# accessory parameter

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## 8 matching pages

##### 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: 31.18 Methods of Computation
The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 2830.
##### 3: 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|. … The eigenvalues $q_{m}$ satisfy the continued-fraction equation …
##### 4: 31.14 General Fuchsian Equation
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 $q_{j}$. …
##### 5: 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. …
##### 6: 31.16 Mathematical Applications
It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. …
##### 7: 31.15 Stieltjes Polynomials
In this case the accessory parameters $q_{j}$ are given by …
##### 8: 31.11 Expansions in Series of Hypergeometric Functions
In this case the accessory parameter $q$ is a root of the continued-fraction equation …