Polygamma function

In mathematics, the polygamma function of order $m$ is a meromorphic function on the complex numbers $\mathbb {C}$ defined as the $(m + 1)$ th derivative of the logarithm of the gamma function:

\psi ^{(m)}(z):={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).

Thus

\psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}

holds where $ψ (z)$ is the digamma function and $Γ(z)$ is the gamma function. They are holomorphic on $\mathbb {C} \backslash \mathbb {Z} _{\leq 0}$ . At all the nonpositive integers these polygamma functions have a pole of order $m + 1$ . The function $ψ (1) (z)$ is sometimes called the trigamma function.

**The logarithm of the gamma function and the first few polygamma functions in the complex plane**

$ln Γ(z)$	$ψ (0) (z)$	$ψ (1) (z)$

$ψ (2) (z)$	$ψ (3) (z)$	$ψ (4) (z)$

Integral representation

When $m > 0$ and $Re z > 0$ , the polygamma function equals

{\begin{aligned}\psi ^{(m)}(z)&=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-zt}}{1-e^{-t}}}\,\mathrm {d} t\\&=-\int _{0}^{1}{\frac {t^{z-1}}{1-t}}(\ln t)^{m}\,\mathrm {d} t\\&=(-1)^{m+1}m!\zeta (m+1,z)\end{aligned}}

where $\zeta (s,q)$ is the Hurwitz zeta function.

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Setting $m = 0$ in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the $m = 0$ case above but which has an extra term $⁠ e - t / t ⁠$ .

Recurrence relation

It satisfies the recurrence relation

\psi ^{(m)}(z+1)=\psi ^{(m)}(z)+{\frac {(-1)^{m}\,m!}{z^{m+1}}}

which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:

{\frac {\psi ^{(m)}(n)}{(-1)^{m+1}\,m!}}=\zeta (1+m)-\sum _{k=1}^{n-1}{\frac {1}{k^{m+1}}}=\sum _{k=n}^{\infty }{\frac {1}{k^{m+1}}}\qquad m\geq 1

and

\psi ^{(0)}(n)=-\gamma \ +\sum _{k=1}^{n-1}{\frac {1}{k}}

for all $n\in \mathbb {N}$ , where $\gamma$ is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain $\mathbb {N}$ uniquely to positive real numbers only due to their recurrence relation and one given function-value, say $ψ (m) (1)$ , except in the case $m = 0$ where the additional condition of strict monotonicity on $\mathbb {R} ^{+}$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on $\mathbb {R} ^{+}$ is demanded additionally. The case $m = 0$ must be treated differently because $ψ (0)$ is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relation

(-1)^{m}\psi ^{(m)}(1-z)-\psi ^{(m)}(z)=\pi {\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\cot {\pi z}=\pi ^{m+1}{\frac {P_{m}(\cos {\pi z})}{\sin ^{m+1}(\pi z)}}

where $P m$ is alternately an odd or even polynomial of degree $| m - 1 |$ with integer coefficients and leading coefficient $(-1) m ⌈2 m - 1 ⌉$ . They obey the recursion equation

{\begin{aligned}P_{0}(x)&=x\\P_{m+1}(x)&=-\left((m+1)xP_{m}(x)+\left(1-x^{2}\right)P'_{m}(x)\right).\end{aligned}}

Multiplication theorem

The multiplication theorem gives

k^{m+1}\psi ^{(m)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m)}\left(z+{\frac {n}{k}}\right)\qquad m\geq 1

and

k\psi ^{(0)}(kz)=k\ln {k}+\sum _{n=0}^{k-1}\psi ^{(0)}\left(z+{\frac {n}{k}}\right)

for the digamma function.

Series representation

The polygamma function has the series representation

\psi ^{(m)}(z)=(-1)^{m+1}\,m!\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}

which holds for integer values of $m > 0$ and any complex $z$ not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

\psi ^{(m)}(z)=(-1)^{m+1}\,m!\,\zeta (m+1,z).

This relation can for example be used to compute the special values^[1]

\psi ^{(2n-1)}\left({\frac {1}{4}}\right)={\frac {4^{2n-1}}{2n}}\left(\pi ^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta (2n)\right);

\psi ^{(2n-1)}\left({\frac {3}{4}}\right)={\frac {4^{2n-1}}{2n}}\left(\pi ^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta (2n)\right);

\psi ^{(2n)}\left({\frac {1}{4}}\right)=-2^{2n-1}\left(\pi ^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta (2n+1)\right);

\psi ^{(2n)}\left({\frac {3}{4}}\right)=2^{2n-1}\left(\pi ^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta (2n+1)\right).

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

One more series may be permitted for the polygamma functions. As given by Schlömilch,

{\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {z}{n}}}.

This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:

\Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{\frac {z}{n}}.

Now, the natural logarithm of the gamma function is easily representable:

\ln \Gamma (z)=-\gamma z-\ln(z)+\sum _{k=1}^{\infty }\left({\frac {z}{k}}-\ln \left(1+{\frac {z}{k}}\right)\right).

Finally, we arrive at a summation representation for the polygamma function:

\psi ^{(n)}(z)={\frac {\mathrm {d} ^{n+1}}{\mathrm {d} z^{n+1}}}\ln \Gamma (z)=-\gamma \delta _{n0}-{\frac {(-1)^{n}n!}{z^{n+1}}}+\sum _{k=1}^{\infty }\left({\frac {1}{k}}\delta _{n0}-{\frac {(-1)^{n}n!}{(k+z)^{n+1}}}\right)

Where $δ n 0$ is the Kronecker delta.

Also the Lerch transcendent

\Phi (-1,m+1,z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(z+k)^{m+1}}}

can be denoted in terms of polygamma function

\Phi (-1,m+1,z)={\frac {1}{(-2)^{m+1}m!}}\left(\psi ^{(m)}\left({\frac {z}{2}}\right)-\psi ^{(m)}\left({\frac {z+1}{2}}\right)\right)

Taylor series

The Taylor series at $z = -1$ is

\psi ^{(m)}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}{\frac {(m+k)!}{k!}}\zeta (m+k+1)z^{k}\qquad m\geq 1

and

\psi ^{(0)}(z+1)=-\gamma +\sum _{k=1}^{\infty }(-1)^{k+1}\zeta (k+1)z^{k}

which converges for $| z | < 1$ . Here, $ζ$ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

Asymptotic expansion

These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:^[2]

\psi ^{(m)}(z)\sim (-1)^{m+1}\sum _{k=0}^{\infty }{\frac {(k+m-1)!}{k!}}{\frac {B_{k}}{z^{k+m}}}\qquad m\geq 1

and

\psi ^{(0)}(z)\sim \ln(z)-\sum _{k=1}^{\infty }{\frac {B_{k}}{kz^{k}}}

where we have chosen $B 1 = ⁠ 1 / 2 ⁠$ , i.e. the Bernoulli numbers of the second kind.

Inequalities

The hyperbolic cotangent satisfies the inequality

{\frac {t}{2}}\operatorname {coth} {\frac {t}{2}}\geq 1,

and this implies that the function

{\frac {t^{m}}{1-e^{-t}}}-\left(t^{m-1}+{\frac {t^{m}}{2}}\right)

is non-negative for all $m \geq 1$ and $t \geq 0$ . It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that

(-1)^{m+1}\psi ^{(m)}(x)-\left({\frac {(m-1)!}{x^{m}}}+{\frac {m!}{2x^{m+1}}}\right)

is completely monotone. The convexity inequality $e t \geq 1 + t$ implies that

\left(t^{m-1}+t^{m}\right)-{\frac {t^{m}}{1-e^{-t}}}

is non-negative for all $m \geq 1$ and $t \geq 0$ , so a similar Laplace transformation argument yields the complete monotonicity of

\left({\frac {(m-1)!}{x^{m}}}+{\frac {m!}{x^{m+1}}}\right)-(-1)^{m+1}\psi ^{(m)}(x).

Therefore, for all $m \geq 1$ and $x > 0$ ,

{\frac {(m-1)!}{x^{m}}}+{\frac {m!}{2x^{m+1}}}\leq (-1)^{m+1}\psi ^{(m)}(x)\leq {\frac {(m-1)!}{x^{m}}}+{\frac {m!}{x^{m+1}}}.

Since both bounds are strictly positive for $x>0$ , we have:

$\ln \Gamma (x)$ is strictly convex.
For $m=0$ , the digamma function, $\psi (x)=\psi ^{(0)}(x)$ , is strictly monotonic increasing and strictly concave.
For $m$ odd, the polygamma functions, $\psi ^{(1)},\psi ^{(3)},\psi ^{(5)},\ldots$ , are strictly positive, strictly monotonic decreasing and strictly convex.
For $m$ even the polygamma functions, $\psi ^{(2)},\psi ^{(4)},\psi ^{(6)},\ldots$ , are strictly negative, strictly monotonic increasing and strictly concave.

This can be seen in the first plot above.

Trigamma bounds and asymptote

For the case of the trigamma function ( $m=1$ ) the final inequality formula above for $x>0$ , can be rewritten as:

{\frac {x+{\frac {1}{2}}}{x^{2}}}\leq \psi ^{(1)}(x)\leq {\frac {x+1}{x^{2}}}

so that for $x\gg 1$ : $\psi ^{(1)}(x)\approx {\frac {1}{x}}$ .

References

^ Kölbig, K. S. (1996). "The polygamma function psi^k(x) for x=1/4 and x=3/4". J. Comput. Appl. Math. 75 (1): 43–46. doi:10.1016/S0377-0427(96)00055-6.
^ Blümlein, J. (2009). "Structural relations of harmonic sums and Mellin transforms up to weight w=5". Comp. Phys. Comm. 180: 2218–2249. arXiv:0901.3106. doi:10.1016/j.cpc.2009.07.004.

Abramowitz, Milton; Stegun, Irene A. (1964). "Section 6.4". Handbook of Mathematical Functions. New York: Dover Publications. ISBN 978-0-486-61272-0.

[1] Kölbig, K. S. (1996). "The polygamma function psi^k(x) for x=1/4 and x=3/4". J. Comput. Appl. Math. 75 (1): 43–46. doi:10.1016/S0377-0427(96)00055-6.

[2] Blümlein, J. (2009). "Structural relations of harmonic sums and Mellin transforms up to weight w=5". Comp. Phys. Comm. 180: 2218–2249. arXiv:0901.3106. doi:10.1016/j.cpc.2009.07.004.

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