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{{Short description|Meromorphic function}}
{{For|Barnes's gamma function|multiple gamma function}}
{{one source|date=August 2021}}{{Use American English|date = March 2019}}
[[File:Mplwp polygamma03.svg|thumb|300px|Graphs of the polygamma functions {{math|''ψ''}}, {{math|''ψ''(1)}}, {{math|''ψ''(2)}} and {{math|''ψ''(3)}} of real arguments]]
[[File:Plot of polygamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1.svg| thumb | alt=Plot of the [[digamma function]], the first polygamma function, in the complex plane, with colors showing one cycle of phase shift around each pole and zero | Plot of the [[digamma function]], the first polygamma function, in the complex plane from −2−2i to 2+2i with colors created by Mathematica's function ComplexPlot3D showing one cycle of phase shift around each pole and the zero]]

In [[mathematics]], the '''polygamma function of order {{mvar|m}}''' is a [[meromorphic function]] on the [[complex numbers]] <math>\mathbb{C}</math> defined as the {{math|(''m'' + 1)}}th [[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 {{math|''ψ''(''z'')}} is the [[digamma function]] and {{math|Γ(''z'')}} is the [[gamma function]]. They are [[Holomorphic function|holomorphic]] on <math>\mathbb{C} \backslash\mathbb{Z}_{\le0}</math>. At all the nonpositive integers these polygamma functions have a [[isolated singularity|pole]] of order {{math|''m'' + 1}}. The function {{math|''ψ''(1)(''z'')}} is sometimes called the [[trigamma function]].

{| style="text-align:center"
|+ '''The logarithm of the gamma function and the first few polygamma functions in the complex plane'''
|[[Image:Complex LogGamma.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 0.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 1.jpg|1000x140px|none]]
|-
|{{math|ln Γ(''z'')}}
|{{math|''ψ''(0)(''z'')}}
|{{math|''ψ''(1)(''z'')}}
|-
|[[Image:Complex Polygamma 2.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 3.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 4.jpg|1000x140px|none]]
|-
|{{math|''ψ''(2)(''z'')}}
|{{math|''ψ''(3)(''z'')}}
|{{math|''ψ''(4)(''z'')}}
|}

==Integral representation==
{{see also|Digamma function#Integral representations}}
When {{math|''m'' > 0}} and {{math|Re ''z'' > 0}}, 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 {{math|{{sfrac|(−1)''m''+1 ''tm''|1 − ''e''−''t''}}}}. It follows from [[Bernstein's theorem on monotone functions]] that, for {{math|''m'' > 0}} and {{math|''x''}} real and non-negative, {{math|(−1)''m''+1 ''ψ''(''m'')(''x'')}} is a completely monotone function.

Setting {{math|''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 {{math|''m'' {{=}} 0}} case above but which has an extra term {{math|{{sfrac|''e''−''t''|''t''}}}}.

==Recurrence relation==
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 [[Natural number|<math>\mathbb{N}</math>]] [[unique (mathematics)|unique]]ly to [[positive real numbers]] only due to their recurrence relation and one given function-value, say {{math|''ψ''(''m'')(1)}}, except in the case {{math|''m'' {{=}} 0}} where the additional condition of strict [[Monotonic function|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 {{math|''m'' {{=}} 0}} must be treated differently because {{math|''ψ''(0)}} is not normalizable at infinity (the sum of the reciprocals doesn't converge).

==Reflection relation==
:<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 {{math|''Pm''}} is alternately an odd or even polynomial of degree {{math|{{abs|''m'' − 1}}}} with integer coefficients and leading coefficient {{math|(−1)''m''⌈2''m'' − 1⌉}}. 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 theorem==
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 representation==
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 {{math|''m'' > 0}} and any complex {{mvar|z}} 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>
{{cite journal|first1=K. S. |last1=Kölbig|year=1996|journal=Journal of Computational and Applied Mathematics |volume=75|number=1|pages=43–46|title=The polygamma function <math>\psi^{(k)}(x)</math> for <math>x=\frac{1}{4}</math> and <math>x=\frac{3}{4}</math>|doi=10.1016/S0377-0427(96)00055-6|doi-access=free}}
</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 {{math|''δ''''n''0}} 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 series==
The [[Taylor series]] at {{math|''z'' {{=}} -1}} 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 {{math|{{abs|''z''}} < 1}}. Here, {{mvar|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:<ref>{{cite journal|first1=J.|last1=Blümlein|journal=Comput. Phys. Commun.|year=2009|volume=180|pages=2218–2249|doi=10.1016/j.cpc.2009.07.004|title=Structural relations of harmonic sums and Mellin transforms up to weight w=5|issue=11 |arxiv=0901.3106|bibcode=2009CoPhC.180.2218B }}</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 {{math|''B''1 {{=}} {{sfrac|1|2}}}}, i.e. the [[Bernoulli numbers]] of the second kind.

==Inequalities==
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 {{math|''m'' ≥ 1}} and {{math|''t'' ≥ 0}}. 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 {{math|''et'' ≥ 1 + ''t''}} implies that
:<math>\left(t^{m-1} + t^m\right) - \frac{t^m}{1 - e^{-t}}</math>
is non-negative for all {{math|''m'' ≥ 1}} and {{math|''t'' ≥ 0}}, 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 {{math|''m'' ≥ 1}} and {{math|''x'' > 0}},
:<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 function|convex]].
* For <math>m=0</math>, the digamma function, <math>\psi(x)=\psi^{(0)}(x)</math>, is strictly monotonic increasing and strictly [[concave function|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 asymptote===
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>.

==See also==
* [[Factorial]]
* [[Gamma function]]
* [[Digamma function]]
* [[Trigamma function]]
* [[Generalized polygamma function]]

==References==
{{Reflist}}
* {{cite book|first1=Milton|last1=Abramowitz|first2=Irene A.|last2=Stegun|title=[[Abramowitz and Stegun|Handbook of Mathematical Functions]]|date=1964|publisher=Dover Publications|location=New York|isbn=978-0-486-61272-0|chapter-url=https://personal.math.ubc.ca/~cbm/aands/page_260.htm|chapter=Section 6.4}}

[[Category:Gamma and related functions]]

Polygamma function - Revision history

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