Template:Short description Template:For Template:One sourceTemplate:Use American English
In mathematics, the polygamma function of order Template:Mvar is a meromorphic function on the complex numbers <math>\mathbb{C}</math> defined as the Template:Mathth derivative of the logarithm of the gamma function:
- <math>\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).</math>
Thus
- <math>\psi^{(0)}(z) = \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}</math>
holds where Template:Math is the digamma function and Template:Math is the gamma function. They are holomorphic on <math>\mathbb{C} \backslash\mathbb{Z}_{\le0}</math>. At all the nonpositive integers these polygamma functions have a pole of order Template:Math. The function Template:Math is sometimes called the trigamma function.
| Template:Math | Template:Math | Template:Math |
| Template:Math | Template:Math | Template:Math |
Integral representationEdit
Template:See also When Template:Math and Template:Math, the polygamma function equals
- <math>\begin{align}
\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{align}</math>
where <math>\zeta(s,q)</math> is the Hurwitz zeta function.
This expresses the polygamma function as the Laplace transform of Template:Math. It follows from Bernstein's theorem on monotone functions that, for Template:Math and Template:Math real and non-negative, Template:Math is a completely monotone function.
Setting Template:Math 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 Template:Math case above but which has an extra term Template:Math.
Recurrence relationEdit
It satisfies the recurrence relation
- <math>\psi^{(m)}(z+1)= \psi^{(m)}(z) + \frac{(-1)^m\,m!}{z^{m+1}}</math>
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
- <math>\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 \ge 1</math>
and
- <math>\psi^{(0)}(n) = -\gamma\ + \sum_{k=1}^{n-1}\frac{1}{k}</math>
for all <math>n \in \mathbb{N}</math>, where <math>\gamma</math> is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain Template:Math uniquely to positive real numbers only due to their recurrence relation and one given function-value, say Template:Math, except in the case Template:Math where the additional condition of strict monotonicity on <math>\mathbb{R}^{+}</math> is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on <math>\mathbb{R}^{+}</math> is demanded additionally. The case Template:Math must be treated differently because Template:Math is not normalizable at infinity (the sum of the reciprocals doesn't converge).
Reflection relationEdit
- <math>(-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)}</math>
where Template:Math is alternately an odd or even polynomial of degree Template:Math with integer coefficients and leading coefficient Template:Math. They obey the recursion equation
- <math>\begin{align} P_0(x) &= x \\ P_{m+1}(x) &= - \left( (m+1)xP_m(x)+\left(1-x^2\right)P'_m(x)\right).\end{align}</math>
Multiplication theoremEdit
The multiplication theorem gives
- <math>k^{m+1} \psi^{(m)}(kz) = \sum_{n=0}^{k-1} \psi^{(m)}\left(z+\frac{n}{k}\right)\qquad m \ge 1</math>
and
- <math>k \psi^{(0)}(kz) = k\ln{k} + \sum_{n=0}^{k-1}
\psi^{(0)}\left(z+\frac{n}{k}\right)</math>
for the digamma function.
Series representationEdit
The polygamma function has the series representation
- <math>\psi^{(m)}(z) = (-1)^{m+1}\, m! \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}</math>
which holds for integer values of Template:Math and any complex Template:Mvar not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as
- <math>\psi^{(m)}(z) = (-1)^{m+1}\, m!\, \zeta (m+1,z).</math>
This relation can for example be used to compute the special values<ref> Template:Cite journal </ref>
- <math>
\psi^{(2n-1)}\left(\frac14\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta(2n)\right); </math>
- <math>
\psi^{(2n-1)}\left(\frac34\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta(2n)\right); </math>
- <math>
\psi^{(2n)}\left(\frac14\right) = -2^{2n-1}\left(\pi^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right); </math>
- <math>
\psi^{(2n)}\left(\frac34\right) = 2^{2n-1}\left(\pi^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right). </math>
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,
- <math>\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}.</math>
This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:
- <math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^\frac{z}{n}.</math>
Now, the natural logarithm of the gamma function is easily representable:
- <math>\ln \Gamma(z) = -\gamma z - \ln(z) + \sum_{k=1}^\infty \left( \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right) \right).</math>
Finally, we arrive at a summation representation for the polygamma function:
- <math>\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)</math>
Where Template:Math is the Kronecker delta.
Also the Lerch transcendent
- <math>\Phi(-1, m+1, z) = \sum_{k=0}^\infty \frac{(-1)^k}{(z+k)^{m+1}}</math>
can be denoted in terms of polygamma function
- <math>\Phi(-1, m+1, z)=\frac1{(-2)^{m+1}m!}\left(\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right)</math>
Taylor seriesEdit
The Taylor series at Template:Math is
- <math>\psi^{(m)}(z+1)= \sum_{k=0}^\infty (-1)^{m+k+1} \frac {(m+k)!}{k!} \zeta (m+k+1) z^k \qquad m \ge 1</math>
and
- <math>\psi^{(0)}(z+1)= -\gamma +\sum_{k=1}^\infty (-1)^{k+1}\zeta (k+1) z^k</math>
which converges for Template:Math. Here, Template:Mvar 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 expansionEdit
These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:<ref>Template:Cite journal</ref>
- <math> \psi^{(m)}(z) \sim (-1)^{m+1}\sum_{k=0}^{\infty}\frac{(k+m-1)!}{k!}\frac{B_k}{z^{k+m}} \qquad m \ge 1</math>
and
- <math> \psi^{(0)}(z) \sim \ln(z) - \sum_{k=1}^\infty \frac{B_k}{k z^k}</math>
where we have chosen Template:Math, i.e. the Bernoulli numbers of the second kind.
InequalitiesEdit
The hyperbolic cotangent satisfies the inequality
- <math>\frac{t}{2}\operatorname{coth}\frac{t}{2} \ge 1,</math>
and this implies that the function
- <math>\frac{t^m}{1 - e^{-t}} - \left(t^{m-1} + \frac{t^m}{2}\right)</math>
is non-negative for all Template:Math and Template:Math. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
- <math>(-1)^{m+1}\psi^{(m)}(x) - \left(\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}}\right)</math>
is completely monotone. The convexity inequality Template:Math implies that
- <math>\left(t^{m-1} + t^m\right) - \frac{t^m}{1 - e^{-t}}</math>
is non-negative for all Template:Math and Template:Math, so a similar Laplace transformation argument yields the complete monotonicity of
- <math>\left(\frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}\right) - (-1)^{m+1}\psi^{(m)}(x).</math>
Therefore, for all Template:Math and Template:Math,
- <math>\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}} \le (-1)^{m+1}\psi^{(m)}(x) \le \frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}.</math>
Since both bounds are strictly positive for <math>x>0</math>, we have:
- <math>\ln\Gamma(x)</math> is strictly convex.
- For <math>m=0</math>, the digamma function, <math>\psi(x)=\psi^{(0)}(x)</math>, is strictly monotonic increasing and strictly concave.
- For <math>m</math> odd, the polygamma functions, <math>\psi^{(1)},\psi^{(3)},\psi^{(5)},\ldots</math>, are strictly positive, strictly monotonic decreasing and strictly convex.
- For <math>m</math> even the polygamma functions, <math>\psi^{(2)},\psi^{(4)},\psi^{(6)},\ldots</math>, are strictly negative, strictly monotonic increasing and strictly concave.
This can be seen in the first plot above.
Trigamma bounds and asymptoteEdit
For the case of the trigamma function (<math>m=1</math>) the final inequality formula above for <math>x>0</math>, can be rewritten as:
- <math>
\frac{x+\frac12}{x^2} \le \psi^{(1)}(x)\le \frac{x+1}{x^2} </math> so that for <math>x\gg1</math>: <math>\psi^{(1)}(x)\approx\frac1x</math>.