# Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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### Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after ...
The generalized polygamma function^{[1]}$$\!^{[2]} is defined as$$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1where ... 1answer 257 views ### Re-Expressing the Digamma I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function \psi^{(0)}(n) in terms of a trigonometric function or a logarithmic ... 1answer 88 views ### Logarithmic Integral I Consider the integral \begin{align} I = \int_{0}^{1} \frac{ \ln^{2}x}{(x^{2} - x + 1)^{2}} \, dx. \end{align} It is speculated that the value is \begin{align} I = \frac{10 \, \pi^{3}}{3^{5} \, \sqrt{3}... 1answer 116 views ### Bounding infinite series derived from polygamma functions Let f(x) = 2 \psi^{(1)}(x+1) + x \psi^{(2)}(x+1) for x > 0 , where \psi^{(i)}(x) is the i^{th} derivative of the digamma function \psi(x). The goal is to prove that f(x) < \frac{... 1answer 52 views ### Solve for X in the given Equation(Gamma Curve) I'm having a set of points in the form: {(\frac A{255})}^{\frac 1x}=\frac B{255} I need to Find x in the Equation.Where A & B are set of constants ranging from 0 to 255. Please ... 1answer 484 views ### Asymptotic behavior of the zeros of the digamma function The gamma function has just one extremum on each interval (k,k+1), where k is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let z_n denote the n-... 6answers 353 views ### Various evaluations of the series \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3} I recently ran into this series:\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$Of course this is just a special case of the Beta Dirichlet Function , for s=3. I had given the following solution:... 3answers 356 views ### Residue of \Gamma^{2} and \Gamma^{3} Based on wiki, the residues of \Gamma at non positive integers are given by:$$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$I have been trying to find residue for \Gamma^{2} ... 0answers 28 views ### How can I get the value of k? I have to get the value of k in this equation: \frac{(\lambda T)^k [ln(\lambda T)-\psi(k+1)]}{\Gamma(k+1)}=0, where \psi is the digamma function. Since \Gamma(k+1) is in the denominator and the ... 1answer 83 views ### An identity on Polygamma I would like to know how to prove that:$$\psi^{(n)}(z)=(-1)^{n+1}n! \sum_{k=0}^{\infty}\frac{1}{\left ( k+z \right )^{n+1}}$$I know that \displaystyle \sum_{n=0}^{\infty}\frac{1}{n+z}=-\psi (z) ... 0answers 68 views ### Are these identities Newton series? Newton series is the following expansion of a function:$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$Now ... 0answers 393 views ### Is this similarity just a coincidence? Here is a graph of the function y=-1/x: If we add infinitely many similar functions with a shift of \pi/2 each in both directions, we get \tan x. But if we do the same only in one direction, we ... 0answers 171 views ### Solving an integral (or series) equations system Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ... 1answer 183 views ### Sum of complex digamma functions It seems that the sum of the digamma function of z and the digamma function of its conjugate z^* is always real-valued.$$\psi(z)+\psi(z^*)=\frac{\Gamma'(z)}{\Gamma(z)}+\frac{\Gamma'(z^*)}{\Gamma(... 1answer 154 views ### Double series of Harmonic Numbers In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}... 1answer 332 views ### Generalized FoxTrot SeriesF(a,b,q,x) = \sum_{k=q}^{\infty} \dfrac {(-1)^{k+1} k^a}{k^b+x}The FoxTrot Series is defined as: $$F = \sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1}.$$ Using partial fraction decomposition we can show that $$F = \frac 13 \left[ 1 - \ln2 + \pi\operatorname{... 1answer 596 views ### Prove \psi^{(1)}\left(\frac 56\right)-\psi^{(1)}\left(\frac 16 \right)=5\left(\psi^{(1)}\left(\frac 23\right)-\psi^{(1)}\left(\frac 13\right)\right) While trying to improve this interesting answer by @Anastasiya-Romanova I noticed that$$\psi^{(1)}\left(\frac 56\right)-\psi^{(1)}\left(\frac 16 \right)=5\left(\psi^{(1)}\left(\frac 23\right)-\psi^{(... 1answer 155 views ### Digamma equation identification I was messing around with the digamma function the other day, and I discovered this identity: $$\psi\left(\frac ab\right)=\sum_{\substack{\large\rho^b=1\\\large\rho\ne1}}(\rho^a-1)\ln(1-\bar\rho)-\... 1answer 379 views ### Bounds on the real and imaginary parts of the digamma function \psi Let \psi be the digamma function given by$$\psi (z)=\left.\frac {d}{dt}\log\Gamma (t)\right|_{t=z}. $$I wonder does anyone know of any lower and/or upper bounds on the real and imaginary parts ... 2answers 186 views ### Special values \psi \left(\frac12\right) and \psi \left(\frac13\right) I wonder if it is easy to prove that$$ \begin{align} \psi \left(\frac12\right) & = -\gamma - 2\ln 2, \\ \psi \left(\frac13\right) & = -\gamma + \frac\pi6\sqrt{3}- \frac32\ln 3, \end{align} $$... 1answer 83 views ### Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm. What function satisfies the following: Let the matrix:$$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ 1&1&... 1answer 760 views ### Evaluating\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dxUsing the method shown here, I have found the following closed form. \int_0^{\!\Large \frac{\pi}{2}}\!\!\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan x}\right)^2\! \mathrm dx= 3\ln2-\frac{4}{\pi}... 4answers 182 views ### Value of \psi\left(\frac{1}{2}\right) I apologise if this is a dumb question, but I have trouble deriving \displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}. I have tried the following. \begin{align} \psi\left(\frac{1}{2}\right) &... 2answers 326 views ### Trigamma identity 4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}. I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]):4\,\... 1answer 295 views ### Looking for an identity connecting polylogarithm and polygamma functions of arguments\frac14$and$\frac34$I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments$\frac14$and$\frac34$. But I don't remember details, and searching my books and the Internet ... 1answer 1k views ### Derivative of Binomial Coefficient$\binom{2N}{N-x}$with respect to$x$I've got$\binom{2N}{N-x}$and I'd like to take the derivative with respect to$x$. I know that I can take the derivative of$\binom{n}{k}$w.r.t. n using logarithmic differentiation, but that's not ... 1answer 824 views ### A closed form of$\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$? The following result $$\sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3)$$ where$\psi^{(1)}$is the polygamma function makes me think there is a nice sum for the series $$\sum_{k=1}^\... 2answers 232 views ### How is \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right) when \alpha is great? Let \psi := \Gamma'/\Gamma denote the digamma function. Could you find, as \alpha tends to +\infty, an equivalent term for the following series?$$ \sum_{n=1}^{\infty} \left( \psi (\alpha ... 1answer 92 views ### Is there a nice solution to the equation$\Psi(x)=\ln(\pi)$with a positive real$x$? I tried to find a nice solution to the following equation: $$\Psi(x)=\ln(\pi)$$ with$x\in\Bbb R_{\ge0}$and where$\Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Is there a nice expression for x satisfying ... 1answer 489 views ### Numeric approximation for fitting a Gamma Distribution with a single parameter Given a series of$N$observations$\left(x_1, \ldots, x_N\right)$that follow a Gamma distribution with a single parameter,$ \text{Gamma}(k, k)$, what is the maximum likelihood estimate of$ k $?. ... 2answers 2k views ### The most complete reference for identities and special values for polylogarithm and polygamma functions I am looking for a book, paper, web site, etc. (or several ones) containing the most complete list of identities and special values for the polylogarithm$\operatorname{Li}_s(z)$and polygamma$\psi^{(...
In the reflection formula for the polygamma function $$\psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z)$$ the right hand side is a polynomial \$(-1)^{n}\...