On the composition conjecture for a class of rigid systems

On the composition conjecture for a class of rigid systems

Zhengxin Zhou

and Yuexin Yan

1School of Mathematical Sciences, Yangzhou University, China

2Guangling College of Yangzhou University, China

Received 10 April 2019, appeared 21 December 2019 Communicated by Armengol Gasull

Abstract. In this paper, we prove that for a class of rigid systems the Composition Con- jecture is correct. We show that the Moments Condition is the sufficient and necessary conditions for these rigid systems to have a center at origin point. By the obtained con- clusions we can derive all the focal values of these higher order polynomial differential systems and their expressions are more succinct and beautiful.

Keywords: Composition Conjecture, rigid system, Center Condition, Moments Condi- tion.

2010 Mathematics Subject Classification: 34C07, 34C05, 34C25, 37G15.

1 Introduction

Consider the planar differential system

(x⁰ = −y+p(x,y),

y⁰ = x+q(x,y) ^(1.1)

where p,qare analytic functions starting with second order terms.

If p andqare polynomials of degreen andyp−xq = 0, the system (1.1) in polar coordi- nates becomes

dr dθ =r²

n−2 i

∑

A_i(θ)rⁱ, (1.2)

where A_i(θ) (i = 0, 1, 2, . . . ,n−2)are 2π-periodic functions. Therefore, the system (1.1) has a center at (0, 0)if and only if all solutionsr(θ)of equation (1.2) near the solutionr = 0 are periodic. In such case it is said that equation (1.2) has a center atr=0 [15,20].

If p andqare homogeneous polynomials of degree n, via the Cherkas [9] transformation equation (1.2) becomes the Abel equation

dρ

dθ =ρ²(A^˜₁(θ) +A^˜₂(θ)ρ)_{, (1.3)}

BCorresponding author. Email: zxzhou@yzu.edu.cn

where ˜A_i(θ) (i=1, 2)are 2π-periodic functions. Thus, finding the center conditions for (1.1) is equivalent to studying when the Abel equation (1.3) has a center atρ=0. This problem has been investigated in [5,8,10,14,20] among other works.

The problem of determining necessary and sufficient conditions onpandqfor system (1.1) to have a center at the origin is known as the center-focus problem. Due to the Hilbert’ basis theorem we know that when pandqare polynomials of a given degree this set of conditions is finite. To get the necessary and sufficient conditions is very complexity, up to now only for a very few families of polynomial system (1.1) the center conditions are known. The problem is solved for quadratic system and some families of cubic systems and systems in the form of the linear center perturbed by homogeneous quartic and quintic nonlinearities, see e.g.

[1,4,9–16,22] and references given there.

Alwash and Lloyd [6,7] give the following simple sufficient condition for the Abel equation to have a center.

Theorem 1.1([6,7]). If there exists a differentiable function u of period2πsuch that A˜₁(θ) =u⁰(θ)A^ˇ₁(u(θ)), A˜₂(θ) =u⁰(θ)A^ˇ₂(u(θ))

for some continuous functionsAˇ₁ and Aˇ₂, then the Abel differential equation(1.3)has a center at the origin.

The following statement presents a generalization of Theorem1.1.

Theorem 1.2([2,22]). If there exists a differentiable function u of period2πsuch that A_i(θ) =u⁰Aˆ_i(u), (i=1, 2, . . . ,n−2)

for some continuous functionsAˆ_i(i=1, 2, . . . ,n−2), then the differential equation(1.2)has a center at r=0.

The condition in Theorem 1.1 (or Theorem 1.2) is called the Composition Condition.

When an Abel equation (or (1.2) ) has a center because its coefficients satisfy the composi- tion condition we will say that this equation has aCC-center. In [7,18] it was shown that this condition is not necessary to have a center.

The Composition Conjecture is that the composition condition in Theorem 1.1 (or The- orem 1.2) is not only the sufficient but also necessary condition for a center. This conjec- ture first appeared in [6] with classes of coefficients which are polynomial functions in t, or trigonometric polynomials. A counterexample was presented in [7,13] to demonstrate that the conjecture is not true. To find the restrictive conditions under which the composition conjecture is true, this is an open problem which has attracted during the last years a wide interest [2–8,10,12,14,17,18,21,22]. The authors in paper [11] give the sufficient and necessary conditions for ther =0 of the Abel equation (1.3) to be a CC-center.

The condition

Z _2π

0

Z _θ

0

A˜₁(τ)dτ k

A˜2(θ)dθ=0 (k ≥0)

is calledMoments Condition[10]. In [18] an example of a polynomial Abel equation satisfy- ing the moments condition and not satisfying the composition condition is given. Later on, in [17] a full algebraic characterization of the moments condition in the polynomial case is done.

In [10] prove that a natural trigonometric analogous to it does not hold.

In [21], it was proved that for the rigid system (x⁰ =−y+xP,

y⁰ = x+yP (1.4)

with P = P₁+P_m, P_m is a homogeneous polynomial of degree m and which is an arbitrary natural number greater than 1, the composition conjecture is true, i.e., its origin point is a center and a CC-center, and shown that for its corresponding 2π-periodic equation

dr

dθ =r(P₁(cosθ, sinθ)r+P_m(cosθ, sinθ)r^m), r =0 is a center if and only if it satisfies the moments conditions:

Z _2π

0

Z _θ

0

P₁(_{cosτ, sin}τ)dτ k

P_m(_{cosθ, sin}θ)dθ =_0, (k =0, 1, 2, . . . ,m)_.

In [1] the authors used the methods of the normal form theory to prove that the rigid system (1.4) has a center if and only if it is reversible. In [19], the author has calculated by computer and obtained the center condition for system (1.4) with P = P_m+P_2m, mis a finite number that does not exceed 5.

In this paper, we will study the rigid system

(x⁰ =−y+x(P₁(x,y) +P_m(x,y) +P_2m+1(x,y)),

y⁰ = _x+_y(_P1(_x,y) +_Pm(_x,y) +_P2m+1(_x,y))_, ^(1.5) where P_k(x,y)is a homogeneous polynomial in x,yof degree kandP₁ 6= 0,mis an arbitrary positive integer greater than 1, and give the necessary and sufficient conditions for the origin of (1.5) to be a center. We prove that under some restrictions conditions the composition conjecture is true for its corresponding periodic differential equation

dr

dθ =r(P₁(cosθ, sinθ)r+Pm(cosθ, sinθ)r^m+P_2m+1(cosθ, sinθ)r^2m+1). (1.6) By this, we can derive all the focal values of system (1.5) and they contain exactly ^[m₂^] +m+2 relations. As m is an arbitrary number, in general, even with the help of computers, it is difficult to get the center conditions. However, in this paper, we have obtained these results only by using the mathematical analysis technique. We firmly believe that the method of this paper can be used to solve the center-focus problem of more high-order polynomial differential systems.

In the following we denote ¯P = Rθ

0 P(cosθ, sinθ)dθ; C^k_n = _k!(n^n!_−k)!(0 ≤ k ≤ n); C_n^k = 0, if k<0 orn<0;∑i+j=k =0, ifk<0.

2 Several lemmas

In order to prove the main result, first of all, we give the following lemmas.

Lemma 2.1([21]). If P1 6=0and for an arbitrary positive integer m, Z _2π

0

P¯₁ⁱ(cosθ, sinθ)Pm(cosθ, sinθ)dθ =0 (i=0, 1, 2, . . . ,m),

then

P_m = P₁

∑

jλ_jP¯₁^j−1, (2.1)

whereλ_j(j=1, 2, . . . ,m)are real numbers.

Lemma 2.2. Suppose that P_m satisfies(2.1)and h⁰₀= P_m, h⁰_k =2P₁

∑

i+j=k−1

P¯₁ⁱh_j+P_mC^m_m+kP¯₁^k, (k =1, 2, 3, . . .) _(2.2) Then

h_k =

∑

h_k^jP¯₁^k−jP¯₁^jP_m =h^ˆ_kP¯₁^m+k+

k−1 j

∑

hˆ_{k j}P¯₁^m+j, (2.3) wherehˆ_{k j}(j=0, 1, 2, . . . ,k−₁)are real numbers,

h_k^j = ² k−j

k−1

∑

h_i^j = (k−j+1)C_m^j _+j−2, (j=0, 1, 2, . . . ,k−1), (2.4)

h^k_k =C^k_m+k−

k−1 j

∑

h^j_k =C^k_m+k−2, (2.5)

hˆ_k =λ_m

∑

m

m+jh_k^j =λ_m

∑

m

m+j(_k−j+₁)_Cm^j _{+j−2. (2.6)} Proof. By (2.1) we get

P¯_m =

Z _θ

0 P_mdθ =λ_mP¯₁^m+

m−1 j

∑

λ_jP¯₁^j, (2.7)

P¯₁^kP_m =

Z _θ

0 P₁

∑

jλ_jP¯₁^j+k−1 = ^m

k+mλ_mP¯₁^m+k+

m−1 j

∑

j

j+kλ_jP¯₁^j+k, (2.8) whereλ_j(j=1, 2, . . . ,m)are real numbers.

By (2.2) and (2.7) and (2.8) we geth₀⁰ =Pm, h₀ =P^¯_m =h⁰₀P¯_m =h^ˆ₀P¯₁^m+

m−1

∑

λ_jP¯₁^j, h⁰₀=_{1, ˆ}h₀=λ_mh⁰₀=λ_m. h₁⁰ =2P₁h₀+P_mC^m_m+1P¯₁,

h₁ =2h⁰₀P¯₁P¯m+ (C¹_m+1−2h⁰₀)P^¯₁Pm = h⁰₁P¯₁P¯m+h¹₁P¯₁Pm= h^ˆ₁P¯₁^m+1+

∑

hˆ_1jP¯₁^j,

where ˆh_1j(j=1, 2, . . . ,m)are real numbers,

h⁰₁ =2h⁰₀, h¹₁=C_m¹₊₁−h⁰₁ =C¹_m−1, hˆ₁=

h⁰₁+h¹₁ m 1+m

λ_m = ^m

2+m+2 m+1 λ_m, this means that the relations (2.3)–(2.6) are valid fork=0, 1. Now suppose that these identifies are valid for integerk, next we will prove they are correct for integerk+_1.

Indeed, by (2.2)–(2.5) and (2.7) and (2.8) we have h⁰_k+1=2P₁

∑

i+j=k

P¯₁ⁱh_j+PmC^m_m+k+1P¯₁^k+1

=2P₁

∑

P¯₁^k−jP¯₁^jP_m

∑

h_i^j+P_mC_m^m_+k+1P¯₁^k+1, solving this equation we get

h_k+1 =

k+1 j

∑

h^j_k+1P¯₁^k+1−jP¯₁^jPm = h^ˆ_k+1P¯₁^m+k+1+

m+k j

∑

hˆ_k+1jP¯₁^j,

where ˆh_k+1j (j=1, 2, . . . ,m+k)are real numbers, hˆ_k+1 =λ_m

k+1

∑

m m+jh^j_k+1,

h_k^j₊₁= ² k+1−j

∑

h^j_i, (j=0, 1, 2, ...,k), h^k_k⁺₊¹₁ =C^k_m⁺₊¹_k+1−

∑

h^j_k+1, By Lemma 3.2 of [21], we get

h^j_k+1 = (k−j+2)C_m^j _+j−2,h^k_k⁺₊¹₁ =C_m^k+₊¹_k−1. By mathematical induction, the present lemma holds.

Similarly, we can prove the following lemma.

Lemma 2.3. If

δ⁰_k =2P₁

∑

i+j=k−1

P¯₁ⁱδ_j+P_2m+1C^2m_2m+⁺_k¹₊₁P¯₁^k, (2.9) then

δ_k =

∑

δ_k^jP¯₁^k−jP¯₁^jP_2m+1, (2.10) where

δ_k^j = ² k−j

k−1

∑

δ_i^j = (k−j+1)C_2m^j _+j−1, (j=0, 1, 2, . . . ,k−1),

δ^k_k =C_2m^2m+₊¹_k+1−

k−1 j

∑

δ_k^j =C^k_2m+k−1,δ₀⁰ =1.

Lemma 2.4. If the conditions of Lemma2.2are satisfied and α⁰₀ =C¹_m+1Pmh0,

α⁰_k =P₁

∑

i+j=k−1

(h_ih_j+2 ¯P₁ⁱα_j) +2P₁

∑

i+j=k−1−m

h_iα_j+P_mC_m¹₊₁

∑

i+j=k

C_m^m−₊¹_i−1P¯₁ⁱh_j +P_m

∑

i+j+l=k−m

C_m²₊₁C_m^m₊⁻²_i−2P¯₁ⁱh_jh_l+P_m

∑

i+j=k−m

C¹_m+1C^m_m+⁻_i¹₋₁P¯₁ⁱα_j,

(2.11)

then

α_k =αˆ_kP¯₁^2m+k+

2m+k−1 j

∑

ˆ

α_{k j}P¯₁^j, (k=0, 1, 2, . . .), (2.12) whereαˆ_{k j}, (j=2, 3, . . . , 2m+k−1)are real numbers and

ˆ

α₀ = ^m+1 2 λ²_m, ˆ

α_k = ¹ 2m+k

∑

i+j=k−1

hˆ_ihˆ_j+2

k−1

∑

ˆ

α_j+2

∑

i+j=k−1−m

hˆ_iαˆ_j+mλ_mC¹_m+1

∑

i+j=k

C_m^m₊⁻¹_i−1hˆ_j +λ_mm

∑

i+j+l=k−m

C²_m+1C_m^m−₊²_i−2hˆ_jhˆ_l+λ_mm

∑

i+j=k−m

C¹_m+1C_m^m−₊¹_i−1αˆ_j

, (k=1, 2, . . .). (2.13)

Proof. Solving equationα₀⁰ =C¹_m+1P_mh₀, by Lemma2.2we get α₀= ¹

2C_m¹₊₁h⁰₀P¯_m² =αˆ₀P¯₁^2m+

2m−1 j

∑

ˆ α_0jP¯₁^j.

where ˆα_0j are real numbers, ˆα₀ = ¹₂C_m¹₊₁λ²_m. Thus, the relation (2.12) is valid fork=0.

Suppose that the conclusion of the present lemma is correct for integerk−1, next we will show that they are held for integerk.

Indeed, by Lemma2.2and (2.11) we get α⁰_k = P₁P¯₁^2m+k−1

∑

i+j=k−1

hˆ_ihˆ_j+2

k−1

∑

ˆ

α_j+2

∑

i+j=k−1−m

hˆ_iαˆ_j+λmC¹_mC¹_m+1

∑

i+j=k

C^m_m+⁻_i¹₋₁hˆ_j +λ_mC¹_m

∑

i+j+l=k−m

C_m²₊₁C^m_m+⁻_i²₋₂hˆ_jhˆ_l+λ_mC¹_m

∑

i+j=k−m

C_m¹₊₁C^m_m+⁻_i¹₋₁αˆ_j

+· · · which implies that the identities (2.12) and (2.13) are valid. By mathematical induction, the present lemma holds.

Lemma 2.5. Suppose that h_kandδ_k are the solutions of equations(2.2)and(2.9)respectively, and β⁰_k =2P₁

∑

i+j=k−1

(P^¯₁ⁱβ_j+h_iδ_j) +P_mC¹_m+1

∑

i+j=k

C_m^m₊⁻¹_i−1P¯₁ⁱδ_j +P_2m+1C_2m¹ ₊₂

∑

i+j=k

C_2m^2m_+iP¯₁ⁱh_j, (k =0, 1, 2, . . . ,m+1) (2.14) Then

β_k =

∑

(β^j_kP¯₁^m+k−jP¯₁^jP_2m+1+β^m_k^+jP¯₁^k−jP¯₁^m+jP_2m+1) +

m+k−1

∑

∑

(β_{kj i}P¯₁^j−iP¯₁ⁱP_2m+1), (2.15) whereβ_{kj i}, (i=0, 1, 2, . . . ,j)are real numbers,

β^j_k =C_2m^j _+j−1C¹_m+k−j+1hˆ_k−j, (j=0, 1, 2, . . . ,k), (2.16) β^m_k^+j = (k−j+₁)β^m_j ^+j, β^m₀ =C¹_m+1hˆ₀, (j=0, 1, 2, . . . ,k−1)_{, (2.17)} β^m_k^+k =

∑

(C_2m¹ ₊₂C_2m^j _+j−C_2m^j _+j−1C_m¹_+k−j+1)h^ˆ_k−j−

k−1 j

∑

(k−j+1)β^m_j ^+j. (2.18)

Proof. By (2.14) we get

β⁰₀ =C¹_m+1Pmδ₀+P2m+₁C¹_2m+2h0, using Lemma2.2and Lemma2.3 we get

h₀ =h⁰₀P¯_m, δ₀ =δ₀⁰P¯_2m+1,h⁰₀ =δ₀⁰=1, thus

β⁰₀ =C¹_m+1P_mδ₀⁰P¯_2m+1+P_2m+1C_2m¹ ₊₂h⁰₀P¯_m,

β₀ =C¹_m+1δ₀⁰P¯_mP¯_2m+1+ (C_2m¹ ₊₂h⁰₀−C¹_m+1h⁰₀)P^¯_mP_2m+1

=β⁰₀P¯₁^mP¯_2m+1+β^m₀P¯₁^mP_2m+1+

m−1

∑

(β_0j0P¯₁^jP¯_2m+1+β_00jP¯₁^jP_2m+1), where β_0j0,β_00j (j=0, 1, 2, . . . ,m−1)are real numbers,

β⁰₀ =λ_mC_m¹₊₁δ⁰₀ =λ_mC¹_m+1, β^m₀ =λ_m(C¹_2m+2h⁰₀−C_m¹₊₁δ⁰₀) =λ_mC_m¹₊₁. Therefore, the conclusion of the present lemma is correct fork =0. Now we suppose that

β_n =

∑

(β^j_nP¯₁^m+n−jP¯₁^jP_2m+1+β^m_n^+jP¯₁^n−jP¯₁^m+jP_2m+1) +· · · (n=0, 1, 2, . . . ,k−1). By this and (2.14) we get

β⁰_k =2P₁

∑

i+j=k−1

∑

(β^l_jP¯₁^m+k−1−lP¯₁^lP2m+₁+β^m_j ^+lP¯₁^k−1−lP¯₁^m+lP2m+₁)

+2P₁

∑

i+j=k−1

hˆ_i

∑

δ^l_jP¯₁^m+k−1−lP¯₁^lP_2m+1+mλmC_m¹₊₁

∑

i+j=k

C_m^m−₊¹_i−1

∑

δ^l_jP¯₁^k−lP¯₁^lP_2m+1

+P_2m+1P¯₁^k+mC¹_2m+2

∑

i+j=k

C^2m_2m+ihˆ_j+· · · , solving this equation we get

β_k =

∑

(β^j_kP¯₁^m+k−jP¯₁^jP_2m+1+β^m_k^+jP¯₁^k−jP¯₁^m+jP_2m+1) +

m+k−1 j

∑

∑

(β_{kj i}P¯₁^j−iP¯₁ⁱP_2m+1), where β_{kj i}, (i=0, 1, 2, . . . ,j)are real numbers,

β^j_k = ¹ m+k−j

2

k−j−1 i

∑

hˆ_iδ_k^j_−1−i+2

k−1

∑

β^j_i+mλ_mC¹_m+1C_2m^j _+j−1C_m^k−₊^j_k−j+1

, (2.19) (j=0, 1, 2, . . . ,k−1)

β^k_k =λ_mC¹_m+1C_2m^k _+k−1, (2.20)

β^m_k^+j = ² k−j

k−1

∑

β^m_i ^+j, (j=0, 1, 2, . . . ,k−1)_{, (2.21)}

β^m_k^+k =C_2m¹ ₊₂

∑

i+j=k

C^2m_2m+ihˆ_j−β^k_k−

k−1 j

∑

(β^j_k+β^m_k^+j). (2.22)

In order to prove the relations (2.16)–(2.18) are valid, first of all, we use mathematical induction to show that

2

∑

hˆ_j+λmmC_m^m_+k+1 = (m+k+1)h^ˆ_k+1 (k=0, 1, 2, . . .). (2.23)

Indeed, by (2.6) we get ˆh₀= λ_m, ˆh₁ =λ_m 2+_m^m₊₁C¹_m−1 , thus

2ˆh₀+λ_mmC_m^m₊₁=λ_m(2+mC¹_m+1) = (m+1)h^ˆ₁,

this means that the relation (2.23) is valid for k = 0. Suppose that the identity (2.23) is valid for positive integerk, next we will show that

2

k+1

∑

hˆ_j+λmmC_m^m_+k+2 = (m+k+2)h^ˆ_k+2. (2.24)

Indeed, from the inductive hypothesis and (2.6) we have 2

k+1 j

∑

hˆ_j = (m+k+1)h^ˆ_k+1−λ_mmC_m^m_+k+1+2ˆh_k+1

=λ_m(m+k+3)

k+₁ j

∑

m

m+j(k+2−j)C_m^j _+j−2−λ_mmC_m^m_+k+1. By this andC_m^j =C_m^j−₋¹₁+C_m^j ₋₁, we get

(m+k+2)h^ˆ_k+2−2

k+₁ j

∑

hˆ_j = (m+k+2)λ_m

k+₂ j

∑

m

m+j(k+3−j)C_m^j _+j−2

−λ_m(m+k+3)

k+1 j

∑

m

m+j(k+2−j)C_m^j _+j−2+λ_mmC_m^m_+k+1

= λmm

k+1 j

∑

C_m^m_+j−2+λmmC_m^k+₊²_k+λmmC^m_m+k+1

= λ_mm

k+2 j

∑

C_m^j _+j−2+λ_mmC_m^k+₊¹_k+1 =mλ_mC^k_m⁺₊²_k+2.

Thus the identity (2.24) is correct. By mathematical induction, the identity (2.23) is valid.

Now, we prove that the relation (2.16) is correct. Indeed, by (2.19) and (2.20) we get β⁰₀ =λ_mC¹_m+1= C_m¹₊₁hˆ₀,

and

β⁰₁ = ¹

m+1(2ˆh0+2β0+λmmC_m+1C_m¹₊₂)

= ¹

m+1C¹_m+2(2+mC_m+1) =C_m¹₊₂hˆ₁, and

β¹₁ =λ_mC_2m¹ C¹_m+1= C_2m¹ C¹_m+1hˆ₀.

Thus, the identity (2.16) is correct fork=0, 1. Suppose that (2.16) is valid for natural numbers less than or equal tok, next we will prove that (2.16) holds fork+1.

In fact, by (2.19) and (2.23) and Lemma2.4we get β^j_k+1 = ¹

m+k+1−j 2

k−j i

∑

hˆ_iδ_k^j_−i+₂

∑

β^j_i+mλ_mC¹_m+1C_2m^j _+j−1C_m^k−₊^j+_k−¹_j+2

!

= ¹

m+k+1−jC_2m^j _+j−1C_k¹_+m−j+2 2

k−j i

∑

hˆ_i+λmmC_m^m_+k+1−j

!

=C_2m^j _+j−1C_k¹_+m−j+2hˆ_k−j+1. So, by mathematical induction the identity (2.16) is correct.

Now we prove that the identity (2.17) is correct.

Asβ^m₀ = C¹_m+1λm, by (2.21) we getβ^m₁ = 2β^m₀, thus, the relation (2.17) is correct for k =1.

Suppose that

β^m_k−⁺₁^j = (k−j)β^m_j ^+j, (j=0, 1, 2, . . . ,k−2), by this and (2.21) we get

β^m_k^+j = ² k−j

k−1

∑

β^m_i ^+j = ² k−j

k−1

∑

(i−j+1)β^m_j^+j = (k−j+1)β^m_j ^+j. Therefore, the identity (2.17) is valid.

Substituting (2.16) and (2.17) into (2.22) implies that the identity (2.18) holds.

In summary, the conclusions of the present lemma is correct.

Lemma 2.6. Ifλ_m≥₀and m≥2, then

β^m_k^+k ≥0, (k=0, 1, 2, . . . ,m+1). (2.25) Proof. Asλ_m ≥0, by (2.6) we get

hˆ_k =λ_m

∑

m

m+j(k−j+1)C_m^j_+j−2 ≥0, (k=0, 1, 2, . . . ,m+1). Using (2.17) we get

β^m₀ =C¹_m+1hˆ0=λ_mC¹_m+1 ≥0.

By this and using (2.18) we have

β¹₁^+m= C_m¹hˆ₁+C_m+1C_2m¹ ₊₂hˆ₀−2β^m₀ =C¹_mhˆ₁+C_m¹C_2m¹ ₊₂hˆ₀ ≥0.

Thus the inequality (2.25) is valid fork=0 andk =1. Now we suppose that inequality (2.25) holds for natural numbers less than k, i.e.,βⁱ_i^+m ≥0, (i=0, 1, 2, . . . ,k−1). Next we will show that β^k_k^+m ≥0.

Indeed, by (2.18) we get β^k_k⁻₋¹₁^+m =

k−1 j

∑

(C_2m¹ ₊₂C_2m^j−1_+j−1+C_2m^j _+j−1C¹_m+2+j−k)h^ˆ_k−1−j−

k−2 j

∑

(k−j)β^m_j ^+j

=

∑

(C_2m¹ ₊₂C_2m^j−2_+j−2+C_2m^j−1_+j−2C_m¹_+1+j−k)h^ˆ_k−j−

k−2 j

∑

(k−j)β^m_j ^+j.