History of Values of the polygamma functions $\psi^{(n)}(x)$ at rational numbers

 Title: Values of the polygamma functions $\psi^{(n)}(x)$ at rational numbers Definition: Listed are values of the polygamma functions $\psi^{(n)}(x)=d^n\psi(x)/dx^n$-  for positive integers $n$, where $\psi(x)$ is the digamma function $\Gamma'(x)/\Gamma(x)$-  CITE{DLMFPolygamma}, at rational arguments $x$.+  of positive integer order $n$ at rational arguments $x$, where $\psi=\Gamma'/\Gamma$+  is the digamma function CITE{DLMFPolygamma}. Keywords: - polygamma function

     display: $x$     constraints: $x\in\mathbb{Q}$ and $x$ is not a non-positive integer-Comments:-  comment-n-zero: The case $n=0$ is the digamma function and is stored in HREF{Values_of_the_digamma_function_at_rational_numbers}[the-    digamma table]. Formulas:   formula-hurwitz: With HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}[$\zeta(s,x)$]     the Hurwitz zeta function, $\psi^{(n)}(x)=(-1)^{n+1}n!\,\zeta(n+1,x)$ for $n\geq1$     CITE{DLMFPolygamma}.+  formula-reflection: $\psi^{(n)}(1-x)+(-1)^{n+1}\psi^{(n)}(x)=(-1)^n\pi\,\dfrac{d^n}{dx^n}\cot(\pi+    x)$ for $n\geq1$ and $x\notin\mathbb{Z}$ CITE{DLMFPolygamma}.   formula-recurrence: $\psi^{(n)}(x+1)=\psi^{(n)}(x)+(-1)^n n!/x^{n+1}$ when both     sides are finite.

 - table: HREF{Values_of_the_digamma_function_at_rational_numbers}[Values of the digamma     function $\psi(x)$ at rational numbers]-  relation: stores the order $0$ case of the same notation+  relation: holds $\psi=\psi^{(0)}$, the case $n=0$, at the same arguments - table: HREF{Zeros_of_the_polygamma_functions}[Zeros of the polygamma functions $\psi^{(n)}$]-  relation: stores the real zeros of the same functions+  relation: stores the real zeros of $\psi^{(n)}$ for even $n$, including those of+    $\psi''$; $\psi'$ has none - table: HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}[Values     of the Hurwitz zeta function at pairs of rational numbers]-  relation: gives the Hurwitz zeta values that express these derivatives+  relation: holds $\zeta(2,x)=\psi'(x)$ and $\zeta(3,x)=-\psi''(x)/2$ for the rationals+    $x$ in $(0,1)$ with denominator at most $5$ - table: HREF{Values_of_the_Gamma_function_at_rational_numbers}[Values of the Gamma     function at rational numbers]

     and $\psi''(1)=-2\zeta(3)$.   complete: 'no'-  complete-note: it holds $\psi^{(n)}(x)$ for $n=1,2$ and every rational argument-    $x=a/b$ in lowest terms with $b\leq12$ and $-4<x\leq4$ at which the function is-    finite+  complete-note: it holds the first two positive orders, $\psi'$ and $\psi''$, for+    every rational argument $x=a/b$ in lowest terms with $b\leq12$ and $-4<x\leq4$+    at which the function is finite; these are the same arguments as HREF{Values_of_the_digamma_function_at_rational_numbers}[the+    table of digamma values], and arguments outside $0<x\leq1$ are included because+    the recurrence CITE{formula-recurrence} makes $\psi^{(n)}(x+k)-\psi^{(n)}(x)$+    rational for positive integers $k$ whenever both sides are finite Display properties:   number-header: $\psi^{(n)}(x)$