Polygamma Function

The polygamma function is sometimes denoted $F_m(z)$, and sometimes $\psi_m(z)$. In $F_m(z)$ notation,

$$F_m(z) \equiv {d^{m+1}\over dz^{m+1}} \ln z!$$ (1)

$$= (-1)^{m+1}m! \sum_{n=0}^\infty {1\over (z+n)^{m+1}}$$ (2)

$$= (-1)^{m+1}m!\zeta(m+1,z),$$ (3)

where $\zeta(a,z)$ is the Hurwitz Zeta Function. In the $\psi_m$ Notation (the form returned by the PolyGamma[m,z] function in Mathematica ${}^{\scriptstyle\circledRsymbol}$; Wolfram Research, Champaign, IL),

$$\psi_m(z) = {d^{m+1}\over dz^{m+1}} \ln[\Gamma(z)]$$

$$= {d^m\over dz^m} {\Gamma'(z)\over\Gamma(z)} = {d^m\over dz^m} \Psi(z),$$ (4)

where $\Gamma(z)$ is the Gamma Function and $\Psi(z)$ is the Digamma Function. $\psi_m(z)$ is therefore related to $F_m(z)$ by

$$\psi_m(z) = F_m(z-1).$$ (5)

The function $\psi_0(z)$ is equivalent to the Digamma Function $\Psi(z)$. Note that Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach's $\psi_m(z)$ is equal to the above $\psi_{m-1}(z)$.

The polygamma function obeys the Recurrence Relation

$$\psi_n(z+1)=\psi_n(z)+(-1)^n n! z^{-n-1},$$ (6)

the reflection Formula

$$\psi_n(1-z)+(-1)^{n+1}\psi_n(z)=(-1)^n \pi {d^n\over dz^n} \cot(\pi z),$$ (7)

and the multiplication Formula,

$$\psi_n(mz)=\delta_{n0}\ln m+{1\over m^{n+1}} \sum_{k=1}^{m-1} \psi_n\left({z+{k\over m}}\right),$$ (8)

where $\delta_{mn}$ is the Kronecker Delta.

In general, special values for integral indices are given by

$$\psi_n(1) = (-1)^{n+1}n! \zeta(n+1)$$ (9)

$$\psi_n({\textstyle{1\over 2}}) = (-1)^{n+1}n!(2^{n+1}-1)\zeta(n+1),$$ (10)

giving

$$\psi_1({\textstyle{1\over 2}}) = {\textstyle{1\over 2}}\pi^2$$ (11)

$$\psi_1(1) = \zeta(2) = {\textstyle{1\over 6}}\pi^2$$ (12)

$$\psi_2(1) = -2\zeta(3),$$ (13)

$$\psi_3({\textstyle{1\over 2}}) = \pi^4$$ (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

$$\psi_1({\textstyle{1\over 3}}) = {\textstyle{2\over 3}}\pi^2+{\textstyle{3\over 2}}\sqrt{3}[\matho... ...textstyle{2\over 3}}\pi)-\mathop{\it Cl}\nolimits _2({\textstyle{4\over 3}}\pi)$$ (15)

$$\psi_1({\textstyle{2\over 3}}) = {\textstyle{2\over 3}}\pi^2-{\textstyle{3\over 2}}\sqrt{3}[\matho... ...textstyle{2\over 3}}\pi)-\mathop{\it Cl}\nolimits _2({\textstyle{4\over 3}}\pi)$$ (16)

$$\psi_1({\textstyle{1\over 4}}) = \pi^2+4[\mathop{\it Cl}\nolimits _2({\textstyle{1\over 2}}\pi)-\mathop{\it Cl}\nolimits _2({\textstyle{3\over 2}}\pi)]$$ (17)

$$\psi_1({\textstyle{3\over 4}}) = \pi^2-4[\mathop{\it Cl}\nolimits _2({\textstyle{1\over 2}}\pi)-\mathop{\it Cl}\nolimits _2({\textstyle{3\over 2}}\pi)]$$ (18)

$$\psi_2({\textstyle{1\over 2}}) = -8[\mathop{\it Cl}\nolimits _3(0)-\mathop{\it Cl}\nolimits _3(\pi)]$$ (19)

$$\psi_2({\textstyle{1\over 3}}) = -{4\pi^3\over 3\sqrt{3}}-18\mathop{\it Cl}\nolimits _3(0)+9[\math... ...extstyle{2\over 3}}\pi)+\mathop{\it Cl}\nolimits _3({\textstyle{4\over 3}}\pi)]$$ (20)

$$\psi_2({\textstyle{2\over 3}}) = {4\pi^3\over 3\sqrt{3}}-18\mathop{\it Cl}\nolimits _3(0)+9[\matho... ...extstyle{2\over 3}}\pi)+\mathop{\it Cl}\nolimits _3({\textstyle{4\over 3}}\pi)]$$ (21)

$$\psi_2({\textstyle{1\over 4}}) = -2\pi^3-32[\mathop{\it Cl}\nolimits _3(0)-\mathop{\it Cl}\nolimits _3(\pi)]$$ (22)

$$\psi_2({\textstyle{3\over 4}}) = 2\pi^3-32[\mathop{\it Cl}\nolimits _3(0)-\mathop{\it Cl}\nolimits _3(\pi)]$$ (23)

$$\psi_3({\textstyle{1\over 3}}) = {\textstyle{8\over 3}}\pi^4+81\sqrt{3}[\mathop{\it Cl}\nolimits _... ...extstyle{2\over 3}}\pi)-\mathop{\it Cl}\nolimits _4({\textstyle{4\over 3}}\pi)]$$ (24)

$$\psi_3({\textstyle{2\over 3}}) = {\textstyle{8\over 3}}\pi^4-81\sqrt{3}[\mathop{\it Cl}\nolimits _... ...extstyle{2\over 3}}\pi)-\mathop{\it Cl}\nolimits _4({\textstyle{4\over 3}}\pi)]$$ (25)

$$\psi_3({\textstyle{1\over 4}}) = 8\pi^4+384[\mathop{\it Cl}\nolimits _4({\textstyle{1\over 2}}\pi)-\mathop{\it Cl}\nolimits _4({\textstyle{3\over 2}}\pi)]$$ (26)

$$\psi_3({\textstyle{3\over 4}}) = 8\pi^4-384[\mathop{\it Cl}\nolimits _4({\textstyle{1\over 2}}\pi)-\mathop{\it Cl}\nolimits _4({\textstyle{3\over 2}}\pi)].$$ (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 $\psi_k(x)$ for $x=1/4$ and $x=3/4$.'' 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.