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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 .
15.10.17 w 3 ( z ) = Γ ( 1 - c ) Γ ( a + b - c + 1 ) Γ ( a - c + 1 ) Γ ( b - c + 1 ) w 1 ( z ) + Γ ( c - 1 ) Γ ( a + b - c + 1 ) Γ ( a ) Γ ( b ) w 2 ( z ) ,