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1: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …For further implications of the parameter constraints see the Note in §18.5(iii).
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
2: 18.5 Explicit Representations
For corresponding formulas for Chebyshev, Legendre, and the Hermite He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
3: 18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) .
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
4: 18.19 Hahn Class: Definitions
Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
p n ( x ) X w x h n
5: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
6: 15.6 Integral Representations
15.6.1 F ( a , b ; c ; z ) = 1 Γ ( b ) Γ ( c - b ) 0 1 t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c > b > 0 .
15.6.6 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( - t ) Γ ( c + t ) ( - z ) t d t , | ph ( - z ) | < π ; a , b 0 , - 1 , - 2 , .
15.6.7 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( c - a - b - t ) Γ ( - t ) ( 1 - z ) t d t , | ph ( 1 - z ) | < π ; a , b , c - a , c - b 0 , - 1 , - 2 , .
15.6.8 F ( a , b ; c ; z ) = 1 Γ ( c - d ) 0 1 F ( a , b ; d ; z t ) t d - 1 ( 1 - t ) c - d - 1 d t , | ph ( 1 - z ) | < π ; c > d > 0 .
15.6.9 F ( a , b ; c ; z ) = 0 1 t d - 1 ( 1 - t ) c - d - 1 ( 1 - z t ) a + b - λ F ( λ - a , λ - b d ; z t ) F ( a + b - λ , λ - d c - d ; ( 1 - t ) z 1 - z t ) d t , | ph ( 1 - z ) | < π ; λ , c > d > 0 .
7: 33.14 Definitions and Basic Properties
33.14.13 0 s ( ϵ 1 , ; r ) s ( ϵ 2 , ; r ) d r = δ ( ϵ 1 - ϵ 2 ) , ϵ 1 , ϵ 2 > 0 ,
8: 10.32 Integral Representations
10.32.13 K ν ( z ) = ( 1 2 z ) ν 4 π i c - i c + i Γ ( t ) Γ ( t - ν ) ( 1 2 z ) - 2 t d t , c > max ( ν , 0 ) , | ph z | < 1 2 π .
9: 19.16 Definitions
Thus R - a ( b ; z ) is symmetric in the variables z j and z if the parameters b j and b are equal. …
19.16.9 R - a ( b ; z ) = 1 B ( a , a ) 0 t a - 1 j = 1 n ( t + z j ) - b j d t = 1 B ( a , a ) 0 t a - 1 j = 1 n ( 1 + t z j ) - b j d t , b 1 + + b n > a > 0 , b j , z j ( - , 0 ] ,
19.16.12 R - a ( b 1 , , b 4 ; c - 1 , c - k 2 , c , c - α 2 ) = 2 ( sin 2 ϕ ) 1 - a B ( a , a ) 0 ϕ ( sin θ ) 2 a - 1 ( sin 2 ϕ - sin 2 θ ) a - 1 ( cos θ ) 1 - 2 b 1 ( 1 - k 2 sin 2 θ ) - b 2 ( 1 - α 2 sin 2 θ ) - b 4 d θ ,
R - a ( b ; z ) is an elliptic integral iff the z ’s are distinct and exactly four of the parameters a , a , b 1 , , b n are half-odd-integers, the rest are integers, and none of a , a , a + a is zero or a negative integer. …
10: 10.40 Asymptotic Expansions for Large Argument
10.40.6 I ν ( z ) K ν ( z ) 1 2 z ( 1 - 1 2 μ - 1 ( 2 z ) 2 + 1 3 2 4 ( μ - 1 ) ( μ - 9 ) ( 2 z ) 4 - ) ,
10.40.7 I ν ( z ) K ν ( z ) - 1 2 z ( 1 + 1 2 μ - 3 ( 2 z ) 2 - 1 2 4 ( μ - 1 ) ( μ - 45 ) ( 2 z ) 4 + ) ,
10.40.9 α k ( ν ) = ( 4 ν 2 - 1 2 ) ( 4 ν 2 - 3 2 ) ( 4 ν 2 - ( 2 k + 1 ) 2 ) ( k + 1 ) ! ( 1 4 ν 2 - 1 2 + 1 4 ν 2 - 3 2 + + 1 4 ν 2 - ( 2 k + 1 ) 2 ) .
10.40.12 𝒱 z , ( t - ) { | z | - , | ph z | 1 2 π , χ ( ) | z | - , 1 2 π | ph z | π , 2 χ ( ) | z | - , π | ph z | < 3 2 π ,