Digital Library of Mathematical Functions
The Gamma Function -- Series Expansions

### Series Expansions

###### Contents

- Maclaurin Series
- Other Series

#### Maclaurin Series

Throughout this subsection ζ ( k ) is

1 Γ ( z ) = ∑ k = 1 ∞ c k z k

where c 1 = 1 , c 2 = γ , and

( k - 1 ) c k = γ c k - 1 - ζ ( 2 ) c k - 2 + ζ ( 3 ) c k - 3 - … + ( - 1 ) k ζ ( k - 1 ) c 1
k ≥ 3 .

For 15D numerical values of c k see Abramowitz and Stegun(1964)(p. 256), and for 31D values see Wrench(1968).

ln Γ ( 1 + z ) = - ln ( 1 + z ) + z ( 1 - γ ) + ∑ k = 2 ∞ ( - 1 ) k ( ζ ( k ) - 1 ) z k k
| z | < 2 .
ψ ( 1 + z ) = - γ + ∑ k = 2 ∞ ( - 1 ) k ζ ( k ) z k - 1
| z | < 1 ,
ψ ( 1 + z ) = 1 2 z - π 2 cot ( π z ) + 1 z 2 - 1 + 1 - γ - ∑ k = 1 ∞ ( ζ ( 2 k + 1 ) - 1 ) z 2 k
| z | < 2 , z ≠ 0 , ± 1 .

For 20D numerical values of the coefficients of the Maclaurin series for Γ ( z + 3 ) see Luke(1969)(p. 299).

When z ≠ 0 , - 1 , - 2 , … ,

ψ ( z ) = - γ - 1 z + ∑ k = 1 ∞ z k ( k + z ) = - γ + ∑ k = 0 ∞ ( 1 k + 1 - 1 k + z )

and

ψ ( z + 1 2 ) - ψ ( z 2 ) = 2 ∑ k = 0 ∞ ( - 1 ) k k + z

Also,

ℑ ψ ( ⅈ y + 1 ) = ∑ k = 1 ∞ y k 2 + y 2