accessory parameter

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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: 31.18 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 28–30.

3: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . … ►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, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ 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 …