# parameters

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##### 1: 31.13 Asymptotic Approximations
###### §31.13 Asymptotic Approximations
For asymptotic approximations for the accessory parameter eigenvalues $q_{m}$, see Fedoryuk (1991) and Slavyanov (1996). …
##### 2: 31.3 Basic Solutions
31.3.5 $z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).$
31.3.6 $\mathit{H\!\ell}\left(1-a,\alpha\beta-q;\alpha,\beta,\delta,\gamma;1-z\right),$
31.3.8 $\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),$
31.3.12 $\mathit{H\!\ell}\left(1/a,q/a;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma-\delta% ;z/a\right),$
##### 3: 29.11 Lamé Wave Equation
29.11.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu(\nu+1)k^{2}{\operatorname{% sn}^{2}}\left(z,k\right)+k^{2}\omega^{2}{\operatorname{sn}^{4}}\left(z,k\right% ))w=0,$
in which $\omega$ is another parameter. … For properties of the solutions of (29.11.1) see Arscott (1956, 1959), Arscott (1964b, Chapter X), Erdélyi et al. (1955, §16.14), Fedoryuk (1989), and Müller (1966a, b, c).
##### 4: 31.14 General Fuchsian Equation
31.14.1 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.
$\alpha\beta=\sum_{j=1}^{N}a_{j}q_{j}.$
The three sets of parameters comprise the singularity parameters $a_{j}$, the exponent parameters $\alpha,\beta,\gamma_{j}$, and the $N-2$ free accessory parameters $q_{j}$. With $a_{1}=0$ and $a_{2}=1$ the total number of free parameters is $3N-3$. …
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
##### 5: 28.17 Stability as $x\to\pm\infty$
###### §28.17 Stability as $x\to\pm\infty$
If all solutions of (28.2.1) are bounded when $x\to\pm\infty$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify
##### 6: 31.1 Special Notation
 $x$, $y$ real variables. … complex parameter, $|a|\geq 1,a\neq 1$. complex parameters.
Sometimes the parameters are suppressed.
##### 7: 34.8 Approximations for Large Parameters
###### §34.8 Approximations for Large Parameters
For large values of the parameters in the $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols, different asymptotic forms are obtained depending on which parameters are large. … and the symbol $o\left(1\right)$ denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). …
##### 8: 28.14 Fourier Series
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$
##### 9: 15.7 Continued Fractions
15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
15.7.5 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
##### 10: 31.6 Path-Multiplicative Solutions
31.6.1 $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$ $m=0,1,2,\dots$,