Polygamma function

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

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

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

This expresses the polygamma function as the Laplace transform of (−1)^m+1 t^m/1 − e^−t. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)^m+1 ψ^(m)(x) is a completely monotone function.

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.

It satisfies the recurrence relation

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

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

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

and

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

for all ${\displaystyle n\in \mathbb {N} }$, where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain ${\displaystyle \mathbb {N} }$ 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 ${\displaystyle \mathbb {R} ^{+}}$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on ${\displaystyle \mathbb {R} ^{+}}$ 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).

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

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

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

The multiplication theorem gives

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

and

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

for the digamma function.

The polygamma function has the series representation

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

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

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

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

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

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

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

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

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,

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

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

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

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

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

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

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

Where δ_n0 is the Kronecker delta.

Also the Lerch transcendent

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

can be denoted in terms of polygamma function

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

The Taylor series at z = -1 is

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

and

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

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.

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

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

and

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

where we have chosen B₁ = 1/2, i.e. the Bernoulli numbers of the second kind.

The hyperbolic cotangent satisfies the inequality

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

and this implies that the function

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

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

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

is completely monotone. The convexity inequality e^t ≥ 1 + t implies that

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

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

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

Therefore, for all m ≥ 1 and x > 0,

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

Since both bounds are strictly positive for ${\displaystyle x>0}$, we have:

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

This can be seen in the first plot above.

• Factorial
• Gamma function
• Digamma function
• Trigamma function
• Generalized polygamma function

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