 1  Error  r_{n}, ω = 0.1  Error  r_{n}, ω = 0.01  Error
1 4.412(+1) 4.017(+0)  3.101(−3)  1.411(−3)  1.212(−5)  1.434(−7)
2 4.395(+0) 5.425(−1)  5.982(−6)  2.023(−6)  3.671(−9)  2.057(−12)
3 5.140(−1) 6.971(−2)  1.097(−8)  3.043(−9)  1.061(−16) 1.892(−15)
4 6.280(−2) 8.698(−2)  1.877(−11) 4.490(−12)  1.838(−21) 4.580(−22)
5 7.701(−3) 1.050(−3)  3.012(−14) 9.457(−15)  2.977(−26) 6.546(−27)
6 9.212(−4) 1.227(−4)  4.569(−17) 8.812(−18)  4.548(−31) 9.015(−32)
9 1.365(−6) 1.635(−7)  1.213(−25) 1.867(−26)  1.224(−45) 1.920(−46)
12 1.613(−9) 1.738(−10) 2.350(−34) 3.067(−35)  2.394(−60) 3.168(−61)

2. Gauss–Radau quadrature rule with single end point

In this section we analyze the remainder term of the Gauss–Radau quadrature rule with the end point −1

∫_{−1}^{1} f(t)ω(t) dt = λ_{0}^{R}f(−1) + ∑_{ν=1}^{n} λ_{ν}^{R}f(τ_{ν}^{R}) + R_{n+1}^{R} (f),

where τ_{ν}^{R} are zeros of π_{n}(·; ω^{R}), the orthogonal polynomial on [−1, 1] with respect to the Chebyshev weight functions ω_{2}(t). The remainder term R_{n+1}^{R} (f) admits the contour integral representation

R_{n+1,r}^{R} (f) = \frac{1}{2πi} ∮_{Γ} K_{n+1,r}^{R} (z; ω)f(z)dz.

The kernel is given by

K_{n+1}^{R} (z; ω) ≡ K_{n+1} (z, ω) = \frac{1}{w_{n+1} (z; ω)} ∫_{−1}^{1} \frac{ω(t)w_{n+1} (z; ω)}{z − t} dt, z ∉ [−1, 1],

where w_{n+1} (z; ω) = ∏ _{i=1}^{n+1} (z − τ_{i}). We take Γ = E_{ρ}, where the ellipse E_{ρ} is given by (1.3).

In [3] Gautschi considered the Gauss–Radau and the Gauss–Lobatto quadrature rules with respect to the Chebyshev weight function ω_{2} and presented the conjectures based on numerical considerations. The case ρ ⩾ ρ_{n} from Gautschi’s conjecture has been already proven [10]. Here, we consider the case ρ < ρ_{n} from Gautschi’s conjecture, which supplements earlier work in [10]. We also give the strong numerical evidence for the precise values of ρ_{n}.

2.1. Maximum modulus of the kernel. Gautschi analyzed the maximum modulus of the kernel K_{n+1} (z; ω_{2}) and made the conjecture [3, p. 224] that the maximum is attained at θ = π if 1 < ρ < ρ_{n} and n ⩾ 4 where ρ_{n} is determined for 4 ⩽ n ⩽ 10.

The kernel K_{n+1} (z; ω_{2}) is given by

K_{n+1} (z; ω_{2}) = \frac{π}{u^{n+1}} × \frac{1 − u^{−2} + α(u + u^{−1} )}{u^{n+2} − u^{−(n+2)} + α[u^{n+1} − u^{−(n+1)} ]} ,

where α = \frac{n+2}{n+1} , z = \frac{1}{2}(u + u^{−1} ) and u = ρe^{iθ}. By introducing some substitutions, we can express the modulus of the kernel in the form

|K_{n+1} (z; ω_{2})| = ( \frac{π^{2}}{ρ^{2n+2}} \frac{γ}{δ})^{1/2} ,

where

γ = |1 − u^{−2} + α(u + u^{−1} )|^{2} = \frac{c}{ρ^{4}} ,
c = γ · ρ^{4} = ρ^{6} · α^{2} + ρ^{5} · 2α cosθ + ρ^{4} · (1 − 2α^{2} cos2θ)
− ρ^{3} · 2α(cosθ + cos3θ) + ρ^{2} · (α^{2} − 2 cos2θ) + ρ · 2α cosθ + 1.

The terms γ and δ are multiplied by ρ^{4} and ρ^{2n+4} respectively

δ = |u^{n+2} − u^{−(n+2)} + α[u^{n+1} − u^{−(n+1)} ]|^{2} = \frac{d}{ρ^{2n+4}} ,
d = δ · ρ^{2n+4} = ρ^{4n+8} + ρ^{4n+7} · 2α cosθ + ρ^{4n+6} · α^{2}
− ρ^{2n+5} · 2α cos(2n + 3)θ − 2ρ^{2n+4} · [cos(2n + 4)θ + α^{2} cos(2n + 2)θ]
− ρ^{2n+3} · 2α cos(2n + 3)θ + ρ^{2} · α^{2} + ρ · 2α cosθ + 1.

We get |K_{n+1} (z; ω_{2})|^{2} = \frac{π^{2}}{ρ^{2}} \frac{c}{d}. By letting C and D denote the values of c and d at θ = π, the square of the modulus of the kernel at θ = π can be expressed as

|K_{n+1} (z; ω_{2})|^{2} = \frac{π^{2}}{ρ^{2}} \frac{C}{D} ,

with appropriate replacements

C = ρ^{6} · α^{2} − ρ^{5} · 2α + ρ^{4} · (1 − 2α^{2})
+ ρ^{3} · 4α + ρ^{2} · (α^{2} − 2) − ρ · 2α + 1,
D = ρ^{4n+8} − ρ^{4n+7} · 2α + ρ^{4n+6} · α^{2} + ρ^{2n+5} · 2α
− 2ρ^{2n+4} · (1 + α^{2}) + ρ^{2n+3} · 2α + ρ^{2} · α^{2} − ρ · 2α + 1.

Our task is to show that this is the maximum value of the modulus for all 1 < ρ < ρ_{n} if n ⩾ 4.

2.2. Gautschi’s conjecture. The conjecture suggests that the maximum modulus of the kernel is attained on the negative real axis for all ρ from the interval (1, ρ_{n}) if n ⩾ 4. Whereas that ρ belongs to the bounded interval, it is not possible to conduct any asymptotic analysis.

Referring to the previously introduced notation, we have to show that \frac{c}{d} ⩽ \frac{C}{D} for each 1 < ρ < ρ_{n} and n ⩾ 4. The previous inequality can be written as

I(ρ) = [cD − Cd] ⩽ 0.

The term I(ρ) is a polynomial in ρ whose coefficients depend only on θ

I(ρ) = [2α(1 + cosθ)(1 − α^{2})] · ρ^{4n+13} + [2α^{2}(1 − cos2θ)] · ρ^{4n+12}
+ 2α[α^{2} + (3α^{2} − 2) cosθ + 2α^{2} cos2θ − cos3θ − 3] · ρ^{4n+11}
+ 4 sin^{2} θ[1 + α^{4} − 4α^{2} cosθ] · ρ^{4n+10}
− 2α[3α^{2} − 2 cos2θ + cosθ(α^{2} + 2α^{2} cos2θ − 3) − 1] · ρ^{4n+9}
+ [4α^{2} sin^{2} θ] · ρ^{4n+8} + [2α(α^{2} − 1)(1 + cosθ)] · ρ^{4n+7} + ⋯

i.e., I(ρ) = ∑ _{i=0}^{4n+13} a_{i}(θ)ρ^{i}, 1 < ρ < ρ_{n}. Modeled on the consideration from the previous section, we can write the polynomial I(ρ) as a polynomial in the terms of positive differences ρ_{n} − ρ, and show nonpositivity of its new coefficients.

In order to ensure nonpositivity for 1 < ρ < ρ_{n}, first of all, we shift the interval ρ ∈ (1, ρ_{n}) iff −ρ ∈ (−ρ_{n}, −1) iff −ρ + ρ_{n} ∈ (0, ρ_{n} − 1). The polynomial I(ρ) can be written in the form

J(ρ) = ∑_{k=0}^{4n+13} β_{k}(θ, ρ_{n})(ρ_{n} − ρ)^{k}.

Its nonpositivity on the interval (0, ρ_{n} −1) is a sufficient condition for nonpositivity of the initial polynomial I(ρ) on the interval (1, ρ_{n}). The β_{i}(θ, ρ_{n}) coefficients can be expressed by applying the transformation ρ ↦ (−1) · ρ + ρ_{n}. Some of them are presented

β_{4n+13} (θ, ρ_{n}) = 2α(cosθ + 1)(1 − α^{2}),
β_{4n+12} (θ, ρ_{n}) = −4α cos^{2} \frac{θ}{2}[(α^{2} − 1)(13 + 4n)ρ_{n} + 2α cosθ − 2α],
β_{4n+11} (θ, ρ_{n}) = 2α[α^{2} − 2 cosθ + 3α^{2} cosθ − 2α^{2} − (3 + n)(13 + 4n)(ρ_{n})^{2}
· (1 + cosθ) + 2α^{2} cos2θ − cos3θ + 8α(3 + n)ρ_{n} sin^{2} θ − 3],
β_{4n+10} (θ, ρ_{n}) = −\frac{4}{3}α(11 + 4n)ρ_{n} cos^{2} \frac{θ}{2}[3(5 + α^{2}) + 2(α^{2} − 1)(3 + n)
· (13 + 4n)(ρ_{n})^{2} − 12(1 + α^{2}) cosθ + 6 cos2θ]
+ 4 sin^{2} θ[1 + α^{4} + 2α^{2}(3 + n)(11 + 4n)(ρ_{n})^{2} − 4α^{2} cosθ],
β_{4n+9} (θ, ρ_{n}) = −\frac{4}{3}α(5 + 2n)(11 + 4n)(ρ_{n})^{2} cos^{2} \frac{θ}{2}[15 + 3α^{2}
+ (α^{2} − 1)(3 + n)(13 + 4n)(ρ_{n})^{2} − 12 cosθ(1 + α^{2}) + 6 cos2θ]
− 2α[3α^{2} − 2 cos2θ + cosθ(α^{2} + 2α^{2} cos2θ − 3) − 1] + \frac{8}{3}ρ_{n} sin^{2} θ
· (5 + 2n)[3 + 3α^{4} + 2α^{2}(3 + n)(11 + 4n)(ρ_{n})^{2} − 12α^{2} cosθ].

Figure 2. The functions β_{49} (θ, ρ_{n}), β_{48} (θ, ρ_{n}), β_{47} (θ, ρ_{n}), β_{46} (θ, ρ_{n}) in
the case n = 9, ρ_{n} = 1.0394

Figure 2 presents the graphs of the previous coefficients in the case n = 9, ρ_{n} = 1.0394.

Figure 3. The functions β_{0}(θ, ρ_{n}), …, β_{29} (θ, ρ_{n}) in the case n = 4 (left) and the functions β_{0}(θ, ρ_{n}), …β_{121} (θ, ρ_{n}) in the case n = 27 (right)

Table 5. The values of ρ_{n} for 4 ⩽ n ⩽ 45

\begin{matrix}n & ρ_{n} & n & ρ_{n} & n & ρ_{n} & n & ρ_{n} & n & ρ_{n} & n & ρ_{n} \\ 4 & 1.2845 & 9 & 1.0394 & 14 & 1.0156 & 19 & 1.0084 & 24 & 1.0052 & 41 & 1.0017 \\ 5 & 1.1517 & 10 & 1.0314 & 15 & 1.0136 & 20 & 1.0075 & 25 & 1.0048 & 42 & 1.0017 \\ 6 & 1.0964 & 11 & 1.0257 & 16 & 1.0119 & 21 & 1.0068 & 26 & 1.0043 & 43 & 1.0016 \\ 7 & 1.0679 & 12 & 1.0215 & 17 & 1.0105 & 22 & 1.0062 & ⋯ & ⋯ & 44 & 1.0015 \\ 8 & 1.0506 & 13 & 1.0182 & 18 & 1.0093 & 23 & 1.0057 & 40 & 1.0018 & 45 & 1.0014 \end{matrix}

The explicit formulae for the coefficients β_{i}(θ, ρ_{n}) are complicated trigonometric terms, inappropriate for a further analytical consideration. Numerical calculations show that all coefficients β_{i}(θ, ρ_{n}), i ⩽ 4n + 13 are strictly under the x-axis. We tested the cases n = 4, 3, …, 45, and some of them are presented (Fig. 3). For fixed n ⩾ 4, we treated the terms J(ρ) and tested the largest possible values of ρ_{n} such that the terms J(ρ) are nonpositive for each 1 < ρ < ρ_{n} (Table 5).

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Higher Medical and Business-Technological School (Received 21 01 2016) of Professional Studies (Revised 15 11 2016) Šabac Serbia

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