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##### 1: 19.2 Definitions
Here $a,b,p$ are real parameters, and $k_{c}$ and $x$ are real or complex variables, with $p\neq 0$, $k_{c}\neq 0$. … If $1, then $k_{c}$ is pure imaginary. …
###### §19.2(iv) A Related Function: $R_{C}\left(x,y\right)$
When $x$ and $y$ are positive, $R_{C}\left(x,y\right)$ is an inverse circular function if $x and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15). …
##### 2: 15.5 Derivatives and Contiguous Functions
The six functions $F\left(a\pm 1,b;c;z\right)$, $F\left(a,b\pm 1;c;z\right)$, $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$. …
15.5.14 $c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-% a)(c-b)zF\left(a,b;c+1;z\right)=0,$
15.5.18 $c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}F\left(a,b;c% ;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0.$
By repeated applications of (15.5.11)–(15.5.18) any function $F\left(a+k,b+\ell;c+m;z\right)$, in which $k,\ell,m$ are integers, can be expressed as a linear combination of $F\left(a,b;c;z\right)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. …
15.5.20 $z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F% \left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z% \right)+(b-c+az)F\left(a,b;c;z\right),$
##### 3: 4.42 Solution of Triangles
where $s=\tfrac{1}{2}(a+b+c)$ (the semiperimeter). …
##### 4: 15.13 Zeros
Let $N(a,b,c)$ denote the number of zeros of $F\left(a,b;c;z\right)$ in the sector $|\operatorname{ph}\left(1-z\right)|<\pi$. If $a$, $b$, $c$ are real, $a$, $b$, $c$, $c-a$, $c-b\neq 0,-1,-2,\dots$, and, without loss of generality, $b\geq a$, $c\geq a+b$ (compare (15.8.1)), then
15.13.1 $N(a,b,c)=\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S)+\left\lfloor a-c+1\right\rfloor S% ,&a<0,c-a<0,\\ \end{cases}$
where $S=\operatorname{sign}\left(\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left% (c-a\right)\Gamma\left(c-b\right)\right)$. … If $a$, $b$, $c$, $c-a$, or $c-b\in\{0,-1,-2,\dots\}$, then $F\left(a,b;c;z\right)$ is not defined, or reduces to a polynomial, or reduces to $(1-z)^{c-a-b}$ times a polynomial. …
##### 5: 18.30 Associated OP’s
In the recurrence relation (18.2.8) assume that the coefficients $A_{n}$, $B_{n}$, and $C_{n+1}$ are defined when $n$ is a continuous nonnegative real variable, and let $c$ be an arbitrary positive constant. …Then the associated orthogonal polynomials $p_{n}(x;c)$ are defined by …  (18.30.3) continues to hold, except that when $n=0$, $B_{c}$ is replaced by an arbitrary real constant. Then the polynomials $p_{n}(x,c)$ generated in this manner are called corecursive associated OP’s. … where $p_{n}(x;c)$ is given by (18.30.2) and (18.30.3), with $A_{n}$, $B_{n}$, and $C_{n}$ as in (18.9.2). …
##### 6: 15.1 Special Notation
 $x$ real variable. … real or complex parameters. …
15.1.1 ${{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),$
##### 7: 15.7 Continued Fractions
$t_{n}=c+n,$
$u_{2n+1}=(a+n)(c-b+n),$
$u_{2n}=(b+n)(c-a+n).$
$v_{n}=c+n+(b-a+n+1)z,$
$w_{n}=(b+n)(c-a+n)z.$
##### 8: 28.14 Fourier Series
The coefficients satisfy
$c_{0}^{\nu}(0)=1,$
##### 9: 15.2 Definitions and Analytical Properties
In general, $F\left(a,b;c;z\right)$ does not exist when $c=0,-1,-2,\dots$. … For all values of $c$
• (c)

Diverges when $\Re\left(c-a-b\right)\leq-1$.

• The principal branch of $\mathbf{F}\left(a,b;c;z\right)$ is an entire function of $a$, $b$, and $c$. …The same properties hold for $F\left(a,b;c;z\right)$, except that as a function of $c$, $F\left(a,b;c;z\right)$ in general has poles at $c=0,-1,-2,\dots$. …
##### 10: 26.11 Integer Partitions: Compositions
$c\left(n\right)$ denotes the number of compositions of $n$, and $c_{m}\left(n\right)$ is the number of compositions into exactly $m$ parts. $c\left(\in\!T,n\right)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\ldots\}$. …
26.11.1 $c\left(0\right)=c\left(\in\!T,0\right)=1.$
26.11.2 $c_{m}\left(0\right)=\delta_{0,m},$
26.11.3 $c_{m}\left(n\right)=\genfrac{(}{)}{0.0pt}{}{n-1}{m-1},$