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

(4) |

where is the Gamma Function and is the Digamma Function. is therefore related to by

(5) |

The polygamma function obeys the Recurrence Relation

(6) |

(7) |

(8) |

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) |

**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

1999-05-25