Digamma Function
The digamma function usually denoted by \(\psi\) is defined as the logarithmic derivative of the gamma function.
Contents
Definition
\[\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \ln \Gamma(z) = \dfrac{\Gamma^\prime (z)}{\Gamma(z)}\]
Functional Equation
\[\psi(s+1)=\psi(s)+\dfrac{1}{s}\]
Consider
\[\Gamma(s+1)=s\Gamma(s).\]
Taking the natural log, we have
\[\ln\big(\Gamma(s+1)\big)=\ln\big(\Gamma(s)\big)+\ln(s).\]
Differentiating with respect to \( s,\) we get
\[\psi(s+1)=\psi(s)+\dfrac{1}{s}.\ _\square\]
Series Representation
\[\psi(s+1)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right)\]
Consider the Weierstrass representation of the gamma function:
\[\Gamma(s)=\dfrac{e^{-\gamma s}}{s} \prod_{k=1}^\infty e^{s/k} \left(1+\dfrac{s}{k}\right)^{-1}. \]
Taking natural logs,
\[\ln\big(\Gamma(s)\big)=-\gamma s-\ln(s)+\sum_{k=1}^\infty \left(\dfrac{s}{k}-\ln \Big(1+\dfrac{s}{k}\Big)\right).\]
Differentiating with respect to \(s,\) we have
\[\psi(s)=-\gamma-\dfrac{1}{s}+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s-1}\right).\]
So, replacing \(s\) with \(s+1\) gives
\[\psi(s+1)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right).\ _\square\]
Integral Representation
\[\psi(s+1) = -\gamma + \int_0^1 \dfrac{1-x^s}{1-x} dx\]
Consider
\[\begin{align} \psi(s+1) &=-\gamma+\sum_{n=1}^\infty \left(\dfrac{1}{n}-\dfrac{1}{n+s}\right)\\ &=-\gamma+\sum_{n=1}^\infty \int_0^1 \big(x^{n-1}-x^{n+s-1}\big)dx\\ &=-\gamma+ \int_0^1\sum_{n=1}^\infty \big(x^{n-1}-x^{n+s-1}\big)dx. \end{align}\]
By applying geometric progression sum, we have
\[\psi(s+1) = -\gamma + \int_0^1 \dfrac{1-x^s}{1-x} dx.\ _\square\]
From this, we can find specific values of the digamma function easily; for example, putting \(s=0,\) we get \(\psi(1)=-\gamma.\)
Also, by the integral representation of harmonic numbers,
\[\psi(s+1) = -\gamma + H_s.\]
Euler's Reflection Formula
By Euler's reflection formula, we have the following relation:
\[\psi(1-z) - \psi(z) = \pi \cot \pi z.\]
Euler's reflection formula is as follows:
\[\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}.\]
Taking natural logarithm and differentiating the above expression, we observe that
\[\ln\big(\Gamma(z)\big)+\ln\big(\Gamma(1-z)\big) = \log \pi - \log \sin \pi z.\]
On differentiating, we get
\[\begin{align} \dfrac{\Gamma^{\prime}(z)}{\Gamma(z)}-\dfrac{\Gamma^{\prime}(1-z)}{\Gamma(1-z)} &= - \dfrac{\pi \cos \pi z}{\sin \pi z}\\\\ \psi(z) - \psi(1-z) &= - \dfrac{\pi \cos \pi z}{\sin \pi z}\\\\ \psi(1-z) - \psi(z) &= \pi \cot \pi z, \end{align}\]
where \(\psi\) denotes the digamma function, which is the logarithmic derivative of the gamma function. \(_\square\)
Legendre Duplication Formula
\[2\psi(2s)=2\ln(2)+\psi(s)+\psi\left(s+\frac12\right)\]
Consider
\[\sqrt{\pi} \ \Gamma(2s)=2^{2s-1} \Gamma(s)\Gamma\left(s+\frac12\right).\]
Taking logs,
\[\ln\big(\sqrt{\pi}\big)+\ln\big(\Gamma(2s)\big)=(2s-1)\ln(2)+\ln\big(\Gamma(s)\big)+\ln\left(\Gamma\Big(s+{\small\frac12}\Big)\right).\]
Differentiating with respect to \(s,\) we have
\[2\psi(2s)=2\ln(2)+\psi(s)+\psi\left(s+\frac12\right).\ _\square\]
Applications in Summations
Prove
\[\sum_{n=0}^\infty \dfrac{1}{n^2+1} =\dfrac{\pi+1}{2} + \dfrac{\pi}{e^{2\pi}-1}.\]
We have
\[ S = \sum_{n=0}^{\infty} \frac{1}{n^2+1} = \frac{1}{2i} \sum_{n=0}^{\infty} \left( \frac{1}{n-i} - \frac{1}{n+i} \right). \]
Rewriting this summation gives
\[\begin{align} 2iS &= \sum_{n=1}^{\infty} \left( \frac{1}{n-1-i} - \frac{1}{n-1+i} \right)\\ &=\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n-1+i} \right)-\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n-1-i} \right). \end{align}\]
By the series representation of the digamma function, this is just
\[\begin{align} 2iS &=\psi(i)-\psi(-i)\\ &=\psi(i)-\psi(1-i)-\dfrac{1}{i}\\ &=-\pi \cot(i\pi)+i\\ &=\pi i \coth(\pi)+i. \end{align}\]
Simplifying further, we get
\[S= \dfrac{1+\pi\coth(\pi)}{2}\implies S=\dfrac{\pi+1}{2} + \dfrac{\pi}{e^{2\pi}-1}.\ _\square\]
Polygamma Functions
The \(n^\text{th}\) polygamma function is given by
\[\psi_n (s)= \dfrac{d^n}{ds^n} \psi(s)=\psi^{(n)}(s).\]
We can get many properties from this; for example, by differentiating the series representation \(n\) times, we have
\[\psi_n(s)=(-1)^{n+1}n! \sum_{k=1}^\infty \dfrac{1}{(k+s-1)^{n+1}}=(-1)^{n+1}n! \sum_{k=0}^\infty \dfrac{1}{(k+s)^{n+1}}=(-1)^{n+1} n!\zeta(n+1,s),\]
where \(\zeta(n+1,s)\) is the Hurwitz zeta function. Putting \(s=1,\) we can get \(\psi_n(1)=(-1)^{n+1} n!\zeta(n+1).\)
From this, we can also get the Taylor series of the digamma function:
\[\begin{align} \psi(s)&=\sum_{n=0}^\infty \dfrac{\psi^{(n)}(1)(s-1)^n}{n!}\\ &=-\gamma+\sum_{n=1}^\infty \dfrac{\psi_n(1)(s-1)^n}{n!}\\ &=-\gamma-\sum_{n=1}^\infty (-1)^n\zeta(n+1)(s-1)^n\\ \psi(s+1)&=-\gamma-\sum_{n=1}^\infty \zeta(n+1)(-s)^n. \end{align}\]
We can differentiate the integral representation \(n\) times to get
\[\psi_n(s+1)=\int_0^1 \dfrac{\ln^n(x) x^s}{x-1}dx.\]
We can also do this to the functional equation to get
\[\psi_n(s+1)=\psi_n(s)+(-1)^nn! z^{-n-1}.\]
Example Problems
\[\sum _{ k=1 }^{ \infty }{ \dfrac { { \psi }^{ 1 }\left( k \right) }{ k } } =\sum _{ n=1 }^{ \infty }{ \dfrac { A }{ { n }^{ B } } } \]
If the equation above holds true for positive integers \(A\) and \(B\), find \(A+B\).
Inspired by Ishan.
Implementation in Mathematica
The digamma functions can be implemented in Mathematica as follows:
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or
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