# polynomial cases

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##### 1: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation $D_{\nu}\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).
##### 2: 18.1 Notation
They are defined in the literature by $C^{(0)}_{0}\left(x\right)=1$ and …
##### 3: 15.2 Definitions and Analytical Properties
For example, when $a=-m$, $m=0,1,2,\dots$, and $c\neq 0,-1,-2,\dots$, $F\left(a,b;c;z\right)$ is a polynomial: …
##### 6: 13.2 Definitions and Basic Properties
13.2.9 $U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma\left(a-n\right)}\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+% \psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+\frac{% 1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k-1)!{\left(1-a+k\right)_{n-k}}}{% (n-k)!}z^{-k}.$
Except when $a=0,-1,\dots$ (polynomial cases), …
##### 7: 12.11 Zeros
Lastly, when $a=-n-\tfrac{1}{2}$, $n=1,2,\dots$ (Hermite polynomial case) $U\left(a,x\right)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}\,]$. …
##### 10: 13.14 Definitions and Basic Properties
Except when $\mu-\kappa=-\frac{1}{2},-\frac{3}{2},\dots$ (polynomial cases), …