# parameter constraints

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##### 2: 18.5 Explicit Representations
For corresponding formulas for Chebyshev, Legendre, and the Hermite $\mathit{He}_{n}$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
##### 3: 18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_{n}\left(x;a,b,c,d\right)$, continuous dual Hahn polynomials $S_{n}\left(x;a,b,c\right)$, Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$, and dual Hahn polynomials $R_{n}\left(x;\gamma,\delta,N\right)$.
##### 5: 20.10 Integrals
###### §20.10(i) Mellin Transforms with respect to the Lattice Parameter
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
Then …
##### 6: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.8 $\mathbf{A}_{\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$,
11.11.10 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$.
11.11.11 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }e^{-\nu\mu}\sum_{k=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{k}}b_{k}(% \lambda)}{\nu^{k}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$,
11.11.14 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi\nu(\lambda-1)},$ $\lambda>1$, $|\operatorname{ph}\nu|\leq\pi-\delta$,
11.11.15 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}\right)^{\nu}\frac{e^{-\nu\sqrt{1% -\lambda^{2}}}}{(1-\lambda^{2})^{1/4}},$ $0<\lambda<1$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$.
##### 7: 8.7 Series Expansions
8.7.6 $\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$ $x>0$, $\Re a<\frac{1}{2}$.
##### 8: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
25.14.6 $\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{% s}}\,\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{% arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\,\mathrm{d}x,$ $\Re a>0$ if $\left|z\right|<1$; $\Re s>1$, $\Re a>0$ if $\left|z\right|=1$.
##### 9: 15.6 Integral Representations
15.6.1 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re b>0$.
15.6.2 $\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $c-b\neq 1,2,3,\dots$, $\Re b>0$.
15.6.6 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t% \right)}(-z)^{t}\,\mathrm{d}t,$ $|\operatorname{ph}\left(-z\right)|<\pi$; $a,b\neq 0,-1,-2,\dots$.
15.6.7 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t\right)\Gamma\left(b+t% \right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1-z)^{t}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $a,b,c-a,c-b\neq 0,-1,-2,\dots$.
15.6.8 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re d>0$.
##### 10: 33.14 Definitions and Basic Properties
33.14.13 $\int_{0}^{\infty}s\left(\epsilon_{1},\ell;r\right)s\left(\epsilon_{2},\ell;r% \right)\,\mathrm{d}r=\delta\left(\epsilon_{1}-\epsilon_{2}\right),$ $\epsilon_{1},\epsilon_{2}>0$,