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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. …
5.15.2 ψ ( n ) ( 1 ) = ( - 1 ) n + 1 n ! ζ ( n + 1 ) ,
5.15.3 ψ ( n ) ( 1 2 ) = ( - 1 ) n + 1 n ! ( 2 n + 1 - 1 ) ζ ( n + 1 ) ,
For B 2 k see §24.2(i). …
2: 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. …
3: 5.16 Sums
§5.16 Sums
4: 5.22 Tables
§5.22(ii) Real Variables
5: Bibliography K
  • E. Konishi (1996) Calculation of complex polygamma functions. Sci. Rep. Hirosaki Univ. 43 (1), pp. 161–183.
  • 6: Bibliography
  • V. S. Adamchik (1998) Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.
  • 7: Bibliography B
  • K. O. Bowman (1984) Computation of the polygamma functions. Comm. Statist. B—Simulation Comput. 13 (3), pp. 409–415.
  • 8: Software Index
    Open Source With Book Commercial
    5.24(iii) ψ ( x ) , ψ ( n ) ( x ) , x
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    9: 5.24 Software
    A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C7). …
    §5.24(iii) ψ ( x ) , ψ ( n ) ( x ) , x
    §5.24(iv) Γ ( z ) , ψ ( z ) , ψ ( n ) ( z ) , z
    No research software has been identified for this function. …