# continuous

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##### 2: 1.4 Calculus of One Variable
###### §1.4(ii) Continuity
For an example, see Figure 1.4.1If $\phi^{\prime}(x)$ is continuous or piecewise continuous, then …
##### 3: 18.30 Associated OP’s
In the recurrence relation (18.2.8) assume that the coefficients $A_{n}$, $B_{n}$, and $C_{n+1}$ are defined when $n$ is a continuous nonnegative real variable, and let $c$ be an arbitrary positive constant. …
18.30.1 $A_{n}A_{n+1}C_{n+1}>0,$ $n\geq 0$.
18.30.4 $P^{(\alpha,\beta)}_{n}\left(x;c\right)=p_{n}(x;c),$ $n=0,1,\dots$,
###### Continuous Phase Mapping
For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. …
##### 5: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. … For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).
##### 6: 18.1 Notation
• Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$.

• Continuous Dual Hahn: $S_{n}\left(x;a,b,c\right)$.

• Continuous $q$-Ultraspherical: $C_{n}\left(x;\beta\,|\,q\right)$.

• Continuous $q$-Hermite: $H_{n}\left(x\,|\,q\right)$.

• Continuous $q^{-1}$-Hermite: $h_{n}\left(x\,|\,q\right)$

• ##### 7: 18.26 Wilson Class: Continued
###### Continuous Dual Hahn
Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …
###### §18.28(vii) Continuous$q^{-1}$-Hermite Polynomials
18.28.18 $h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}% \frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}e^{(n-2\ell)t}=e^{% nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t}\right)={\mathrm{i}}^{-n}H% _{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).$
For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. …
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 $R_{n}\left(x;\gamma,\delta,N\right)$. …
Table 18.25.2 provides the leading coefficients $k_{n}$18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … for all functions $\phi(x)$ that are continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$. … More generally, assume $\phi(x)$ is piecewise continuous1.4(ii)) when $x\in[-c,c]$ for any finite positive real value of $c$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$. … provided that $\phi(x)$ is continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). … provided that $\phi(x)$ is continuous and of period $2\pi$; see §1.8(ii). …