# parameters

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## 11—20 of 379 matching pages

##### 11: 31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values $q_{m}$, $m=0,1,2,\dots$, of the accessory parameter $q$, the function $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic at $z=1$, and hence also throughout the disk $|z|. …
31.4.1 $(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$.
The eigenvalues $q_{m}$ satisfy the continued-fraction equation
31.4.3 $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$,
##### 13: 31.2 Differential Equations
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\mathbb{C}\cup\{\infty\}$, can be transformed into (31.2.1). The parameters play different roles: $a$ is the singularity parameter; $\alpha,\beta,\gamma,\delta,\epsilon$ are exponent parameters; $q$ is the accessory parameter. The total number of free parameters is six. … Then (suppressing the parameter $k$) … Except for the identity automorphism, each alters the parameters.
##### 14: 8.17 Incomplete Beta Functions
Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\leq x\leq 1$. …
8.17.4 $I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right).$
With $a>0$, $b>0$, and $0, …
8.17.13 $(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),$
##### 15: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …
##### 17: 31.12 Confluent Forms of Heun’s Equation
31.12.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.$
31.12.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$
31.12.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$
##### 18: 32.1 Special Notation
 $m,n$ integers. … real parameter.
##### 19: 34.14 Tables
Tables of exact values of the squares of the $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols on pp. … Tables of $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 17/2$ are given in Appel (1968) to 6D. … In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters $\leq 5$ and numerical values for all parameters $\leq 3$ are given on pp. …
##### 20: 15.5 Derivatives and Contiguous Functions
15.5.1 $\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1% ;c+1;z\right),$
15.5.2 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a% \right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z% \right).$
15.5.3 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).$
15.5.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)% \right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right).$
15.5.11 $(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z% -1)F\left(a+1,b;c;z\right)=0,$