Questions tagged [polygamma]

For questions about, or related to the polygamma function.

18 questions with no upvoted or accepted answers
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15
votes
0answers
392 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 ...
10
votes
0answers
220 views

Asymptotic behavior of the generalized polygamma function

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),\tag1$$ where ...
5
votes
0answers
120 views

Closed form for $\sum_{k\geq 1}\frac{1}{(2^k-1)^a}$

Is there a closed form for $$f(a)=\sum_{k=1}^\infty\frac{1}{(2^k-1)^a},$$ where $0<a\in\mathbb{R}$. My attempts so far have considered $a\in\mathbb{N}$, which appears to give finite sums of the q-...
4
votes
0answers
165 views

Recurrence relation for polygamma reflection polynomials

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}\...
2
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0answers
65 views

Generalized hypergeometric function at unity

I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$: $${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$ Specifically, I would like to have a formula in ...
2
votes
1answer
57 views

Path containing zeros of all derivatives

Following @mercio's comments, I've rewritten my question in terms of zeros instead of saddles. Also, after more careful consideration, I've decided that perhaps the path I seek might not depend on the ...
2
votes
0answers
95 views

Relation between the derivative of hurwitz zeta and trigamma

How to prove that $$\zeta '(-1, 1/3)- \zeta '(-1, 2/3) = \frac{\psi ' (1/3)}{6 \sqrt{3} \pi}-\frac{\pi}{9 \sqrt{3}}$$ Does there exist a general formula for $$\zeta '(-1, x)- \zeta '(-1, 1-x) $$
1
vote
0answers
39 views

Digamma function question

I just learned online about Polygamma functions, and I want to know what $x$ equals (and how to get it) when $\psi(x)=1$ and $x>1$.
1
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0answers
67 views

Infinite Gamma Derivative Identity

We have $$ \Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\;dx \tag{1} $$ We also have $$ \frac{d^n}{dz^n}\, x^{z-1}e^{-x}=\log(x)^n e^{-x}x^{z-1}, \;\; z>1 \tag{2} $$ If we create an operator $$ \hat{O}=\...
1
vote
0answers
104 views

Convexity of reciprocal polygamma

Is the reciprocal of polygamma functions of odd order convex on $ \mathbb{R^+}$, while that of even order above 0 concave? Plotting the functions suggest so, but I've been trying for days to come up ...
1
vote
0answers
68 views

Does this reduce down to the PolyGamma function?

Does this reduce down to the PolyGamma function? $H_n=$ $$\lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k \left(\...
1
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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 ...
1
vote
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 ...
0
votes
1answer
37 views

Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
0
votes
0answers
25 views

How to sum this vaguely zeta-looking function to QGammafunctions?

For $q>1$ and $s\ge{}1$, I'm trying to express $$\sum_{n=1}^\infty \dfrac{1}{(1-q^n)^s}$$ in terms of the Qpolygamma function. Just from its definition, it's clear to see that for $s$ the ...
0
votes
0answers
61 views

Does the following digamma/trigamma inequality hold? And can it be formally shown?

Suppose $x > \frac{1}{2}$. Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions, respectively. Does the following inequality hold for any such $x$? $\psi^{(0)}(x) - \psi^{(0)...
0
votes
0answers
27 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 ...
-1
votes
1answer
527 views

Solve the system of equations in the maximum likelihood estimation of Gamma distribution parameters

I'm trying to calculate the two parameters of the Gamma distribution by solving the system of two equations obtained by differentiating the ...