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##### 1: 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. If the other exponent is not a positive integer, that is, if $\gamma\neq 0,-1,-2,\dots$, then from §2.7(i) it follows that $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ exists, is analytic in the disk $|z|<1$, and has the Maclaurin expansion … Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. …For example, $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is equal to … The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
##### 3: 31.7 Relations to Other Functions
31.7.1 ${{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).$
Other reductions of $\mathit{H\!\ell}$ to a ${{}_{2}F_{1}}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha\beta p$. …
31.7.2 $\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),$
31.7.3 $\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(% \alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta% ;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),$
31.7.4 $\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},\alpha\beta(\tfrac{1}{% 2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(\alpha+\beta+1),\tfrac{1}{3% }(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}% \beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{% 3}}{2}\right)z\right)^{3}\right).$
##### 5: 31.1 Special Notation
The main functions treated in this chapter are $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$, $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, and the polynomial $\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$. …
##### 6: 31.9 Orthogonality
31.9.3 $\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},$
$f_{0}(q_{m},z)=\mathit{H\!\ell}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$
$f_{1}(q_{m},z)=\mathit{H\!\ell}\left(1-a,\alpha\beta-q_{m};\alpha,\beta,\delta% ,\gamma;1-z\right),$
31.9.6 $\rho(s,t)=(s-t)(st)^{\gamma-1}\left((s-1)(t-1)\right)^{\delta-1}\*\left((s-a)(% t-a)\right)^{\epsilon-1},$
##### 7: 2.6 Distributional Methods
Let $f(t)$ be locally integrable on $[0,\infty)$. The Stieltjes transform of $f(t)$ is defined by …Since $f(t)$ is locally integrable on $[0,\infty)$, it defines a distribution by … In terms of the convolution product …of two locally integrable functions on $[0,\infty)$, (2.6.33) can be written …