The Method and General Results of this Essay. The F-equation
10
(4)
Reduction to the F-Equation
14
(25)
Reduction to the F-equation
14
(2)
Notation
16
(1)
Solutions of the F-equation Derived from Familiar Functions
17
(6)
Functions which Satisfy also an Ordinary Differential or Difference Equation of the Second Order
23
(5)
The Uniqueness of the Reductions of §3
28
(7)
An Integral Equation which in Some Cases is Equivalent to the Equation (3.4)
35
(4)
Existence and Uniqueness Theorems
39
(16)
The Existence and Uniqueness of Solutions of a More General Functional Equation
39
(11)
Some Observations on the Existence Theorems
50
(5)
Methods of Treating Special Functions Based on the Uniqueness Theorem for the Condition F(zo, α) = φ(alpha;)
55
(82)
Power Series Solutions
55
(14)
Factorial and Newton Series Solutions
69
(5)
Contour Integral Solutions
74
(8)
Taylor's Theorem. Generating Functions
82
(23)
Certain Definite Integrals
105
(11)
Relations Among Various Solutions of the F-equation
116
(21)
Remarks on Solutions Such That F(z, αo) = ψ(z)
137
(14)
The Uniqueness of Solutions such that F(z, αo) = ψ(z)
137
(1)
Differential Formulas
138
(6)
Integral Formulas
144
(7)
Conclusions and Unsolved Problems
151
(12)
Historical Note
151
(6)
Some Conclusions from this Essay
157
(1)
Some Unanswered Questions
158
(5)
Appendix I. Special Functions
163
(11)
Appendix II. Operators
174
(1)
Appendix III. Examples of Equations of Type (3.4) Not Reducible to the F-Equation
175
(2)
Bibliography
177
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