Polygamma Function

A special function mostly commonly denoted , , or which is given by the st derivative of the logarithm of the gamma function (or, depending on the definition, of the factorial ). This is equivalent to the th normal derivative of the logarithmic derivative of (or ) and, in the former case, to the th normal derivative of the digamma function . Because of this ambiguity in definition, two different notations are sometimes (but not always) used, namely

			(1)
			(2)
			(3)

which, for can be written as

			(4)
			(5)

where is the Hurwitz zeta function.

The alternate notation

(6)

is sometimes used, with the two notations connected by

(7)

Unfortunately, Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach's "" is equal to in the usual notation. Also note that the function is equivalent to the digamma function and is sometimes known as the trigamma function.

is implemented in the Wolfram Language as PolyGamma[n, z] for positive integer . In fact, PolyGamma[nu, z] is supported for all complex (Grossman 1976; Espinosa and Moll 2004).

The polygamma function obeys the recurrence relation

(8)

the reflection formula

(9)

and the multiplication formula,

(10)

where is the Kronecker delta.

The polygamma function is related to the Riemann zeta function and the generalized harmonic numbers by

(11)

for , 2, ..., and in terms of the Hurwitz zeta function as

(12)

The Euler-Mascheroni constant is a special value of the digamma function , with

			(13)
			(14)

In general, special values for integral indices are given by

			(15)
			(16)

giving the digamma function, trigamma function, and tetragamma function identities

			(17)
			(18)
			(19)
			(20)
			(21)

and so on.

The polygamma function can be expressed in terms of Clausen functions for rational arguments and integer indices. Special cases are given by

			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)

where is Catalan's constant, is the Riemann zeta function, and is the Dirichlet beta function.