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Theory and Modern Applications

Table 5 Some numerical results of \(\eta_{2}\) and \(\varGamma_{q}(\alpha_{2} - 1)\) from (17) in Example 1 for \(q \in \{ \frac {1}{8}, \frac{1}{2}, \frac{8}{9} \}\). One can heck that \(\frac {\eta_{2}}{\varGamma_{q}(\alpha_{2} - 1)}\) by approximation is smaller than 1

From: Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus

n

\(\vphantom{\sum_{A_{A}}}q =\frac{1}{8}\)

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

\(q =\frac{8}{9}\)

\(\eta_{2}\)

\(\varGamma_{q}(\alpha_{2} - 1)\)

\(\vphantom{\sum_{A_{A}}}\frac{\eta_{2}}{\varGamma_{q}(\alpha _{2} - 1)}\)

\(\eta_{2}\)

\(\varGamma_{q}(\alpha_{2} - 1)\)

\(\frac{\eta_{2}}{\varGamma_{q}(\alpha_{2} - 1)}\)

\(\eta_{2}\)

\(\varGamma_{q}(\alpha_{2} - 1)\)

\(\frac{\eta_{2}}{\varGamma_{q}(\alpha_{2} - 1)}\)

1

0.5097

1.0329

0.4934

0.5010

1.0233

0.4896

0.3771

0.7200

0.5238

2

0.5101

1.0339

0.4933

0.5136

1.0534

0.4876

0.4010

0.7854

0.5106

3

0.5101

1.0341

0.4933

0.5198

1.0679

0.4867

0.4201

0.8356

0.5028

4

0.5101

1.0341

0.4933

0.5228

1.075

0.4863

0.4358

0.8757

0.4977

5

0.5101

1.0341

0.4933

0.5243

1.0786

0.4861

0.4490

0.9085

0.4942

6

0.5101

1.0341

0.4933

0.5250

1.0803

0.4860

0.4601

0.9359

0.4916

7

0.5101

1.0341

0.4933

0.5254

1.0812

0.486

0.4696

0.9589

0.4897

8

0.5101

1.0341

0.4933

0.5256

1.0816

0.4859

0.4778

0.9784

0.4883

9

0.5101

1.0341

0.4933

0.5257

1.0818

0.4859

0.4848

0.9952

0.4872

10

0.5101

1.0341

0.4933

0.5257

1.0819

0.4859

0.4910

1.0096

0.4863

11

0.5101

1.0341

0.4933

0.5258

1.0820

0.4859

0.4963

1.0221

0.4856

34

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5336

1.1071

0.4820

35

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5339

1.1077

0.4819

36

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5341

1.1083

0.4819

37

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5343

1.1088

0.4819

38

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5345

1.1092

0.4819

39

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5347

1.1096

0.4819

40

0.5101

1.0341

0.4933

0.5258

1.0821

0.4859

0.5348

1.1099

0.4819