# Book:Earl David Rainville/Special Functions

## Earl David Rainville: Special Functions

Published $1960$, The Macmillan Company, New York.

### Contents

Chapter 1: INFINITE PRODUCTS
1. Introduction
2. Definition of an infinite product
3. A necessary condition for convergence
4. The associated series of logarithms
5. Absolute convergence
6. Uniform convergence
Chapter 2: THE GAMMA AND BETA FUNCTIONS
7. The Euler or Mascheroni constant $\gamma$
$7.(1)$
$7.(2)$
7.(3)
8. The Gamma function
$8.(1)$
9. A series for $\dfrac{\Gamma'(z)}{\Gamma(z)}$
10. Evaluation of $\Gamma(1)$ and $\Gamma'(1)$
11. The Euler product for $\Gamma(z)$
12. The difference equation $\Gamma(z+1)=z\Gamma(z)$
13. The order symbols $o$ and $O$
14. Evaluation of certain infinite products
15. Euler's integral for $\Gamma(z)$
$15.(1)$
16. The Beta function
17. The value of $\Gamma(z)\Gamma(1-z)$
18. The factorial function
$(1)$
19. Legendre's duplication formula
20. Gauss' multiplication theorem
21. A summation formula due to Euler
22. The behavior of $\log \Gamma(z)$ for large $|z|$
Chapter 3: ASYMPTOTIC SERIES
23. Definition of an asymptotic expansion
25. Algebraic properties
26. Term-by-term integration
27. Uniqueness
28. Watson's lemma
Chapter 4: THE HYPERGEOMETRIC FUNCTION
29. The function $F(a,b;c;z)$
30. A simple integral form
31. $F(a,b;c;1)$ as a function of the parameters
32. Evaluation of $F(a,b;c;1)$
33. The contiguous function relations
34. The hypergeometric differential equation
35. Logarithmic solutions of the hypergeometric equation
36. $F(a,b;c;z)$ as a function of its parameters
37. Elementary series manipulations
38. Simple transformations
39. Relation between functions of $z$ and $1-z$
42. A theorem due to Kummer
Chapter 5: GENERALIZED HYPERGEOMETRIC FUNCTIONS
44. The function ${}_pF_q$
45. The exponential and binomial functions
46. A differential equation
47. Other solutions of the differential equation
48. The contiguous function relations
49. A simple integral
50. The ${}_pF_q$ with unit argument
51. Saalschütz' theorem
52. Whipple's theorem
53. Dixon's theorem
54. Contour integrals of Barnes' type
55. The Barnes integrals and the function ${}_pF_q$
56. A useful integral
Chapter 6: BESSEL FUNCTIONS
57. Remarks
58. Definition of $J_n(z)$
59. Bessel's differential equation
60. Differential recurrence relations
61. A pure recurrence relation
62. A generating function
63. Bessel's integral
64. Index half an odd integer
65. Modified Bessel functions
66. Neumann Polynomials
67. Neumann series
Chapter 7: THE CONFLUENT HYPERGEOMETRIC FUNCTION
68. Basic properties of ${}_1F_1$
69. Kummer's first formula
70. Kummer's second formula
Chapter 8: GENERATING FUNCTIONS
71. The generating function concept
72. Generating functions of the form $G(2xt-t^2)$
73. Sets generated by $e^t \psi(xt)$
74. The generating functions $A(t) \exp [-xt/(1-t)]$
75. Another class of generating functions
76. Boas and Buck generating functions
77. An extension
Chapter 9: ORTHOGONAL POLYNOMIALS
78. Simple sets of polynomials
79. Orthogonality
80. An equivalent condition for orthogonality
81. Zeros of orthogonal polynomials
82. Expansion of polynomials
83. The three-term recurrence relation
84. The Christoffel-Darboux formula
85. Normalization; Bessel's inequality
Chapter 10: LEGENDRE POLYNOMIALS
86. A generating function
87. Differential recurrence relation
88. The pure recurrence relation
89. Legendre's differential equation
90. The Rodrigues formula
91. Bateman's generating function
93. Hypergeometric forms of $P_n(x)$
94. Brafman's generating functions
95. Special properties of $P_n(x)$
96. More generating functions
97. Laplace's first integral form
98. Some bounds on $P_n(x)$
99. Orthogonality
100. An expansion theorem
101. Expansion of $x^n$
102. Expansion of analytic functions
Chapter 11: HERMITE POLYNOMIALS
103. Definition of $H_n(x)$
104. Recurrence relations
105. The Rodrigues formula
106. Other generating functions
107. Integrals
108. The Hermite polynomial as a ${}_2F_0$
109. Orthogonality
110. Expansion of polynomials
111. More generating functions
Chapter 12: LAGUERRE POLYNOMIALS
112. The polynomial $L_n^{(\alpha)}(x)$
113. Generating functions
114. Recurrence relations
115. The Rodrigues formula
116. The differential equation
117. Orthogonality
118. Expansion of polynomials
119. Special properties
120. Other generating functions
121. The simple Laguerre polynomials
Chapter 13: THE SHEFFER CLASSIFICATION AND RELATED TOPICS
122. Differential operators and polynomial sets
123. Sheffer's $A$-type classification
124. Polynomials of Sheffer $A$-type zero
125. An extension of Sheffer's classification
126. Polynomials of $\sigma$-type zero
Chapter 14: PURE RECURRENCE RELATIONS
127. Sister Celine's technique
128. A mild extension
Chapter 15: SYMBOLIC RELATIONS
129. Notation
130. Symbolic relations among classical polynomials
131. Polynomials of symbolic form $L_n(y(x))$
Chapter 16: JACOBI POLYNOMIALS
132. The Jacobi polynomials
133. Bateman's generating function
134. The Rodrigues formula
135. Orthogonality
136. Differential recurrence relations
137. The pure recurrence relation
138. Mixed relations
139. Appell's functions of two variables
140. An elementary generating function
141. Brafman's generating functions
142. Expansion in series of polynomials
Chapter 17: ULTRASPHERICAL AND GEGENBAUER POLYNOMIALS
143. Definitions
144. The Gegenbauer polynomials
145. The ultraspherical polynomials
Chapter 18: OTHER POLYNOMIAL SETS
146. Bateman's $Z_n(x)$
147. Rice's $H_n(\zeta,p,v)$
148. Bateman's $F_n(z)$
$(1)$
$(2)$
$(3)$
149. Sister Celine's polynomials
150. Bessel polynomials
151. Bedient's polynomials
152. Shively's pseudo-Laguerre and other polynomials
153. Bernoulli polynomials
154. Euler polynomials
155. Tchebicheff polynomials
Chapter 19: ELLIPTIC FUNCTIONS
156. Doubly periodic functions
157. Elliptic functions
158. Elementary properties
159. Order of an elliptic function
160. The Weierstrass function $P(z)$
161. Other elliptic functions
162. A differential equation for $P(z)$
163. Connections with elliptic integrals
Chapter 20: THETA FUNCTIONS
164. Definitions
$(1)$
$(2)$
$(3)$
$(4)$
165. Elementary properties
166. The basic property table
167. Location of zeros
168. Relations among squares of theta functions
171. The relation $\theta_1'=\theta_2 \theta_3 \theta_4$
173. The value of $G$
175. The function $\mathrm{sn}(u)$
176. The functions $\mathrm{cn}(u)$ and $\mathrm{dn}(u)$