The polygamma function is sometimes denoted , and sometimes . In notation,
(1) | |||
(2) | |||
(3) |
(4) |
(5) |
The polygamma function obeys the Recurrence Relation
(6) |
(7) |
(8) |
In general, special values for integral indices are given by
(9) | |||
(10) |
(11) | |||
(12) | |||
(13) | |||
(14) |
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 for and .'' J. Comp. Appl. Math. 75, 43-46, 1996.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 422-424, 1953.
© 1996-9 Eric W. Weisstein