Polygamma function#Inequalities

{{see also|Digamma function#Integral representations}}

When {{math|m > 0}} and {{math|Re z > 0}}, the polygamma function equals

:$\backslash begin\{align\}$

\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}

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

This expresses the polygamma function as the Laplace transform of {{math|{{sfrac|(−1)^m+1 t^m|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}}}}.

It satisfies the recurrence relation

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

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

:$\backslash frac\{\backslash psi^\{(m)\}(n)\}\{(-1)^\{m+1\}\backslash ,m!\}\; =\; \backslash zeta(1+m)\; -\; \backslash sum\_\{k=1\}^\{n-1\}\; \backslash frac\{1\}\{k^\{m+1\}\}\; =\; \backslash sum\_\{k=n\}^\backslash infty\; \backslash frac\{1\}\{k^\{m+1\}\}\; \backslash qquad\; m\; \backslash ge\; 1$

and

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

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

:$(-1)^m\; \backslash psi^\{(m)\}\; (1-z)\; -\; \backslash psi^\{(m)\}\; (z)\; =\; \backslash pi\; \backslash frac\{\backslash mathrm\{d\}^m\}\{\backslash mathrm\{d\}\; z^m\}\; \backslash cot\{\backslash pi\; z\}\; =\; \backslash pi^\{m+1\}\; \backslash frac\{P\_m(\backslash cos\{\backslash pi\; z\})\}\{\backslash sin^\{m+1\}(\backslash pi\; z)\}$

where {{math|P_m}} 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

:

The multiplication theorem gives

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

and

:$k\; \backslash psi^\{(0)\}(kz)\; =\; k\backslash ln\{k\}\; +\; \backslash sum\_\{n=0\}^\{k-1\}$

\psi^{(0)}\left(z+\frac{n}{k}\right)

for the digamma function.

The polygamma function has the series representation

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

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

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

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

{{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 $\backslash psi^\{(k)\}(x)$ for $x=\backslash frac\{1\}\{4\}$ and $x=\backslash frac\{3\}\{4\}$|doi=10.1016/S0377-0427(96)00055-6|doi-access=free}}

:$$

\psi^{(2n-1)}\left(\frac14\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta(2n)\right);

:$$

\psi^{(2n-1)}\left(\frac34\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta(2n)\right);

:$$

\psi^{(2n)}\left(\frac14\right) = -2^{2n-1}\left(\pi^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right);

:$$

\psi^{(2n)}\left(\frac34\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,

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

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

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

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

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

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

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

Where {{math|δ_n0}} is the Kronecker delta.

Also the Lerch transcendent

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

can be denoted in terms of polygamma function

:$\backslash Phi(-1,\; m+1,\; z)=\backslash frac1\{(-2)^\{m+1\}m!\}\backslash left(\backslash psi^\{(m)\}\backslash left(\backslash frac\{z\}\{2\}\backslash right)-\backslash psi^\{(m)\}\backslash left(\backslash frac\{z+1\}\{2\}\backslash right)\backslash right)$

The Taylor series at {{math|z {{=}} -1}} is

:$\backslash psi^\{(m)\}(z+1)=\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^\{m+k+1\}\; \backslash frac\; \{(m+k)!\}\{k!\}\; \backslash zeta\; (m+k+1)\; z^k\; \backslash qquad\; m\; \backslash ge\; 1$

and

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

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.

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

: $\backslash psi^\{(m)\}(z)\; \backslash sim\; (-1)^\{m+1\}\backslash sum\_\{k=0\}^\{\backslash infty\}\backslash frac\{(k+m-1)!\}\{k!\}\backslash frac\{B\_k\}\{z^\{k+m\}\}\; \backslash qquad\; m\; \backslash ge\; 1$

and

:$\backslash psi^\{(0)\}(z)\; \backslash sim\; \backslash ln(z)\; -\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{B\_k\}\{k\; z^k\}$

where we have chosen {{math|B₁ {{=}} {{sfrac|1|2}}}}, i.e. the Bernoulli numbers of the second kind.

The hyperbolic cotangent satisfies the inequality

:$\backslash frac\{t\}\{2\}\backslash operatorname\{coth\}\backslash frac\{t\}\{2\}\; \backslash ge\; 1,$

and this implies that the function

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

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

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

is completely monotone. The convexity inequality {{math|e^t ≥ 1 + t}} implies that

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

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

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

Therefore, for all {{math|m ≥ 1}} and {{math|x > 0}},

:$\backslash frac\{(m-1)!\}\{x^m\}\; +\; \backslash frac\{m!\}\{2x^\{m+1\}\}\; \backslash le\; (-1)^\{m+1\}\backslash psi^\{(m)\}(x)\; \backslash le\; \backslash frac\{(m-1)!\}\{x^m\}\; +\; \backslash frac\{m!\}\{x^\{m+1\}\}.$

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

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

This can be seen in the first plot above.

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

:$$

\frac{x+\frac12}{x^2} \le \psi^{(1)}(x)\le \frac{x+1}{x^2}

so that for $x\backslash gg1$: $\backslash psi^\{(1)\}(x)\backslash approx\backslash frac1x$.

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

{{Reflist}}

• {{cite book|first1=Milton|last1=Abramowitz|first2=Irene A.|last2=Stegun|title=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