# singularity parameter

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##### 1: 31.14 General Fuchsian Equation
31.14.1 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.
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 $q_{j}$. …
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
31.14.4 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,$ $\sum_{j=1}^{N}\tilde{q}_{j}=0$,
##### 2: 31.2 Differential Equations
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
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. …
##### 3: 31.15 Stieltjes Polynomials
31.15.3 $\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.$
31.15.6 $a_{j} $j=1,2,\dots,N-1$,
If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$, where each $m_{j}$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_{j}$ zeros in the open interval $(a_{j},a_{j+1})$ for each $j=1,2,\dots,N-1$. …
31.15.8 $S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),$ $z_{j}\in(a_{j},a_{j+1})$,
31.15.9 $S_{\mathbf{l}}(z_{1})S_{\mathbf{l}}(z_{2})\cdots S_{\mathbf{l}}(z_{N-1}),$ $z_{j}\in(a_{j},a_{j+1})$,
##### 4: 31.13 Asymptotic Approximations
###### §31.13 Asymptotic Approximations
For asymptotic approximations for the accessory parameter eigenvalues $q_{m}$, see Fedoryuk (1991) and Slavyanov (1996). For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
##### 5: 2.4 Contour Integrals
For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …
##### 6: 31.1 Special Notation
Sometimes the parameters are suppressed.
##### 7: 30.2 Differential Equations
This equation has regular singularities at $z=\pm 1$ with exponents $\pm\frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\infty$ (if $\gamma\neq 0$). The equation contains three real parameters $\lambda$, $\gamma^{2}$, and $\mu$. … …
30.2.2 $\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}t}^{2}}+\left(\lambda+\frac{1}{4}+\gamma^{% 2}{\sin}^{2}t-\frac{\mu^{2}-\frac{1}{4}}{{\sin}^{2}t}\right)g=0,$
With $\zeta=\gamma z$ Equation (30.2.1) changes to …
##### 8: 31.11 Expansions in Series of Hypergeometric Functions
31.11.3 $\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon.$
and (31.11.1) converges outside the ellipse $\mathcal{E}$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at $z=a$. Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities $z=0$ and $z=1$. In this case the accessory parameter $q$ is a root of the continued-fraction equation …
##### 9: 31.4 Solutions Analytic at Two Singularities: Heun Functions
###### §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 $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$,
##### 10: 31.16 Mathematical Applications
thesis “Inversion problem for a second-order linear differential equation with four singular points”. It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. …
31.16.3 $A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,$
31.16.6 $Q_{j}=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma+\delta-% 1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1)j(j+% \delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta-2)},$