Questions tagged [polygamma]

For questions about, or related to the polygamma function.

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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 ...
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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$$where ... 0answers 120 views Closed form for \sum_{k\geq 1}\frac{1}{(2^k-1)^a} Is there a closed form for$$f(a)=\sum_{k=1}^\infty\frac{1}{(2^k-1)^a},$$where 0<a\in\mathbb{R}. My attempts so far have considered a\in\mathbb{N}, which appears to give finite sums of the q-... 0answers 165 views Recurrence relation for polygamma reflection polynomials In the reflection formula for the polygamma function$$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$the right hand side is a polynomial (-1)^{n}\... 0answers 65 views Generalized hypergeometric function at unity I wonder if there is a closed formula for the following generalized hypergeometric function at z=1:$${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$Specifically, I would like to have a formula in ... 1answer 57 views Path containing zeros of all derivatives Following @mercio's comments, I've rewritten my question in terms of zeros instead of saddles. Also, after more careful consideration, I've decided that perhaps the path I seek might not depend on the ... 0answers 95 views Relation between the derivative of hurwitz zeta and trigamma How to prove that$$\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) $$0answers 39 views Digamma function question I just learned online about Polygamma functions, and I want to know what x equals (and how to get it) when \psi(x)=1 and x>1. 0answers 67 views Infinite Gamma Derivative Identity We have$$ \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}=\... 0answers 104 views Convexity of reciprocal polygamma Is the reciprocal of polygamma functions of odd order convex on$ \mathbb{R^+}$, while that of even order above 0 concave? Plotting the functions suggest so, but I've been trying for days to come up ... 0answers 68 views Does this reduce down to the PolyGamma function? Does this reduce down to the PolyGamma function?$H_n=$$$\lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k \left(\... 0answers 68 views Are these identities Newton series? Newton series is the following expansion of a function:$$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)$$Now ... 0answers 171 views Solving an integral (or series) equations system Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ... 1answer 37 views Dimensional regularization and expansion of gamma function In my calculations, I used dimensional regularization, i.e. replace d\rightarrow d-\epsilon and calculated the divergent integral. Then, I would like to expand the answer into seriers by \epsilon ... 0answers 25 views How to sum this vaguely zeta-looking function to QGammafunctions? For q>1 and s\ge{}1, I'm trying to express$$\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 ... 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)...
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 ...