Table 4 Some numerical results of the \(\Delta _{i}\) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)

\(q =\frac{1}{7}\)

2.4207

1.1167

1.0666

4.0611

1.6404

2.4207

1.1167

1.0993

4.425

1.6906

2.418

1.1141

1.1047

4.4214

1.6901

2.4176

1.1137

1.1055

4.4209

1.69

2.4176

1.1136

1.1056

4.4208

1.69

2.4176

1.1136

1.1057

4.4208

1.69

2.4176

1.1136

1.1057

4.4208

1.69

2.4176

1.1136

1.1057

4.4208

1.69

\(q =\frac{1}{2}\)

2.9236

1.2296

0.3506

5.7039

2.4014

2.8254

1.1556

0.4124

5.5329

2.3413

2.7809

1.1221

0.4461

5.4554

2.3141

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⋮

2.7394

1.0909

0.4813

5.3831

2.2887

2.7393

1.0907

0.4814

5.3828

2.2886

2.7392

1.0907

0.4815

5.3827

2.2885

2.7391

1.0907

0.4815

5.3826

2.2885

2.7391

1.0906

0.4816

5.3826

2.2885

2.7391

1.0906

0.4816

5.3826

2.2885

2.7391

1.0906

0.4816

5.3826

2.2885

\(q =\frac{7}{8}\)

5.978

2.8238

0.0053

12.7182

5.9654

5.1343

2.3155

0.0102

10.9018

5.102

4.6238

2.0079

0.0168

9.8013

4.5783

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3.0752

1.075

0.191

6.4539

2.9802

3.0751

1.075

0.191

6.4539

2.9802

3.0751

1.075

0.191

6.4539

2.9802

3.0751

1.0749

0.191

6.4538

2.9802

3.0751

1.0749

0.191

6.4538

2.9802

3.0751

1.0749

0.191

6.4538

2.9802

3.0751

1.0749

0.191

6.4538

2.9802

3.0751

1.0749

0.191

6.4538

2.9802

3.0751

1.0749

0.191

6.4538

2.9801

3.0751

1.0749

0.1911

6.4538

2.9801

3.0751

1.0749

0.1911

6.4538

2.9801

3.0751

1.0749

0.1911

6.4538

2.9801

3.0751

1.0749

0.1911

6.4538

2.9801

3.0751

1.0749

0.1911

6.4537

2.9801

3.0751

1.0749

0.1911

6.4537

2.9801

3.0751

1.0749

0.1911

6.4537

2.9801