degrees two, three, four

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1: 1.11 Zeros of Polynomials

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§1.11(iii) Polynomials of Degrees Two, Three, and Four

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2: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. These eight further families can be grouped in two classes of OP’s: ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). …

3: Bille C. Carlson

… ►Carlson, in 1981, and is survived by his companion, Jody Stadler, two children, Marian Carlson and John Carlson, and four grandchildren. … ►After the war he returned to Harvard and completed Bachelor’s and Master’s degrees in physics and mathematics. He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. After four years in the Physics Department at Princeton, he went to the Ames Laboratory and Iowa State University in 1954, where he was a Professor in the Physics and Mathematics Departments. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

4: 14.15 Uniform Asymptotic Approximations

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14.15.3 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

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14.15.5

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Symbols:

$\mu$: general order, $\nu$: general degree and $\alpha$

Referenced by:

§14.15(ii)

Permalink:

http://dlmf.nist.gov/14.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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§14.15(iii) Large $\nu$, Fixed $\mu$

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14.15.19

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Symbols:

$\mu$: general order, $\nu$: general degree and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

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5: 22.19 Physical Applications

… ►The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta \in (0,\pm \alpha )$ and $\theta \in (\pm \alpha ,0)$ have the same “quarter periods” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►For $\beta$ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions. …The subsequent position as a function of time, $x(t)$, for the three cases is given with results expressed in terms of $a$ and the dimensionless parameter $\eta =\frac{1}{2}\beta a^2$. … ►Two types of oscillatory motion are possible. … ►Hyperelliptic functions $u(z)$ are solutions of the equation $z={\int }_0^u{(f(x))}^{-1/2}dx$, where $f(x)$ is a polynomial of degree higher than 4. …