I want to preface this with the fact that I am WAY out of my depth with my mathematical familiarity with these topics.
While trying to figure out how to compute the solution to the polygamma function of order 1, $\psi_{1}(x)$, I realized that the solutions are all of the form $\frac{\pi^2}{6} - a$, where $a$ changes depending on the value of $x$.
For example for $x$ in [1,6] the solutions for $\psi_{1}(x)$ are:
1: $\frac{\pi^2}{6}-0$
2: $\frac{\pi^2}{6}-1$
3: $\frac{\pi^2}{6}-\frac{5}{4}$
4: $\frac{\pi^2}{6}-\frac{49}{36}$
5: $\frac{\pi^2}{6}-\frac{205}{144}$
6: $\frac{\pi^2}{6}-\frac{5269}{3600}$
It seems to me that there is some pattern in how the $a$ term is calculated, as a function of $x$. I would like to know what this function would be (as I am assuming it would be known, but I can't find any info online or in my books that cover this), or if there is no function for this term, that insight would be appreciated.
Some additional information:
I am using the common integral representation for the polygamma function and an integral calculator to solve the integrals as I have no idea how one solves them by hand.
The Motivation for this is that, as part of some code, I need to compute the $\psi_{1}(x)$ for a large number of $x$, and if the solutions can be found by means of just some $\frac{\pi^2}{6} - a(x)$, rather then having to compute the integral every time, it would save me a lot of computation power and help simplify my code a bit.
As I said early, I am way out of my depth of understanding here. I am just an undergraduate physics student who came across a mathematical method that I would like to use as part of a research project but that utilized gamma and polygamma functions (which I had never used before). Everything I understand about these topics has come from Wikipedia pages in the past few days. It's highly possible that this topic is covered in the material I have been looking for but has eluded my understanding.
If $m$ is a positive integer then $$ \psi_1(m) = \sum\limits_{n = 0}^\infty {\frac{1}{{(n + m)^2 }}} = \sum\limits_{k = m}^\infty {\frac{1}{{k^2 }}} = \sum\limits_{k = 1}^\infty {\frac{1}{{k^2 }}} - \sum\limits_{k = 1}^{m - 1} {\frac{1}{{k^2 }}} = \frac{{\pi ^2 }}{6} - \sum\limits_{k = 1}^{m - 1} {\frac{1}{{k^2 }}} . $$ Another known identity for positive integer $m$ is $$ \psi _1 \!\left( {m - \tfrac{1}{2}} \right) = \frac{{\pi ^2 }}{2} - 4\sum\limits_{k = 1}^{m - 1} {\frac{1}{{(2k - 1)^2 }}} . $$ An expansion that converges for all real $x>0$ is $$ \psi _1 (x) = \frac{1}{x} + \frac{1}{{2x^2 }} + \frac{1}{2}\sum\limits_{k = 2}^\infty {\frac{{(k - 1)!\left( {1 - \frac{2}{{k + 1}}} \right)}}{{x(x + 1) \cdots (x + k)}}} . $$ This is particularly useful for large values of $x$. Finally, here is an asymptotic expansion in terms of Bernoulli numbers.