BY 4.0 license Open Access Pre-published online by Oldenbourg Wissenschaftsverlag September 25, 2021

1,2,3, some inductive real sequences and a beautiful algebraic pattern

Sarsengali Abdygalievich Abdymanapov, Serik Altynbek, Anton Begehr and Heinrich Begehr ORCID logo
From the journal Analysis

Abstract

By rewriting the relation 1+2=3 as 12+22=32, a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with 12+22=32 as initial one converging to the segment [0,1] of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra.

MSC 2010: 40A05

1 An inductive sequence of positive numbers

With the initial numbers m1=3 and m2=23 a real sequence is defined inductively via

mk+2=αkmk+1+βkαk+βkmk+1,αk=mkmk+1-1,βk=mk-mk+1,k.

By introducing positive numbers rk by rk2=mk2-1, k, a sequence of circles |z-mk|=rk in the complex plane is given which is part of some parqueting of the complex plane [4]. The parqueting-reflection principle is always initiating iteratively given sequences of complex numbers; see [2] for another sample. The sequence presented here served in [1, 3] for treating problems in a hyperbolic domain. The triangles appear by taking one tangent to the unit circle through the center mk of a circle, the radius of the unit circle perpendicular to the tangent, and the segment [0,mk] as the hypotenuse; see Figure 1.

Figure 1 First triangles of the sequence.

Figure 1

First triangles of the sequence.

Lemma 1.1.

The sequence {mk} is monotonic decreasing with limit 1. In particular,

0<mk+2-1q1k+1(m1-1),k,

where

q1=m1+1m1-1m2-1m2+1=3-13+1<1.

Proof.

From

mk+2±1=αk±βkαk+βkmk+1(mk+1±1)

and

qk=αk-βkαk+βk=mk+1mk-1mk+1-1mk+1+1,

the estimates

0<(mk+1)(mk+1-1)2αk+βkmk+1=mk+2-1αk-βkαk+βk(mk+1-1)

and in particular

mk+2-1mk+2+1=αk-βkαk+βkmk+1-1mk+1+1=qkmk+1-1mk+1+1

follow. The last equation shows qk=qk+1, and hence the qk do not depend on k:

qk=q1=m1+1m1-1m2-1m2+1=3-13+1<1.

The inequalities mk+2-1q1(mk+1-1) imply

mk+2-1q1k+1(m2-1)q1k+1(m1-1)

for any k.

The monotonicity is seen from

(1.1)mk+2-mk+1=βk(1-mk+12)αk+βkmk+1<0

together with m2<m1. ∎

The sequence allows a simpler representation.

Lemma 1.2.

For any kN,

αkβk=m1,mk+1=1+m1mkm1+mk

hold.

Proof.

By simple verification,

mk+1=αkmk+βkαk+βkmk

is seen. Combining this with

mk+2=αkmk+1+βkαk+βkmk+1

in the definitions for αk+1 and βk+1 shows

αk+1(αk+mkβk)(αk+mk+1βk)=rk2rk+12αk,βk+1(αk+mkβk)(αk+mk+1βk)=rk2rk+12βk.

Thus αkβk is independent of k, and hence coincides with the value m1 for k=1. ∎

Remark 1.3.

Obviously, α1=m1β1=1. Lemma 1.2 suggests a new definition of the sequence with just one initial value m1=3 and

mk+1=1+m1mkm1+mk,k.

The first members of the sequences {mk} and {rk} are presented in Section A.

2 A recurrence relation

With the sequence {mk} and its related coefficients αk=mkmk+1-1,βk=mk-mk+1, the system of recurrence relations

(2.1)α2kδ2k-1+β2kγ2k-1=δ2k+1,α2kγ2k-1+β2kδ2k-1=γ2k+1,
(2.2)α2k+1δ2k+β2k+1γ2k=δ2k+2,α2k+1γ2k+β2k+1δ2k=γ2k+2

defines two new sequences with the initial values γ0=0, δ0=1, γ1=1, δ1=m1. The first further members of the sequences γk,δk are listed in Section A.

By using the relation αk=m1βk according to Lemma 1.2, these sequences are given in a shorter form as

δk+2=βk+1[m1δk+γk],γk+2=βk+1[m1γk+δk],k0.

Theorem 2.1.

The sequences {γk},{δk} defined by equations (2.1) and (2.2) with the initial numbers γ0=0, δ0=1, γ1=1, δ1=m1 are given as

γ2k=λ=1k1κ1<<κ2λ-12k(-1)k+μ=12λ-1κμμ=12λ-1mκμ,
δ2k=λ=1k1κ1<<κ2λ2k(-1)k+μ=12λκμμ=12λmκμ+(-1)k,
γ2k+1=λ=1k1κ1<<κ2λ2k+1(-1)k+μ=12λκμμ=12λmκμ+(-1)k,
δ2k+1=λ=0k1κ1<<κ2λ+12k+1(-1)k+1+μ=12λ+1κμμ=12λ+1mκμ

for kN0.

Proof.

In order to show that these formulas present solutions to the recurrence relations, equation (2.1) for δ2k+1 is proved, assuming the expressions for δ2k-1 and γ2k-1 are verified already. Then

δ=α2kδ2k-1+β2kγ2k-1
=(m2km2k+1-1)λ=0k-11κ1<<κ2λ+12k-1(-1)k+μ=12λ+1κμμ=12λ+1mκμ
+(m2k-m2k+1)[λ=1k-11κ1<<κ2λ2k-1(-1)k-1+μ=12λκμμ=12λmκμ+(-1)k-1].

Splitting the (λ=0)-term from the first sum and multiplying shows

δ=κ=12k-1(-1)k+1+κmκ+(-1)k-1m2k+(-1)km2k+1+κ=12k-1(-1)k+κmκm2km2k+1
+λ=1k-11κ1<<κ2λ+12k-1(-1)k+μ=12λ+1κμμ=12λ+1mκμm2km2k+1
+λ=1k-11κ1<<κ2λ+12k-1(-1)k+1+μ=12λ+1κμμ=12λ+1mκμ
+λ=1k-11κ1<<κ2λ2k-1(-1)k-1+μ=12λκμμ=12λmκμm2k
+λ=1k-11κ1<<κ2λ2k-1(-1)k+μ=12λκμμ=12λmκμm2k+1.

The first three terms on the right-hand side form

κ=12k+1(-1)k+1+κmκ.

The next two sums are composed to

λ=0k-11κ1<<κ2λ+12k-1<κ2λ+2=2k<κ2λ+3=2k+1(-1)k+1+μ=12λ+3κμμ=12λ+3mκμ
=λ=1k-11κ1<<κ2λ-12k-1<κ2λ=2k<κ2λ+1=2k+1(-1)k+1+μ=12λ+1κμμ=12λ+1mκμ+μ=12k+1mμ.

By leaving the next sum unchanged, the last two become

λ=1k-1[1κ1<<κ2λ2k-1<κ2λ+1=2k+1κ1<<κ2λ2k-1<κ2λ+1=2k+1](-1)k+1+μ=12λ+1κμμ=12λ+1mκμ.

This proves

δ=κ=12k+1(-1)k+1+κmκ+λ=1k-11κ1<<κ2λ+12k+1(-1)k+1+μ=12λ+1κμμ=12λ+1mκμ+μ=12k+1mμ
=λ=0k1κ1<<κ2λ+12k+1(-1)k+1+μ=12λ+1κμμ=12λ+1mκμ
=δ2k+1.

In the same manner, the other three formulas can be verified. ∎

Remark 2.2.

The expressions for γk and δk in Theorem 2.1 show that these quantities are combinations of products of the mk. That they in fact form very regular and beautiful algebraic combinations can be seen by writing these expressions down explicitly. These formulas of arbitrary order can even be produced by computer manipulations on the basis of the recurrence relations (2.1) and (2.2). Moreover, the γk and δk can in the same way be expressed through the αk and βk. Also, these formulas show a very regular and beautiful algebraic structure; see A.3 and A.4.

As the sequences {αk}, {βk}, also {γk}, {δk} converge to 0, but only the first two are monotone decreasing.

Theorem 2.3.

The sequences {αk} and {βk} are monotone. They and the sequences {γk} and {δk} converge to 0.

Proof.

From the monotonicity of {mk}, the estimations

m1(βk+1-βk)=αk+1-αk=mk+1(mk+2-mk)<0

are obvious. That {αk} and {βk} are null-sequences follows from limkmk=1. The other two sequences consist of non-negative numbers. This is seen from (2.1) and (2.2) as the initial values are non-negative and the αk,βk are positive numbers. From the assumption

γk,δkrk2(m1+1)m1+mk,

by using (1.1) reformulated as

βk+1=rk+12m1+mk+1,

both right-hand sides from

δk+2=βk+1[m1δk+γk],γk+2=βk+1[m1γk+δk],

can be estimated from above by

rk+12(m1+1)m1+mk+1rk2(m1+1)m1+mk.

The factor

rk2(m1+1)m1+mk

is less than 1 for 1<k, as can be seen from

mk2+mk(mkm1-1)<m22+m2(m2m1-1)=m22+m2<2m1+1.

The assumptions made are readily satisfied for k=1,2. ∎

3 A second recurrence relation

The recurrence relations (2.1) and (2.2) are providing some other recurrence relations.

For k0, the terms

Δ2k=m1δ2k+γ2k,Γ2k=m1γ2k+δ2k,Δ2k+1=m1δ2k+1-γ2k+1,Γ2k+1=m1γ2k+1-δ2k+1

satisfy the systems

(3.1)Δ2k+2=α2k+1Δ2k+β2k+1Γ2k,Γ2k+2=α2k+1Γ2k+β2k+1Δ2k,
(3.2)Δ2k+1=α2kΔ2k-1+β2kΓ2k-1,Γ2k+1=α2kΓ2k-1+β2kΔ2k-1.

This is easily deduced from (2.1) and (2.2). The first members of these sequences are also listed in Section A.

A direct consequence from Theorem 2.1 is the next statement.

Theorem 3.1.

With the initial data γ0=0, δ0=1, γ1=1, δ1=m1, i.e. Δ0=m1, Γ0=1, Δ1=2, Γ1=0, systems (3.1) and (3.2) have the solutions

Δ2k=2λ=1k2κ1<κ2<<κ2λ-12k(-1)k+1+μ=12λ-1κμμ=12λ-1mκμ,
Γ2k=2λ=1k-12κ1<κ2<<κ2λ2k(-1)k+1+μ=12λκμμ=12λmκμ+(-1)k+12,
Δ2k+1=2λ=1k2κ1<κ2<<κ2λ2k+1(-1)k+μ=12λκμμ=12λmκμ+(-1)k2,
Γ2k+1=2λ=1k2κ1<κ2<<κ2λ-12k+1(-1)k+1+μ=12λ-1κμμ=12λ-1mκμ.

Proof.

Exemplarily, the second formula will be verified from

Γ2k=m1γ2k+δ2k.

Splitting

m1γ2k=m1λ=1k1κ1<<κ2λ-12k(-1)k+μ=12λ-1κμμ=12λ-1mκμ
=m1[κ=12k(-1)k+κmκ+λ=2k(1=κ1<κ2<<κ2λ-12k+2κ1<κ2<<κ2λ-12k)(-1)k+μ=12λ-1κμμ=12λ-1mκμ],

separating the factor m1 and renaming summation indices lead to

m1γ2k=m12[(-1)k+1-λ=1k-12κ1<<κ2λ2k(-1)k+μ=12λκμμ=12λmκμ]+m1[λ=1k2κ1<<κ2λ-12k(-1)k+μ=12λ-1κμμ=12λ-1mκμ].

In a similar way, δ2k is split:

δ2k=(-1)k+λ=1k1κ1<<κ2λ2k(-1)k+μ=12λκμμ=12λmκμ
=(-1)k+λ=1k-12κ1<<κ2λ2k(-1)k+μ=12λκμμ=12λmκμ-m1λ=1k2κ1<<κ2λ-12k(-1)k+μ=12λ-1κμμ=12λ-1mκμ.

Adding the two formulas gives the expression for Γ2k. ∎

As the {γk}, {δk}, also {Γk}, {Δk} converge to 0.

Remark 3.2.

Remark 2.2 also applies for the Γk and Δk. They are expressible either through the mk or the αk and βk; see A.5 and A.6. But neither the formulas in Theorem 2.1 nor those in Theorem 3.1 reveal the beauty and regularity of the algebraic pattern. But writing these formulas out or simpler using some computer algebra to develop the single representations on the basis of the respective recurrence relations (2.1) and (2.2) or (3.1) and (3.2) unveils their symmetry.

Lemma 3.3.

For kN,

Γ4k-1=2λ=0k-1ϱ=12k-2λ-2α2μϱτ=12λ+1β2ντ,Δ4k-1=2λ=0k-1ϱ=12k-2λ-1α2μϱτ=12λβ2ντ

with

1μ1<μ2<<μ2k-2λ-22k-1,
1ν1<ν2<<ν2λ+12k-1,
μϱντ,

and

1μ1<μ2<<μ2k-2λ-12k-1,
1ν1<ν2<<ν2λ2k-1,
μϱντ,

respectively. Also,

(3.3)Γ4k+1=2λ=0kϱ=12k-2λ-1α2μϱτ=12λ+1β2ντ,Δ4k+1=2λ=0kϱ=12k-2λα2μϱτ=12λβ2ντ,

where

1μ1<μ2<<μ2k-2λ-12k,
1ν1<ν2<<ν2λ+12k,
μϱντ,

and

1μ1<μ2<<μ2k-2λ2k,
1ν1<ν2<<ν2λ2k,
μϱντ,

respectively.

Proof.

By the relations

Γ1=m1γ1-δ1=0,Δ1=m1δ1-γ1=m12-1=2,

m1 is eliminated as parameter. From (3.2), then

Γ3=α2Γ1+β2Δ1=2β2,Δ3=α2Δ1+β2Γ1=2α2

follow. By assuming equations (3.3) to hold, formula (3.2) implies

Δ4k+3=2λ=0k[ϱ=12k-2λ+1α2μϱτ=12λβ2ντ+ϱ=12k-2λ-1α2μϱτ=12λ+2β2ντ]

with

1μ1<<μ2k-2λ+1=2k+1,
1ν1<<ν2λ<2k+1,

and

1μ1<<μ2k-2λ-1<2k+1,
1ν1<<ν2λ+2=2k+1,
μϱντ,

respectively. Shifting the summation for the second term on the right-hand side replacing λ by λ-1 gives

Δ4k+3=2λ=0k+1ϱ=12k-2λ+1α2μϱτ=12λβ2ντ

with

1μ1<<μ2k-2λ+12k+1,
1ν1<<ν2λ2k+1,
μϱντ.

In the same way, the part for Γ4k+3 can be handled. By repeating the procedure, the respective formulas for the index 4k+5 can be achieved, completing the proof. ∎

Lemma 3.4.

For kN,

(3.4)Γ4k=2λ=0k-1[m2τ=12λα2ντ+1ϱ=12k-2λ-1β2μϱ+1+ϱ=12k-2λ-1α2μϱ+1τ=12λβ2ντ+1],
(3.5)Δ4k=2λ=0k-1[m2ϱ=12k-2λ-1α2μϱ+1τ=12λβ2ντ+1+τ=12λα2ντ+1ϱ=12k-2λ-1β2μϱ+1].

Here the indices involved are partitions of the set

{1,2,,2k-1}={μ1,μ2,,μ2k-2λ-1}{ν1,ν2,,ν2λ}

satisfying

1μ1<<μ2k-2λ-12k-1,1ν1<<ν2λ2k-1.

Moreover,

(3.6)Γ4k+2=2λ=0k-1m2τ=12λ+1α2ντ+1ϱ=12k-2λ-1β2μϱ+1+2λ=0kϱ=12k-2λα2μϱ+1τ=12λβ2ντ+1,
(3.7)Δ4k+2=2λ=0km2ϱ=12k-2λα2μϱ+1τ=12λβ2ντ+1+2λ=0k-1τ=12λ+1α2ντ+1ϱ=12k-2λ-1β2μϱ+1,

where the indices again are decompositions of {1,2,,2k}, each subset ordered according to size.

Proof.

Starting from

Γ0=δ0=1,Δ0=m1δ1=m1

or from

Γ2=m1γ2+δ2=m12-1=2,Δ2=m1δ2+γ2=2m2

creates similar but different expressions. By neglecting the right-hand sides from the last two equations, m1 can be kept as parameter. By using the latter relations for starting, from formulas (3.1),

Γ4=2m2β3+2α3,Δ4=2m2α3+2β3

follow. By assuming (3.4) and (3.5) to hold, from (3.1) follow

Δ4k+2=α4k+1Δ4k+β4k+1Γ4k=2m2Σ1+2Σ2,

where

Σ1=λ=0k-1[ϱ=12k-2λα2μϱ+1τ=12λβ2ντ+1+τ=12λα2ντ+1ϱ=12k-2λβ2μϱ+1]

with

1μ1<<μ2k-2λ=2k,1ν1<<ν2λ<2k,

and

Σ2=λ=0k-1[ϱ=12k-2λ-1β2μϱ+1τ=12λ+1α2ντ+1+τ=12λ+1β2ντ+1ϱ=12k-2λ-1α2μϱ+1]

with

1μ1<<μ2k-2λ-1<2k,1ν1<<ν2λ+1=2k.

Reflecting the summation index, i.e. interchanging λ with k-λ, in the second part of Σ1 gives for this part the same expression as in the first part, but the summation is taken between λ=1 and λ=k. The indices now vary according to

1μ1<<μ2λ-1<2k,1ν1<<ν2k-2λ+1=2k.

Thus,

Σ1=λ=0kϱ=12k-2λα2μϱ+1τ=12λβ2ντ+1

with

1μ1<<μ2k-2λ2k,1ν1<<ν2λ2k.

In the same way, Σ2 is handled, where in the second part besides reflecting also shifting the summation index is used. Finally, (3.5) is attained. By the symmetry, thus also (3.4) is proved.

To finish the proof, the procedure has to be repeated to get the formulas for Γ4(k+1) and Δ4(k+1) from (3.6) and (3.7). This part is skipped. ∎

In order to give an impression of the latter pattern, the first elements of the sequences {Γ2k}, {Γ2k+1}, {Δ2k}, {Δ2k+1} are also listed in Section A.

A Visualization of the sequences

A.1.

{mk}:

3,233,353,7123,11193,26453,41713,971683,1532653,3626273,.

A.2.

{rk}:

2,33,25,312,219,345,271,3168,2265,3627,.

A.3.

{γk},{δk} expressed through {mk}:

γ0=0,
δ0=1,
γ1=1,
δ1=m1,
γ2=m1-m2,
δ2=α1=m1m2-1,
γ3=m1m2-m1m3+m2m3-1,
δ3=m1m2m3-m1+m2-m3,
γ4=m1m2m3-m1m2m4+m1m3m4-m2m3m4-m1+m2-m3+m4,
δ4=m1m2m3m4-m1m2+m1m3-m1m4-m2m3+m2m4-m3m4+1,
γ5=m1m2m3m4-m1m2m3m5+m1m2m4m5-m1m3m4m5
+m2m3m4m5-m1m2+m1m3-m1m4+m1m5
-m2m3+m2m4-m2m5-m3m4+m3m5+1,
δ5=m1m2m3m4m5-m1m2m3+m1m2m4-m1m2m5-m1m3m4
+m1m3m5-m1m4m5+m2m3m4-m2m3m5+m2m4m5
-m3m4m5+m1-m2+m3-m4+m5,

A.4.

{γk},{δk} expressed through {αk},{βk}:

γ0=0,
δ0=1,
γ2=β1,
δ2=α1,
γ4=α3β1+β3α1,
δ4=α3α1+β3β1,
γ6=α5α3β1+α5β3α1+β5α3α1+β5β3β1
δ6=α5α3α1+α5β3β1+β5α3β1+β5β3α1,
γ8=α7α5α3β1+α7α5β3α1+α7β5α3α1+β7α5α3α1
+α7β5β3β1+β7α5β3β1+β7β5α3β1+β7β5β3α1,
δ8=α7α5α3α1+α7α5β3β1+α7β5α3β1+α7β5β3α1
+β7α5α3β1+β7α5β3α1+β7β5α3α1+β7β5β3β1,
γ10=α9α7α5α3β1+α9α7α5β3α1+α9α7β5α3α1+α9β7α5α3α1
+β9α7α5α3α1+α9α7β5β3β1+α9β7α5β3β1+α9β7β5α3β1
+α9β7β5β3α1+β9α7α5β3β1+β9α7β5α3β1+β9α7β5β3α1
+β9β7α5α3β1+β9β7α5β3α1+β9β7β5α3α1+β9β7β5β3β1
δ10=α9α7α5α3α1+α9α7α5β3β1+α9α7β5α3β1+α9α7β5β3α1
+α9β7α5α3β1+α9β7α5β3α1+α9β7β5α3α1+β9α7α5α3β1
+β9α7α5β3α1+β9α7β5α3α1+β9β7α5α3α1+α9β7β5β3β1
+β9α7β5β3β1+β9β7α5β3β1+β9β7β5α3β1+β9β7β5β3α1
γ1=1,
δ1=m1,
γ3=α2+β2m1,
δ3=α2m1+β2,
γ5=(α4β2+β4α2)m1+α4α2+β4β2,
δ5=(α4α2+β4β2)m1+α4β2+β4α2,
γ7=(α6α4β2+α6β4α2+β6α4α2+β6β4β2)m1+α6α4α2+α6β4β2+β6α4β2+β6β4α2,
δ7=(α6α4α2+α6β4β2+β6α4β2+β6β4α2)m1+α6α4β2+α6β4α2+β6α4α2+β6β4β2,
γ9=(α8α6α4β2+α8α6β4α2+α8β6α4α2+β8α6α4α2
+α8β6β4β2+β8α6β4β+β8β6α4β2+β8β6β4α2)m1
+α8α6α4α2+α8α6β4β2+α8β6α4β2+β8α6α4β2
+α8β6β4α2+β8α6β4α2+β8β6α4α2+β8β6β4β2,
δ9=(α8α6α4α2+α8α6β4β2+α8β6α4β2+β8α6α4β2
+α8β6β4α2+β8α6β4α2+β8β6α4α2+β8β6β4β2)m1
+α8α6α4β2+α8α6β4α2+α8β6α4α2+β8α6α4α2
+α8β6β4β2+β8α6β4β+β8β6α4β2+β8β6β4α2,

A.5.

{Γk},{Δk} expressed through {mk}:

Γ0=1,
Δ0=m1,
Γ2=2,
Δ2=2m2,
Γ4=2(m2m3-m2m4+m3m4-1),
Δ4=2(m2m3m4-m2+m3-m4),
Γ6=2(m2m3m4m5-m2m3m4m6+m2m3m5m6-m2m4m5m6+m3m4m5m6-m2m3+m2m4
-m2m5+m2m6-m3m4+m3m5-m3m6+m4m5-m4m6+m5m6+1),
Δ6=2(m2m3m4m5m6-m2m3m4+m2m3m5-m2m3m6+m2m4m5-m2m5m6
+m3m4m5-m3m4m6+m3m5m6-m4m5m6+m2-m3+m4-m5+m6),
Γ1=0,
Δ1=2,
Γ3=2(m2-m3),
Δ3=2(m2m3-1),
Γ5=2(m2m3m4-m2m3m5+m2m4m5-m3m4m5-m2+m3-m4+m5),
Δ5=2(m2m3m5m5-m2m3+m2m4-m2m5-m3m4+m3m5-m4m5+1),
Γ7=2(m2m3m4m5m6-m2m3m4m5m7+m2m3m4m6m7-m2m3m5m6m7
+m2m4m5m6m7-m3m4m5m6m7-m2m3m4+m2m3m5-m2m3m6
+m2m3m7-m2m4m5+m2m4m6-m2m4m7-m2m5m6+m2m5m7
-m2m6m7+m3m4m5-m3m4m6+m3m4m7+m3m5m6-m3m5m7
+m3m6m7-m4m5m6+m4m5m7-m4m6m7+m5m6m7
+m2-m3+m4-m5+m6-m7),
Δ7=2(m2m3m4m5m6m7-m2m3m4m5+m2m3m4m6-m2m3m4m7-m2m3m5m6
+m2m3m5m7-m2m3m6m7+m2m4m5m6-m2m4m5m7+m2m4m6m7
-m2m5m6m7-m3m4m5m6+m3m4m5m7-m3m4m6m7+m3m5m6m7
-m4m5m6m7+m2m3-m2m4+m2m5-m2m6+m2m7+m3m4-m3m5
+m3m6-m3m7+m4m5-m4m6+m4m7+m5m6-m5m7+m6m7-1),

A.6.

{Γk},{Δk} expressed through {αk},{βk}:

Γ0=1,
Δ0=m1,
Γ2=2,
Δ2=2m2,
Γ4=2[m2β3+α3],
Δ4=2[m2α3+β3],
Γ6=2[m2(α3β5+β3α5)+α3α5+β3β5],
Δ6=2[m2(α3α5+β3β5)+α3β5+β3α5],
Γ8=2[m2(α3α5β7+α3β5α7+β3α5α7+β3β5β7)+α3α5α7+α3β5β7+β3α5β7+β3β5α7],
Δ8=2[m2(α3α5α7+α3β5β7+β3α5β7+β3β5α7)+α3α5β7+α3β5α7+β3α5α7+β3β5β7],
Γ1=0,
Δ1=2,
Γ3=2β2,
Δ3=2α2,
Γ5=2(α2β4+β2α4),Δ5=(α2α4+β2β4),
Γ7=2(α2α4β6+α2β4α6+β2α4α6+β2β4β6),
Δ7=2(α2α4α6+α2β4β6+β2α4β6+β2β4α6),
Γ9=2(α2α4α6β8+α2α4β6α8+α2β4α6α8+β2α4α6α8+α2β4β6β8+β2α4β6β8+β2β4α6β8+β2β4β6α8),
Δ9=2(α2α4α6α8+α2α4β6β8+α2β4α6β8+α2β4β6α8+β2α4α6β8+β2α4β6α8+β2β4α6α8+β2β4β6α8),

References

[1] M. Akel and H. Begehr, Neumann function for a hyperbolic strip and a class of related plane domains, Math. Nachr. 290 (2017), 490–506. Search in Google Scholar

[2] S. Altynbek and H. Begehr, A pair of rational double sequences, Georgian Math. J., to appear. Search in Google Scholar

[3] H. Begehr, Green function for a hyperbolic strip and a class of related plane domains, Appl. Anal. 93 (2014), 2370–2385. Search in Google Scholar

[4] H. Begehr, H. Lin, H. Liu and B. Shupeyeva, An iterative real sequence based on 2 and 3 providing a plane parqueting and harmonic Green functions, Complex Var. Elliptic Equ. 66 (2021), no. 6–7, 988–1013. Search in Google Scholar

Received: 2020-04-12
Accepted: 2021-08-11
Published Online: 2021-09-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.