# classification of parameters

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## 8 matching pages

##### 1: 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). …
##### 2: 31.14 General Fuchsian Equation
$\alpha\beta=\sum_{j=1}^{N}a_{j}q_{j}.$
##### 4: 18.27 $q$-Hahn Class
A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949). …These families depend on further parameters, in addition to $q$. The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. … Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. …
##### 5: 16.21 Differential Equation
16.21.1 $\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,$
With the classification of §16.8(i), when $p the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\infty$. …
16.21.2 ${G^{1,p}_{p,q}}\left(z{\mathrm{e}}^{(p-m-n-1)\pi\mathrm{i}};{a_{1},\dots,a_{p}% \atop b_{j},b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\right),$ $j=1,\dots,q$.
##### 6: 16.4 Argument Unity
###### §16.4(i) Classification
The last condition is equivalent to the sum of the top parameters plus $2$ equals the sum of the bottom parameters, that is, the series is 2-balanced. … The function ${{}_{3}F_{2}}\left(a,b,c;d,e;1\right)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … These series contain $\mathit{6j}$ symbols as special cases when the parameters are integers; compare §34.4. … Contiguous balanced series have parameters shifted by an integer but still balanced. …
##### 7: 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$
##### 8: 16.8 Differential Equations
###### §16.8(i) Classification of Singularities
16.8.3 $\left(\vartheta(\vartheta+b_{1}-1)\cdots(\vartheta+b_{q}-1)-z(\vartheta+a_{1})% \cdots(\vartheta+a_{p})\right)w=0.$
16.8.7 $\widetilde{w}_{j}(z)=(-z)^{-a_{j}}{{}_{q+1}F_{q}}\left({a_{j},1-b_{1}+a_{j},% \dots,1-b_{q}+a_{j}\atop 1-a_{1}+a_{j},\ldots*\dots,1-a_{q+1}+a_{j}};\frac{1}{% z}\right),$ $j=1,\dots,q+1$,
16.8.10 $\lim_{|\alpha|\to\infty}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{p},\alpha\atop b_% {1},\dots,b_{q}};\frac{z}{\alpha}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};z\right).$
16.8.11 $\lim_{|\beta|\to\infty}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q},\beta};\beta z\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};z\right),$