# exponent parameters

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##### 1: 31.8 Solutions via Quadratures
For half-odd-integer values of the exponent parameters: … The curve $\Gamma$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_{j}\in\mathbb{Z}$. …
##### 2: 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 $q_{j}$. …
##### 3: 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. …
##### 4: 27.2 Functions
is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …
##### 5: 31.15 Stieltjes Polynomials
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 $j=1,2,\dots,N-1$. …
##### 6: 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
30.2.4 $(\zeta^{2}-\gamma^{2})\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+2\zeta% \frac{\mathrm{d}w}{\mathrm{d}\zeta}+\left(\zeta^{2}-\lambda-\gamma^{2}-\frac{% \gamma^{2}\mu^{2}}{\zeta^{2}-\gamma^{2}}\right)w=0.$
##### 7: 31.3 Basic Solutions
$\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … Similarly, if $\gamma\neq 1,2,3,\dots$, then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at $z=0$ is … Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\epsilon$ at $z=a$ are respectively, … Solutions of (31.2.1) corresponding to the exponents $\alpha$ and $\beta$ at $z=\infty$ are respectively, …
##### 8: 31.11 Expansions in Series of Hypergeometric Functions
31.11.3 $\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon.$
31.11.7 $L_{j}=a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+% \lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta% +\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)},$
Then the Fuchs–Frobenius solution at $\infty$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\infty$. …In this case the accessory parameter $q$ is a root of the continued-fraction equation …
##### 9: 28.29 Definitions and Basic Properties
28.29.9 $2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).$