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1: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation D ν ( z ) is useful when ν is a nonnegative integer (Hermite polynomial case).
2: 18.1 Notation
They are defined in the literature by C 0 ( 0 ) ( x ) = 1 and …
3: 15.2 Definitions and Analytical Properties
For example, when a = - m , m = 0 , 1 , 2 , , and c 0 , - 1 , - 2 , , F ( a , b ; c ; z ) is a polynomial: …
4: 16.2 Definition and Analytic Properties
Polynomials
5: 12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
6: 12.11 Zeros
Lastly, when a = - n - 1 2 , n = 1 , 2 , (Hermite polynomial case) U ( a , x ) has n zeros and they lie in the interval [ - 2 - a , 2 - a ] . …
7: 13.2 Definitions and Basic Properties
13.2.9 U ( a , n + 1 , z ) = ( - 1 ) n + 1 n ! Γ ( a - n ) k = 0 ( a ) k ( n + 1 ) k k ! z k ( ln z + ψ ( a + k ) - ψ ( 1 + k ) - ψ ( n + k + 1 ) ) + 1 Γ ( a ) k = 1 n ( k - 1 ) ! ( 1 - a + k ) n - k ( n - k ) ! z - k .
Except when a = 0 , - 1 , (polynomial cases), …
8: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33(iv) Special Cases
9: 18.21 Hahn Class: Interrelations
§18.21(ii) Limit Relations and Special Cases
See accompanying text
Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify
10: 13.18 Relations to Other Functions
Special cases are the error functions …
§13.18(v) Orthogonal Polynomials
Special cases of §13.18(iv) are as follows. …
Hermite Polynomials
Laguerre Polynomials