# parameter constraint

(0.001 seconds)

## 1—10 of 14 matching pages

##### 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: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
##### 6: 15.6 Integral Representations
15.6.1 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re b>0$.
15.6.6 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t% \right)}(-z)^{t}\mathrm{d}t,$ $|\operatorname{ph}\left(-z\right)|<\pi$; $a,b\neq 0,-1,-2,\dots$.
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 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re d>0$.
15.6.9 $\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^% {a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}% \left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\lambda\in\mathbb{C}$, $\Re c>\Re d>0$.
##### 7: 33.14 Definitions and Basic Properties
33.14.13 $\int_{0}^{\infty}s\left(\epsilon_{1},\ell;r\right)s\left(\epsilon_{2},\ell;r% \right)\mathrm{d}r=\delta\left(\epsilon_{1}-\epsilon_{2}\right),$ $\epsilon_{1},\epsilon_{2}>0$,
##### 8: 10.32 Integral Representations
10.32.13 $K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int_{c-i\infty}^{c+i% \infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}z)^{-2t}% \mathrm{d}t,$ $c>\max(\Re\nu,0),|\operatorname{ph}z|<\frac{1}{2}\pi$.
##### 9: 19.16 Definitions
Thus $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is symmetric in the variables $z_{j}$ and $z_{\ell}$ if the parameters $b_{j}$ and $b_{\ell}$ are equal. …
19.16.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}% \mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a% -1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,$ $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$, $z_{j}\in\mathbb{C}\setminus(-\infty,0]$,
19.16.12 $R_{-a}\left(b_{1},\dots,b_{4};c-1,c-k^{2},c,c-\alpha^{2}\right)=\frac{2({\sin^% {2}}\phi)^{1-a^{\prime}}}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\phi}(% \sin\theta)^{2a-1}{({\sin^{2}}\phi-{\sin^{2}}\theta)}^{a^{\prime}-1}\*(\cos% \theta)^{1-2b_{1}}{(1-k^{2}{\sin^{2}}\theta)}^{-b_{2}}{(1-\alpha^{2}{\sin^{2}}% \theta)}^{-b_{4}}\mathrm{d}\theta,$
$R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime},b_{1},\dots,b_{n}$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime}$, $a+a^{\prime}$ is zero or a negative integer. …
##### 10: 10.40 Asymptotic Expansions for Large Argument
10.40.3 $I_{\nu}'\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$,
10.40.9 $\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{% 2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+% \frac{1}{4\nu^{2}-(2k+1)^{2}}\right).$
10.40.12 $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}$