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1: 5.15 Polygamma Functions
§5.15 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. …
5.15.2 ψ ( n ) ( 1 ) = ( - 1 ) n + 1 n ! ζ ( n + 1 ) ,
For B 2 k see §24.2(i). …
2: 5.24 Software
§5.24(iii) ψ ( x ) , ψ ( n ) ( x ) , x
§5.24(iv) Γ ( z ) , ψ ( z ) , ψ ( n ) ( z ) , z
3: 5.21 Methods of Computation
For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). Similarly for ln Γ ( z ) , ψ ( z ) , and the polygamma functions. …
4: 5.16 Sums
§5.16 Sums
5: 5.22 Tables
§5.22(ii) Real Variables
Abramowitz and Stegun (1964, Chapter 6) tabulates Γ ( x ) , ln Γ ( x ) , ψ ( x ) , and ψ ( x ) for x = 1 ( .005 ) 2 to 10D; ψ ′′ ( x ) and ψ ( 3 ) ( x ) for x = 1 ( .01 ) 2 to 10D; Γ ( n ) , 1 / Γ ( n ) , Γ ( n + 1 2 ) , ψ ( n ) , log 10 Γ ( n ) , log 10 Γ ( n + 1 3 ) , log 10 Γ ( n + 1 2 ) , and log 10 Γ ( n + 2 3 ) for n = 1 ( 1 ) 101 to 8–11S; Γ ( n + 1 ) for n = 100 ( 100 ) 1000 to 20S. …
6: Software Index
7: Bibliography
  • V. S. Adamchik (1998) Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.
  • 8: Bibliography K
  • E. Konishi (1996) Calculation of complex polygamma functions. Sci. Rep. Hirosaki Univ. 43 (1), pp. 161–183.
  • 9: Bibliography B
  • K. O. Bowman (1984) Computation of the polygamma functions. Comm. Statist. B—Simulation Comput. 13 (3), pp. 409–415.