Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

250

9) For all non-integer values z ∈ C the following is true Γ ( z ) Γ ( 1 − z )= π sin ( π z ) This relation is known as the reflection theorem . 10) For half-integer arguments Γ ( n / 2 ) , n ∈ N has a special form Γ n 2 = ( n − 2 ) !! √ π 2 ( n − 1 ) / 2 i Γ ( z ) Γ ( − z )= −

π z sin ( π z )

(5.153)

.

(5.154)

where n !! denotes the double factorial, defined by n !! =  

n · ( n − 2 ) ··· 5 · 3 · 1 , n > 0 , odd n · ( n − 2 ) ··· 6 · 4 · 2 , n > 0 , even 1 , n = 0 , − 1 .

(5.155)

The proof of these properties can be found in [38]. For different values of parameter α , from (5.154), we obtain values for the gamma function shown in the following table

1 2

3 2

5 2

7 2

9 2

11 2

13 2

α

Γ ( α ) √ π 1

2 √ π

3 2 √ π

5 2 √ π

7 2 √ π

9 2 √ π

11 2 √ π

Table 5.1: Some values of the gamma function.

Similarly, bearing in mind the definition of the gamma function, the generalized binomial coefficients can be defined as follows

Definition Generalized binomial coefficients α

k for α ∈ R , k ∈ N 0 , are defined by = α ( α − 1 )( α − 2 ) ··· ( α − k + 1 ) k !

α

k

( − 1 ) k − 1 Γ ( k − α ) Γ ( 1 − α ) Γ ( k + 1 )

(5.156)

=

Combining the ninth and the fourth property we obtain

Γ ( 1 − z ) − z

π sin ( π z )

Γ ( − z ) Γ ( z + 1 )=

Γ ( z ) z = − Γ ( 1 − z ) Γ ( z )= −

(5.157)

,

or more generally, for k ∈ N 0

( − 1 ) k + 1 Γ ( z − k ) Γ ( k + 1 − z )= Γ ( − z ) Γ ( z + 1 ) .

(5.158)

As already pointed out, the gamma function can be expressed as

Γ ( z + n ) z ( z + 1 ) ··· ( z + n − 1 ) .

Γ ( z )=

(5.159)

12 Weierstrass