Digital Library of Mathematical Functions
The Gamma Function -- Polygamma Functions

### Polygamma Functions

The functions ψ ( n ) ( z ) , n = 1 , 2 , … , are called the polygamma functions. In particular, ψ ′ ( z ) is the trigamma function; ψ ′ ′ , ψ ( 3 ) , ψ ( 4 ) are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. This includes asymptotic expansions.

In the second and third equations, n = 1 , 2 , 3 , … ; for ζ ( n + 1 )

ψ ′ ( z ) = ∑ k = 0 ∞ 1 ( k + z ) 2
z ≠ 0 , - 1 , - 2 , …
ψ ( n ) ( 1 ) = ( - 1 ) n + 1 n ! ζ ( n + 1 )
ψ ( n ) ( 1 2 ) = ( - 1 ) n + 1 n ! ( 2 n + 1 - 1 ) ζ ( n + 1 )
ψ ′ ( n + 1 2 ) = 1 2 π 2 - 4 ∑ k = 1 n 1 ( 2 k - 1 ) 2

As z → ∞ in | ph z | ≤ π - δ ( < π )

ψ ′ ( z ) ∼ 1 z + 1 2 z 2 + ∑ k = 1 ∞ B 2 k z 2 k + 1