Polygamma Function
The polygamma function is sometimes denoted 
, and sometimes 
. In notation,
 |  |  | (1) |
 |  | (2) | |
|  | (3) |
where
is the Hurwitz Zeta Function. In the 
Notation (the form returned by the PolyGamma[m,z] function in Mathematica
; Wolfram Research, Champaign, IL), | |  | |
|  | (4) |
where
is the Gamma Function and 
is the Digamma Function. is therefore related to by | (5) |
is equivalent to the Digamma Function . Note that Morse and Feshbach (1953) adopta notation no longer in standard use in which Morse and Feshbach's is equal to the above 
.The polygamma function obeys the Recurrence Relation
 | (6) |
 | (7) |
 | (8) |
is the Kronecker Delta.In general, special values for integral indices are given by
 | |  | (9) |
 | |  | (10) |
giving
 | |  | (11) |
 | |  | (12) |
 | |  | (13) |
 | |  | (14) |
and so on.
R. Manzoni has shown that the polygamma function can be expressed in terms of Clausen Functions for Rational arguments and integer index. Special cases are given by
 | |  | (15) |
 | |  | (16) |
 | |  | (17) |
 | |  | (18) |
 | |  | (19) |
 | |  | |
| (20) | |||
 | |  | |
| (21) | |||
 | |  | (22) |
 | |  | (23) |
 | |  | (24) |
 | |  | (25) |
 | |  | (26) |
 | |  | (27) |
See also Clausen Function, Digamma Function, Gamma Function, Stirling's Series
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Polygamma Functions.'' §6.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 260, 1972. Adamchik, V. S. ``Polygamma Functions of Negative Order.'' Submitted to J. Symb. Comput. Arfken, G. ``Digamma and Polygamma Functions.'' §10.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 549-555, 1985. Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933. Kolbig, V. ``The Polygamma Function Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 422-424, 1953.
for 
and 
.'' J. Comp. Appl. Math. 75, 43-46, 1996.
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