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1: 12.4 Power-Series Expansions
12.4.1 U ( a , z ) = U ( a , 0 ) u 1 ( a , z ) + U ( a , 0 ) u 2 ( a , z ) ,
12.4.2 V ( a , z ) = V ( a , 0 ) u 1 ( a , z ) + V ( a , 0 ) u 2 ( a , z ) ,
12.4.3 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a + 1 2 ) z 2 2 ! + ( a + 1 2 ) ( a + 5 2 ) z 4 4 ! + ) ,
12.4.5 u 1 ( a , z ) = e 1 4 z 2 ( 1 + ( a 1 2 ) z 2 2 ! + ( a 1 2 ) ( a 5 2 ) z 4 4 ! + ) ,
12.4.6 u 2 ( a , z ) = e 1 4 z 2 ( z + ( a 3 2 ) z 3 3 ! + ( a 3 2 ) ( a 7 2 ) z 5 5 ! + ) .
2: 15.7 Continued Fractions
15.7.1 𝐅 ( a , b ; c ; z ) 𝐅 ( a , b + 1 ; c + 1 ; z ) = t 0 u 1 z t 1 u 2 z t 2 u 3 z t 3 ,
3: 15.5 Derivatives and Contiguous Functions
15.5.11 ( c a ) F ( a 1 , b ; c ; z ) + ( 2 a c + ( b a ) z ) F ( a , b ; c ; z ) + a ( z 1 ) F ( a + 1 , b ; c ; z ) = 0 ,
15.5.12 ( b a ) F ( a , b ; c ; z ) + a F ( a + 1 , b ; c ; z ) b F ( a , b + 1 ; c ; z ) = 0 ,
15.5.13 ( c a b ) F ( a , b ; c ; z ) + a ( 1 z ) F ( a + 1 , b ; c ; z ) ( c b ) F ( a , b 1 ; c ; z ) = 0 ,
15.5.14 c ( a + ( b c ) z ) F ( a , b ; c ; z ) a c ( 1 z ) F ( a + 1 , b ; c ; z ) + ( c a ) ( c b ) z F ( a , b ; c + 1 ; z ) = 0 ,
15.5.15 ( c a 1 ) F ( a , b ; c ; z ) + a F ( a + 1 , b ; c ; z ) ( c 1 ) F ( a , b ; c 1 ; z ) = 0 ,
4: 5.1 Special Notation
j , m , n nonnegative integers.
a , b , q , s , w real or complex variables with | q | < 1 .
5: 8.8 Recurrence Relations and Derivatives
8.8.3 w ( a + 2 , z ) ( a + 1 + z ) w ( a + 1 , z ) + a z w ( a , z ) = 0 .
8.8.11 P ( a + n , z ) = P ( a , z ) z a e z k = 0 n 1 z k Γ ( a + k + 1 ) ,
8.8.12 Q ( a + n , z ) = Q ( a , z ) + z a e z k = 0 n 1 z k Γ ( a + k + 1 ) .
8.8.15 d n d z n ( z a γ ( a , z ) ) = ( 1 ) n z a n γ ( a + n , z ) ,
8.8.16 d n d z n ( z a Γ ( a , z ) ) = ( 1 ) n z a n Γ ( a + n , z ) ,
6: 15.1 Special Notation
7: 12.8 Recurrence Relations and Derivatives
12.8.1 z U ( a , z ) U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.2 U ( a , z ) + 1 2 z U ( a , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.3 U ( a , z ) 1 2 z U ( a , z ) + U ( a 1 , z ) = 0 ,
12.8.4 2 U ( a , z ) + U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 .
12.8.5 z V ( a , z ) V ( a + 1 , z ) + ( a 1 2 ) V ( a 1 , z ) = 0 ,
8: 31.17 Physical Applications
31.17.2 x s 2 z k + x t 2 z k 1 + x u 2 z k a = 0 , k = 1 , 2 ,
31.17.4 Ψ ( 𝐱 ) = ( z 1 z 2 ) s 1 4 ( ( z 1 1 ) ( z 2 1 ) ) t 1 4 ( ( z 1 a ) ( z 2 a ) ) u 1 4 w ( z 1 ) w ( z 2 ) ,
9: 15.11 Riemann’s Differential Equation
15.11.3 w = P { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } .
15.11.7 P { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } = ( z α z γ ) a 1 ( z β z γ ) b 1 P { 0 1 0 0 a 1 + b 1 + c 1 ( z α ) ( β γ ) ( z γ ) ( β α ) a 2 a 1 b 2 b 1 a 1 + b 1 + c 2 } .
15.11.8 z λ ( 1 z ) μ P { 0 1 a 1 b 1 c 1 z a 2 b 2 c 2 } = P { 0 1 a 1 + λ b 1 + μ c 1 λ μ z a 2 + λ b 2 + μ c 2 λ μ } ,
10: 15.10 Hypergeometric Differential Equation
15.10.1 z ( 1 z ) d 2 w d z 2 + ( c ( a + b + 1 ) z ) d w d z a b w = 0 .
15.10.3 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( 1 c ) z c ( 1 z ) c a b 1 .
15.10.5 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( a + b c ) z c ( 1 z ) c a b 1 .
15.10.7 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( a b ) z c ( z 1 ) c a b 1 .
15.10.21 w 1 ( z ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) w 3 ( z ) + Γ ( c ) Γ ( a + b c ) Γ ( a ) Γ ( b ) w 4 ( z ) ,